SSC CGL formula sheet

Every formula and shortcut trick from the notes, on one page. Print it (Ctrl/Cmd + P) or save as PDF — it's formatted for paper.

Quantitative Aptitude

Number System

Number types, place value & counting

Multiples of k up to n

\left\lfloor \dfrac{n}{k} \right\rfloor

Multiples of k from a to b

\left\lfloor \dfrac{b}{k} \right\rfloor - \left\lfloor \dfrac{a-1}{k} \right\rfloor

Inclusion-exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B)

the overlap holds multiples of the LCM

Sum of first n natural numbers

\dfrac{n(n+1)}{2}

Sum of squares

1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}

Sum of cubes

1^3 + 2^3 + \cdots + n^3 = \left[\dfrac{n(n+1)}{2}\right]^2

First n odd or even numbers

1 + 3 + \cdots + (2n-1) = n^2, \quad 2 + 4 + \cdots + 2n = n(n+1)

Any arithmetic series

S = \dfrac{n}{2}(a + l)

terms times average of first and last

Digit reversal

(10a + b) - (10b + a) = 9(a - b), \quad (10a + b) + (10b + a) = 11(a + b)

⚡ Floor-division counting. Count multiples of k up to the end of the range, then subtract the multiples before the start. No listing needed.

⚡ Place value in one product. Place value minus face value equals digit times (position value minus 1). The 9999 pattern does it in one line.

⚡ Middle term times count. For any evenly spaced list, the sum equals the number of terms times the middle term. The middle term is the average of first and last.

Divisibility rules

Divisibility by 11

\left(\sum \text{odd-place digits}\right) - \left(\sum \text{even-place digits}\right) \in \{0, \pm 11, \pm 22, \ldots\}

Composite divisor

pq \mid N \iff p \mid N \text{ and } q \mid N, \quad \gcd(p, q) = 1

split into co-prime parts only

Difference of powers

(a - b) \mid (a^n - b^n) \text{ for all } n

Difference of even powers

(a + b) \mid (a^n - b^n) \text{ when } n \text{ is even}

Sum of odd powers

(a + b) \mid (a^n + b^n) \text{ when } n \text{ is odd}

Repeated block

\overline{abcabc} = \overline{abc} \times 1001 = \overline{abc} \times 7 \times 11 \times 13

Divisibility by seven

N \to \left\lfloor \dfrac{N}{10} \right\rfloor - 2 \times (N \bmod 10)

repeat until small

⚡ Split into co-prime factors. Break the divisor into parts with no common factor and test each part separately. Order the tests from the most restrictive first.

⚡ The repeated-block family. Any six-digit number abcabc equals abc times 1001, so it is always divisible by 7, 11 and 13.

⚡ Sum of odd powers. For a to the n plus b to the n with odd n, check the options against a plus b first. No expansion is ever needed.

Remainders & remainder theorem

Division algorithm

N = d \times q + r, \quad 0 \le r < d

Product rule

\mathrm{rem}\left(\dfrac{a \times b}{d}\right) = \mathrm{rem}\left(\dfrac{R_a \times R_b}{d}\right)

same for sums

Fermat's little theorem

a^{p-1} \equiv 1 \pmod{p}, \quad p \text{ prime}, \ \gcd(a, p) = 1

Divisor-multiple rule

N \equiv r \pmod{D},\ d \mid D \ \Rightarrow\ N \equiv r \pmod{d}

one-way rule

Base one more than divisor

(ad + 1)^n \equiv 1 \pmod{d}

Base one less than divisor

(ad - 1)^n \equiv (-1)^n \pmod{d}

1 if n even, d - 1 if n odd

⚡ Make the base plus or minus one. Write the base as (multiple of divisor) plus or minus 1. The whole power then collapses to plus or minus 1.

⚡ Negative remainders for products. When factors sit just below the divisor, replace each by a small negative remainder and multiply those.

⚡ Divisor is a factor, just reduce. If the new divisor divides the old one, reduce the old remainder by the new divisor. If it does not divide it, this shortcut is not available.

Unit digit & cyclicity

Cyclicity rule

\text{unit}(a^n) = \text{unit}(a^r), \quad r = n \bmod 4 \ (r = 0 \Rightarrow r = 4)

Cycles of two, three, seven, eight

2: 2,4,8,6 \quad 3: 3,9,7,1 \quad 7: 7,9,3,1 \quad 8: 8,4,2,6

Factorials

n! \equiv 0 \pmod{10} \text{ for } n \ge 5

Last two digits, base ending in one

(10a + 1)^n \text{ ends in } \left[(a \cdot n) \bmod 10\right] 1

⚡ Last two digits of the exponent. For division by four, only the last two digits of the exponent matter, because 100 is a multiple of 4.

⚡ Even times five ends in zero. If a product contains an even factor and a factor ending in 5, the unit digit is 0 without any cycle work.

⚡ Factorial sums stop at four terms. From 5! on, every factorial ends in 0, so a factorial sum's unit digit comes from the first four terms alone.

Factors, prime factorisation & trailing zeros

Number of factors

d(N) = (a + 1)(b + 1)(c + 1)

Sum of factors

\sigma(N) = (1 + p + \cdots + p^a)(1 + q + \cdots + q^b) \cdots

Even factors

a \times (b + 1)(c + 1)

when 2 has exponent a

Product of all factors

N^{d(N)/2}

Trailing zeros in n factorial

\left\lfloor \dfrac{n}{5} \right\rfloor + \left\lfloor \dfrac{n}{25} \right\rfloor + \left\lfloor \dfrac{n}{125} \right\rfloor + \cdots

Highest power of a prime in n factorial

\sum_{k \ge 1} \left\lfloor \dfrac{n}{p^k} \right\rfloor

⚡ Even factors: force one two. If N has two to the power a in it, even factors equal a times the factor count of the odd part.

⚡ Successive division by five. For trailing zeros, keep dividing n by 5 and add the quotients until the quotient is 0.

⚡ Sum of factors as brackets. One bracket per prime, each running from 1 up to the full power. Multiply the brackets.

Fractions, decimals & recurring decimals

Pure repeating decimal

0.\overline{ab} = \dfrac{ab}{99}

Mixed repeating decimal

0.a\overline{bc} = \dfrac{abc - a}{990}

Terminating test

\dfrac{p}{q} \text{ terminates} \iff q = 2^m \times 5^n

q in lowest terms

⚡ Nines for pure repeats. One repeating digit gives 9 in the denominator, two give 99, three give 999.

⚡ Zeros after nines for mixed repeats. Denominator: one 9 per repeating digit, then one 0 per non-repeating digit. Numerator: all digits minus the non-repeating block.

⚡ Compare what is missing. For fractions close to 1, compare the shortfalls instead of the fractions.

Simplification

BODMAS, 'of', brackets & vinculum

'Of' before division

a \div b \text{ of } c = a \div (b \times c)

Division and multiplication left to right

a \div b \times c = \dfrac{a}{b} \times c

Minus before a bracket

a - (b - c) = a - b + c

Continued fraction

a + \dfrac{1}{b + \dfrac{1}{c}} = a + \dfrac{c}{bc + 1}

⚡ Bracket every 'of'. Rewrite each 'of' as a bracketed product before doing any division. The bracket makes the grouping visible and kills the trap.

⚡ Continued fractions from the bottom. Simplify the deepest fraction first, invert it, add the next layer, and repeat upward.

⚡ Digit-sum check. Casting out nines: digit sums must match on both sides of the equals sign. It kills wrong options without computing the product.

Algebraic identities in numerical simplification

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Squares combine

(a+b)^2 + (a-b)^2 = 2(a^2+b^2)

Squares subtract

(a+b)^2 - (a-b)^2 = 4ab

Zero-sum cubes

a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc

Square-sum from pair facts

a^2 + b^2 = (a+b)^2 - 2ab

⚡ Pattern-match the fraction. Cubes on top and three terms with a middle product below mean a sum or difference of cubes. The fraction is just a plus or minus of the bases.

⚡ Zero-sum check before cubing. Add the three bases. If they sum to zero, the cubes sum to three times their product, signs included.

⚡ Products around a round number. Write the pair as centre minus gap and centre plus gap. The product is centre squared minus gap squared.

Surds & indices

Same base

a^m \cdot a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}

Power of power

(a^m)^n = a^{mn},\quad (ab)^n = a^n b^n

Zero and negative index

a^0 = 1,\quad a^{-n} = \frac{1}{a^n}

Fractional index

a^{p/q} = \sqrt[q]{a^p}

Rationalisation

\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a - b}

Root of a surd

\sqrt{a \pm 2\sqrt{b}} = \sqrt{x} \pm \sqrt{y},\ x + y = a,\ xy = b

Reciprocal of a unit surd

x = a + \sqrt{b},\ a^2 - b = 1 \Rightarrow \tfrac{1}{x} = a - \sqrt{b}

⚡ Common base, then equate powers. Write both sides as powers of the same prime. The equation becomes a linear equation in the exponent.

⚡ Split a + 2 root b. Force the middle term into the 2 root b shape, then find two numbers with the given sum and product.

⚡ Reciprocal by the a squared minus b check. For x = a + sqrt(b), test a squared minus b. If it equals 1, the reciprocal is a minus sqrt(b) with no work.

Square roots & cube roots

Product rule

\sqrt{ab} = \sqrt{a}\,\sqrt{b},\quad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}

Infinite radical, plus

\sqrt{n + \sqrt{n + \cdots}} = \dfrac{1 + \sqrt{1 + 4n}}{2}

Infinite radical, minus

\sqrt{n - \sqrt{n - \cdots}} = \dfrac{-1 + \sqrt{1 + 4n}}{2}

Infinite nested product

\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x

Least add or subtract

\text{subtract } N - k^2,\quad \text{add } (k+1)^2 - N

⚡ Cube root by unit digit and leading group. The last digit of the cube gives the last digit of the root; the leading group, compared with the small cubes, gives the first digit.

⚡ Square root by unit digit and size. Digit pairs fix the number of digits and the first digit; the last-digit map gives two candidates; one size check picks between them.

⚡ n equals k times k plus one radicals. Factor n into two consecutive integers. Plus signs give the larger one, minus signs the smaller.

Approximation

Percentage swap

x\% \text{ of } y = y\% \text{ of } x

Near-square root

\sqrt{n^2 + k} \approx n + \frac{k}{2n}

Percent-fraction anchors

12.5\% = \tfrac{1}{8},\quad 16.7\% = \tfrac{1}{6},\quad 33.3\% = \tfrac{1}{3},\quad 37.5\% = \tfrac{3}{8}

⚡ Swap the percentage. x percent of y equals y percent of x. Flip to whichever side has the nicer multiplier.

⚡ Balanced rounding for products. Round one factor up and the other down. The two errors cancel and the product stays close.

⚡ Anchors before bases. Turn the percent into its fraction anchor first, then apply it to the rounded base.

HCF & LCM

HCF & LCM: definitions and core relations

Product relation (two numbers)

\text{HCF} \times \text{LCM} = a \times b

Other number

b = \dfrac{\text{HCF} \times \text{LCM}}{a}

Co-prime case

\gcd(a, b) = 1 \Rightarrow \text{LCM}(a, b) = ab

HCF divides every difference

\gcd(a, b) \mid (a - b)

LCM is a multiple of the HCF

\gcd(a, b) \mid \text{LCM}(a, b)

Ratio pair

\text{numbers} = hm,\ hn; \quad \text{LCM} = hmn

⚡ Ratio split. Numbers in ratio a : b (co-prime parts) with HCF h are ha and hb. LCM = h times a times b; sum = h times (a plus b).

⚡ Product divided by HCF. Given any two of product, HCF, LCM, the product relation settles the third in one division.

⚡ Co-prime pairs from the quotient. For pair-count questions, divide the product by the square of the HCF, then count co-prime factor pairs of the result.

Finding HCF & LCM (incl. fractions and decimals)

HCF by factors

\text{HCF} = p_1^{\min} \times p_2^{\min} \times \cdots

common primes, lowest powers

LCM by factors

\text{LCM} = p_1^{\max} \times p_2^{\max} \times \cdots

all primes, highest powers

HCF of fractions

\text{HCF}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\gcd(a, c)}{\text{LCM}(b, d)}

LCM of fractions

\text{LCM}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\text{LCM}(a, c)}{\gcd(b, d)}

LCM from the HCF

\text{LCM}(a, b) = \dfrac{a \times b}{\gcd(a, b)}

⚡ Row of prime powers. Write each number as a row of prime powers in a small grid. LCM reads the column maxima, HCF the common minima. Three numbers take under thirty seconds.

⚡ Cross rule for fractions. HCF of fractions: HCF of tops over LCM of bottoms. LCM of fractions: LCM of tops over HCF of bottoms.

⚡ Shift the decimal point. Multiply by the power of 10 that clears every decimal, solve as integers, then shift the point back by the same number of places.

Standard word problems (tiles, bells, groups, divisible numbers)

Same remainder r

N = \text{LCM}(d_1, d_2, \ldots) \times k + r

Remainder is divisor minus c

r_i = d_i - c \Rightarrow N = \text{LCM} \times k - c

Divides with remainders

\text{answer} = \gcd(a - r_1,\ b - r_2,\ c - r_3)

Largest tile count

\text{tiles} = \dfrac{L \times W}{h^2}, \quad h = \gcd(L, W)

⚡ Subtract remainders, then HCF. For 'greatest number dividing a and b leaving remainders r1 and r2', the answer is the HCF of (a - r1) and (b - r2).

⚡ LCM plus r. For 'least number leaving remainder r with each divisor', add r to the LCM of the divisors.

⚡ Bells: add the LCM to the clock. Convert all intervals to one unit, take the LCM, and add it to the given start time.

Two-step LCM/HCF cases (extra condition, N-digit bounds)

Extra divisibility condition

N = Lk + r, \quad N \equiv 0 \pmod{p} \ \Rightarrow\ Lk \equiv -r \pmod{p}

Reconstruction from HCF and LCM

ab = \dfrac{\text{LCM}}{h}, \quad \gcd(a, b) = 1, \quad \text{numbers} = ha,\ hb

Pair count

\#\{(a, b): ab = M,\ \gcd(a, b) = 1,\ a \le b\}

Same unknown remainder

\text{answer} = \gcd(a - b,\ b - c,\ a - c)

⚡ Test k until it clicks. Once N = LCM x k + r is written, only k remains. Test k = 1, 2, 3 against the extra condition; small values click fast.

⚡ n-digit multiple scan. For the least n-digit multiple, divide the smallest n-digit number by the LCM and step up to the next multiple. For the greatest, subtract the remainder from the largest n-digit number.

⚡ Sum or difference picks the pair. With HCF, LCM and a sum or difference given, list co-prime pairs of LCM / HCF and match h times the parts to the sum.

Percentage

Percentage meaning & conversions

x% of a number

\frac{x}{100} \times N

Turn the percent into a fraction first; cancel where you can.

Whole from a part

\text{whole} = \frac{\text{part} \times 100}{x}

x is the percent that the part stands for.

A as a percent of B

\frac{A}{B} \times 100\%

B is the quantity that follows the word 'of'.

Remaining part

\text{rest} = 100\% - \text{given}\%

Only when both percentages are of the same whole.

⚡ Let the fraction table do the division. Replace the percent with its fraction from the table. 12.5\% becomes \dfrac{1}{8}, so the sum turns into a simple division.

⚡ Build the percent from 10% and 1% pieces. Move the decimal point for 10\% and 1\%, then add pieces. Works without pen-and-paper multiplication.

⚡ Swap x% of y into y% of x. x\% of y always equals y\% of x. Swap when one side turns into a friendly percent.

Percentage increase / decrease & successive change

Per cent change

\frac{\text{new} - \text{old}}{\text{old}} \times 100

Old value below the line.

Multiplier

\text{new} = \text{old} \times \frac{100 \pm x}{100}

Plus for a rise, minus for a fall.

Successive changes

a + b + \frac{ab}{100}

a and b carry their own signs.

Same x% up and down

\text{net loss} = \frac{x^2}{100}\%

x% rise followed by x% cut.

Fraction with both parts changed

\text{new} = \text{old} \times \frac{\text{top chip}}{\text{bottom chip}}

⚡ One multiplication per change. Replace every rise or fall with its multiplier fraction and multiply once. This replaces three lines of working.

⚡ a + b + ab/100 for two changes. Two changes in a row combine by this one formula. Keep the signs of both percents.

⚡ Round trip loses x squared over 100. A rise of x% followed by a cut of x% always ends below the start, by exactly \dfrac{x^2}{100} per cent.

'x% more than' ↔ 'x% less than' & chains

x% more, reversed

\frac{100x}{100+x}\% \text{ less}

x% more than becomes this much less than.

x% less, reversed

\frac{100x}{100-x}\% \text{ more}

x% less than becomes this much more than.

Per cent of a per cent

\frac{a}{100} \times \frac{b}{100} = \frac{ab}{10000}

Each 'of' multiplies.

⚡ Memorise the standard pairs. Four pairs cover most papers: 25/20, 20/16\dfrac{2}{3}, 50/33\dfrac{1}{3}, and 100/50. Read the question, write the partner, done.

⚡ Base-100 when memory fails. Put the base quantity at 100, write the other quantity, and divide by the new base. Slow but never wrong.

⚡ Awkward percents: use the fraction. 16\dfrac{2}{3}\% is 1/6, so 'more by 1/6' reverses to 'less by 1/7' of the bigger number.

Population growth, depreciation & elections

Growth for n years

P\left(1 + \frac{r}{100}\right)^n

r% added every year, compounding.

Depreciation for n years

P\left(1 - \frac{r}{100}\right)^n

r% of value lost every year.

Value n years ago

\frac{\text{now}}{\left(1 \pm \frac{r}{100}\right)^n}

Divide by the chip power to go back.

Election margin

\text{margin votes} = (\text{winner}\% - \text{rival}\%) \times \text{valid votes}

Percents on valid votes only.

⚡ Chip powers beat formulas. Write one multiplier per year and multiply. Squares like \left(\dfrac{5}{4}\right)^2 = \dfrac{25}{16} are worth remembering.

⚡ Remove invalid votes first. In elections, bring every statement onto valid votes before touching the margin.

⚡ Roll, cast, valid: walk the chain. Election numbers form a chain: roll, votes cast, valid votes, then candidates. Convert one link at a time with chips.

Marks, income–expenditure–savings & price–consumption

Maximum marks (two candidates)

M = \frac{(a+b) \times 100}{y - x}

a = shortfall below pass, b = marks above pass, x and y are the two percents.

Pass mark from failing

\text{pass} = \frac{x}{100}M + a

a is the shortfall.

Consumption cut after a price rise

\frac{100x}{100 + x}\%

Budget unchanged, price up x%.

Extra quantity after a price cut

\frac{100c}{100 - c}\%

Budget unchanged, price down c%.

⚡ Span of marks over span of percents. Two candidates give one clean fraction: marks apart over percents apart. Multiply by 100 for maximum marks.

⚡ Price rise to consumption cut is a flip. Reuse the more/less flip: price up x\% with the same budget means consumption down \dfrac{100x}{100+x}\%.

⚡ Extra kilograms reveal the price. After a price cut, extra kilograms equal money times the difference of the chips, divided by the new price. Solve for the price.

Ratio, Proportion, Partnership & Ages

Ratio basics & dividing amounts

Share of the total

\text{share} = \frac{\text{ratio term}}{\text{sum of terms}} \times N

Combining ratios

A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq

Cross-multiplication test

a:b > c:d \iff ad > bc

Multiplier method

\text{shares } ax, bx,\ (b-a)x = \text{given difference}

Duplicate / sub-duplicate

a:b \Rightarrow a^2:b^2 \text{ (duplicate)},\ \sqrt{a}:\sqrt{b} \text{ (sub-duplicate)}

⚡ x solves everything. Translate 'ratio a : b' into ax and bx. Every extra fact becomes one small equation in x.

⚡ LCM bridging for three terms. Scale A : B so its B-term matches the B-term of B : C, then read off A : B : C.

⚡ One share known, get the rest. Share = fraction × total. Recover the total from one share, then build any other share.

Proportion & proportional division of terms

Basic proportion

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Fourth proportional

d = \frac{bc}{a}

Third proportional

c = \frac{b^2}{a}

Mean proportional

\text{mean} = \sqrt{ab}

Componendo & dividendo

\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d}

Invertendo / alternando

\frac{b}{a} = \frac{d}{c},\quad \frac{a}{c} = \frac{b}{d}

⚡ Extremes times means. Set up the proportion in the exact order given, cross-multiply, done.

⚡ Mean proportional = geometric mean. Multiply the two numbers and take the square root. Exam numbers make it a perfect square.

⚡ Componendo-dividendo jump. When (x + y) and (x - y) both appear, jump straight to x/y by adding and subtracting the given ratio's terms.

Partnership

Simple partnership

P_1 : P_2 = C_1 : C_2 \quad (\text{same time})

Compound partnership

P_1 : P_2 : P_3 = C_1T_1 : C_2T_2 : C_3T_3

Working partner

\text{profit} = \text{manager's cut} + \text{residual split by } C_iT_i

Capital change mid-year

\text{effective capital} = C_a t_1 + C_b t_2 + \cdots

⚡ One row per partner. Write capital × months for each partner; the profit ratio is the row ratio.

⚡ Cut comes off the top. Salary or a per cent for the manager is removed from the whole profit before the capital-months split.

⚡ Equal shares, find the capital. Equal shares mean equal capital-months. Equate the products and solve.

Problems on ages

Present = ax, bx

\frac{ax \pm n}{bx \pm n} = \frac{p}{q}

Ratio after (or before) n years.

Constant difference

A - B \text{ is the same at every time}

Sum grows by 2 per year

(A + B)_{t+n} = (A + B)_t + 2n

Multiple of age

F = kS \Rightarrow F \pm n = k'(S \pm n)

k falls over time for elder-younger pairs.

⚡ Multiplier plus shift. Ages ax, bx now; shift both by n; equate the new ratio; solve for x.

⚡ Use the constant difference. The gap between two ages never changes — compute it once and reuse it at any time point.

⚡ Bracket the present with two ratios. With 'n years ago' and 'n years hence' ratios, both snapshots share one x — the 2n shift separates them.

Money ratios: income–expenditure, coins & mixed amounts

Savings equations

ax - py = s_1,\quad bx - qy = s_2

Two multipliers: x for incomes, y for expenditures.

Coin value

\text{total value} = k \sum_i (n_i \times d_i)

Price one set, then scale.

Wages together

\text{shares} \propto \frac{1}{\text{days alone}}

Difference of shares

\text{given excess} = (b - a)x

⚡ Subtract the savings equations. Write both savings equations and subtract — equal savings kill the constant and one variable.

⚡ Value per set for coins. Take one full set of coins in the given ratio, price it in paise, and scale.

⚡ Base the chain on the last person. For 'A gets half of B, B gets two-thirds of C', give the last-named person the unit and read the ratio off.

Average

Average & the sum bridge

Definition

\bar{x} = \frac{\sum x_i}{n} \iff \sum x_i = n\bar{x}

The sum form is the one you use.

Shift property

\overline{x_i + k} = \bar{x} + k,\quad \overline{k\, x_i} = k\bar{x}

Adding k shifts the average by k; multiplying by k scales it.

First n naturals / odd / even

\frac{n+1}{2},\quad n,\quad n+1

n = how many terms.

Squares / cubes

\frac{(n+1)(2n+1)}{6},\quad \frac{n(n+1)^2}{4}

Equal gaps

\text{average} = \frac{\text{first} + \text{last}}{2} = \text{middle term}

⚡ Flip to the sum first. Never push averages around. Change them to sums, adjust, divide back at the end.

⚡ Same operation on every value. Do to the average exactly what is done to each value, in the same order.

⚡ Equal gaps: jump from the middle. The average of evenly spaced numbers is the middle term. Step out from it instead of adding everything.

Members joining or leaving

Joining member

\text{value} = A' + n(A' - A)

n = old count; A' = new average.

Leaving member

\text{value} = A' + n(A - A')

A' = average of the remaining n members.

Replacement

\text{new} = \text{old} + n(A' - A)

The count does not change.

Count change both ways

\text{total of newcomers} = \text{new total} - \text{old total}

⚡ Joining: own share plus extras. New value = new average + (old count) × (average jump).

⚡ Replacement: the jump times n. Only the swapped item changes the sum, and it changes by n × (average jump).

⚡ Exclusion: subtract the totals. Old total − (new count × new average) = the removed value.

Weighted average & two-group problems

Weighted mean

\bar{x} = \frac{\sum n_i \bar{x}_i}{\sum n_i}

Missing group average

\bar{x}_2 = \frac{N\bar{x} - n_1\bar{x}_1}{n_2}

N = total count, overall average known.

Sizes from distances

\frac{n_1}{n_2} = \frac{\bar{x}_2 - \bar{x}}{\bar{x} - \bar{x}_1}

Reverse ratio of the distances.

⚡ Totals, not averages. Turn each group into a total, add, divide by the combined count.

⚡ Feel the balance point. The combined average splits the gap between the group averages in the ratio n₂ : n₁, the reverse of the sizes.

⚡ One newcomer is a tiny second group. A single new member changing a group average is just a weighted average with k = 1. Use value = B + n(B − A).

Batsman problems, overlapping sums & multi-step sets

Batsman score

\text{score} = A' + (n-1)d

A' = new average, d = rise.

Batsman new average

A' = \frac{(n-1)A + \text{score}}{n} = x - (n-1)d

Overlap subtraction

\text{Thu} - \text{Mon} = 3(b - a)

Three-day windows; multiply by the overlap length.

Shared middle item

\text{shared} = S_1 + S_2 - S_{\text{total}}

Split sums

n\bar{x} = \sum_{\text{chunks}} (\text{chunk total})

One equation per chunk.

⚡ Batsman: stay in totals. Old total + new score = new count × new average.

⚡ Overlaps: subtract the sums. Shared days cancel, leaving only the difference of the end days.

⚡ One variable for the unknown chunk. Name the smallest unknown x, express the rest through it, and close the equation with the leftover sum.

Interest (SI & CI)

Simple Interest

Simple interest

SI = \frac{PRT}{100}

Any one of the four recovers from the other three.

Amount

A = P + SI = P\left(1 + \frac{RT}{100}\right)

'Amounts to' includes the principal.

Recovering inputs

P = \frac{100\,SI}{RT},\quad R = \frac{100\,SI}{PT},\quad T = \frac{100\,SI}{PR}

n-times in T years

R = \frac{100(n-1)}{T}

Interest is only (n−1)P.

Equal yearly interest

SI_{\text{per year}} = \frac{SI_{\text{total}}}{T}

Same rupees every year.

⚡ Cover the unknown. Write SI = \dfrac{PRT}{100} and cover the letter you want. That covered formula is the whole equation.

⚡ n-times to rate. 'Becomes n times in T years' means the interest earned is (n-1)P. The P cancels.

⚡ One year at a time. Divide the total interest by the years; every year carries exactly that much.

Compound Interest

Compound amount

A = P\left(1 + \frac{R}{100}\right)^T

One chip per year.

Compound interest

CI = A - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]

Year-wise multipliers

A = P \times \frac{100 + r_1}{100} \times \frac{100 + r_2}{100} \times \cdots

For rates that change yearly.

Half-yearly compounding

A = P\left(1 + \frac{R}{200}\right)^{2T}

Rate halves, periods double.

Quarterly compounding

A = P\left(1 + \frac{R}{400}\right)^{4T}

Rate quarters, periods quadruple.

⚡ Multiply the chips. One chip per year, multiplied. Fraction chips cancel before you multiply.

⚡ Divide the chips to get P. An amount after T years walks back to the principal by dividing by the chip T times.

⚡ Half-yearly: halve rate, double count. Convert first, then run the chips on the periods.

CI vs SI: differences & doubling

2-year difference

CI - SI = P\left(\frac{R}{100}\right)^2

3-year difference

CI - SI = P\left(\frac{R}{100}\right)^2\left(3 + \frac{R}{100}\right)

CI doubling chain

2\times \text{ in } T \Rightarrow 2^k\times \text{ in } kT

SI multiple pace

n\times \text{ in } T \Rightarrow n'\times \text{ in } T',\ (n'-1) = (n-1)\frac{T'}{T}

Linear, not powers.

Rate from both figures

R = \frac{200 \times (CI_2 - SI_2)}{SI_2}

Two-year case.

⚡ The rate-fraction squared gap. Two equations — the SI and the difference — hand you the rate and the principal.

⚡ Count the doublings. Every T years at CI the money multiplies by the same factor, so count powers.

⚡ First-year interest lent again. For 2 years, think of the difference as the first year's interest earning R% once more.

Equal annual instalments

Present value (CI)

P = \sum_{k=1}^{n} \frac{x}{\left(1 + \frac{r}{100}\right)^k}

One term per instalment.

Simple interest instalment

P = nx - \frac{x\,r}{100} \cdot \frac{n(n-1)}{2}

Interest the early payments save.

Tabular step

\text{debt}_{k+1} = \text{debt}_k\left(1 + \frac{r}{100}\right) - x

Must end at zero.

⚡ Grow, subtract, repeat. Two or three rows of the debt table finish any small question.

⚡ Present-value one-liner. Discount each instalment by the chip power for its year and add.

⚡ SI instalment shortcut. Use P = nx - \dfrac{xr}{100} \cdot \dfrac{n(n-1)}{2} straight off for simple-interest loans.

Finding P, R, T from amount data

Principal from amount

P = \frac{A}{\left(1 + \frac{r}{100}\right)^T}

CI: divide by the chip power.

Rate from consecutive amounts

1 + \frac{r}{100} = \frac{A_{t+1}}{A_t} \quad (\text{CI})

Divide at CI.

Yearly SI from amounts

SI_{\text{year}} = A_{t+1} - A_t \quad (\text{SI})

Subtract at SI.

Two-amount system

\frac{A_2}{A_1} = 1 + \frac{r}{100} \Rightarrow P = \frac{A_1}{1 + r/100}

⚡ Ratio first, then walk back. The ratio of consecutive CI amounts exposes the chip; divide once more for P.

⚡ Equal yearly steps at SI. Check the amounts differ by a constant — that constant is the yearly interest.

⚡ Two amounts, two unknowns. Dividing the amounts kills P and hands you the chip.

Profit, Loss & Discount

Profit, Loss and CP–SP Basics

Profit / loss per cent

\text{P\%} = \frac{SP - CP}{CP} \times 100

Always on CP.

Selling price

SP = CP \times \frac{100 \pm x}{100}

Plus for profit, minus for loss.

Cost price from SP

CP = SP \times \frac{100}{100 \pm x}

Divide by the chip.

Equal profit and loss at two prices

CP = \frac{S_1 + S_2}{2}

When profit at S1 equals loss at S2.

⚡ Fraction swap for the per cent. Turn the profit into a fraction of CP and the per cent falls out. 170/850 is clearly 1/5.

⚡ Divide by the chip to get CP. A loss of 4\% means the chip 96/100. Divide the SP by the chip, nothing else.

⚡ Same per cent scales linearly. When the profit per cent is fixed, CP and SP scale together. Article counts often hide a clean ratio.

Successive Changes & Equivalent Single Change

Two successive changes

\text{net} = a + b + \frac{ab}{100}

Use negative b for a fall.

Same change twice (rise)

2x + \frac{x^2}{100}

Same change twice (fall)

2x - \frac{x^2}{100}

Equivalent single discount

D = d_1 + d_2 - \frac{d_1 d_2}{100}

For two discounts.

Kept fraction

\text{kept} = \prod \frac{100 - d_i}{100}

Any number of discounts; customer pays this share.

⚡ Multiply the chips. Chips handle any mix of rises and falls, in any order, and extend to three or more changes.

⚡ One line for two discounts. Two discounts collapse with d_1 + d_2 - \dfrac{d_1 d_2}{100}. One line, no chips.

⚡ Work backwards for the missing change. Net chip divided by the known chip gives the unknown chip.

Marked Price, Discount and the CP–MP–SP Chain

Discount per cent

\text{D\%} = \frac{MP - SP}{MP} \times 100

On the marked price.

Marked price for a target gain

MP = CP \times \frac{100 + g}{100 - d}

g = gain%, d = discount%.

SP through the chain

SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)

m = markup%.

Markup per cent

\text{markup\%} = \left(\frac{100+g}{100-d} - 1\right) \times 100

Multi-level markups

\text{final} = \text{cost} \times \prod \frac{100 + x_k}{100}

One chip per trader.

Free-item discount

\text{discount\%} = \frac{\text{free}}{\text{received}} \times 100

⚡ Build the MP-over-CP ratio. Gain g\% and discount d\% together fix MP as a multiple of CP: \dfrac{100+g}{100-d}.

⚡ Markup chip then discount chip. Two fractions give the whole story from CP to SP in one line.

⚡ Trade chains multiply. One chip per middleman. Work backwards from the final price by dividing in reverse order.

Dishonest Dealer, False Weights & Claims

False weight gain

\text{gain\%} = \frac{W - w}{w} \times 100

W = true weight, w = weight used; divide by w.

Weight from a given gain

w = W \times \frac{100}{100 + \text{gain\%}}

Claimed loss with short weight

\text{multiplier} = \frac{100 - L}{100 - c}

L = claimed loss%, c = short weight%.

Cheat at both ends

\text{multiplier} = \frac{100 + b}{100 - s}

b = extra taken while buying, s = shortfall while selling.

True gain from multiplier

\text{gain\%} = (\text{multiplier} - 1) \times 100

⚡ The W minus w over w rule. At cost price, the only question is which weight goes below the line. It is the weight he gives.

⚡ A claimed loss can still be a gain. Judge money per true gram, not per claimed gram.

⚡ Both-ends multiplier. Buy-side chip over sell-side chip. Two fractions, one division.

Same Selling Price: One Profit, One Loss

Net loss for equal plus-minus x%

\text{net loss\%} = \frac{x^2}{100}

Per cent of total CP.

Reconstructed costs

CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}

S = common selling price.

Loss in rupees

\text{loss} = CP_1 + CP_2 - 2S

x from the loss per cent

x = 10\sqrt{\text{loss\%}}

⚡ The x squared over 100 reflex. Same SP, opposite equal percents: write the loss per cent straight away.

⚡ Rebuild both cost prices. Divide the common SP by each chip, add the CPs, compare with twice the SP.

Mixtures & Alligation

Mixture Concentration & Amounts

Amount from ratio

\text{part} = \text{total} \times \frac{\text{share}}{\text{sum of shares}}

Concentration

C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%

Mean price

\text{mean} = \frac{q_1 p_1 + q_2 p_2}{q_1 + q_2}

Adding water

C' = \frac{A}{M + w}

A = amount of the other ingredient, unchanged.

Removing mixture

\text{each ingredient shrinks in its own share}

A uniform draw keeps the ratio.

⚡ Freeze the unchanged ingredient. Water added means milk unchanged. Hang the whole question on the ingredient that does not move.

⚡ Rebuild the per cent after dilution. Milk is fixed, the total grows — divide again.

⚡ One ratio change, one equation. Set the frozen ingredient against the wanted ratio and solve for the addition.

Rule of Alligation

Alligation ratio

\frac{\text{cheaper}}{\text{dearer}} = \frac{D - M}{M - C}

Weighted average

\bar{v} = \frac{n_1 v_1 + n_2 v_2}{n_1 + n_2}

Water as the free ingredient

\text{milk} : \text{water} = (m - 0) : (c - m),\ c > m

⚡ Cross the arms. Dearer-minus-mean and mean-minus-cheaper swap sides. Each quantity takes the other side's arm.

⚡ Water has price zero. Dilution is alligation with one strength at 0.

⚡ Averages alligate too. Salary, age, marks, weight — any average splits into an alligation.

Replacement & Repeated Operations

Repeated equal replacement

\text{left} = C\left(1 - \frac{x}{C}\right)^n

Milk : water after n rounds

\left(1-\frac{x}{C}\right)^n : \left[1-\left(1-\frac{x}{C}\right)^n\right]

Proportional removal

\text{lost} = \text{drawn volume} \times \frac{\text{ingredient share}}{\text{total}}

Unequal draws

C \prod_k \left(1 - \frac{x_k}{C}\right)

Multiply one factor per round when the draws differ.

⚡ One fraction per round. Each round multiplies the original liquid by (1 − x/C). Multiply the fractions; never subtract x twice.

⚡ Two rounds: square the kept fraction. The complement of the milk is the water — no separate calculation.

⚡ Draws keep the inside ratio. A uniform draw removes both liquids in proportion, so the ratio survives the withdrawal. Only the refill changes it.

Milk–Water Ratio & Profit by Adulteration

Adulteration gain

\text{gain\%} = \frac{\text{water}}{\text{milk}} \times 100

Target ratio

W : M = g : 100

For a gain of g% when sold at cost price.

Blend cost price

CP = \frac{q_1 c_1 + q_2 c_2}{q_1 + q_2}, \quad SP = CP\left(1 + \frac{g}{100}\right)

Alloy rebuild

\text{metal} = \text{alloy weight} \times \frac{\text{share}}{\text{sum of shares}}

⚡ Water over milk is the gain. Free litres per honest litre — that ratio, in per cent, is the profit.

⚡ Hit the target ratio by addition. Keep the bigger quantity constant and solve for the addition.

⚡ Rebuild the alloy, then change one metal. Ratio to grams, adjust one column, re-ratio.

Mixing Two Mixtures

Blend of two mixtures

f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}

Volumes for a target fraction

\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}

Fraction from ratio

f_{\text{milk}} = \frac{m}{m + w}

Equal volumes

\bar{f} = \frac{f_1 + f_2 + f_3}{3}

Only when every vessel holds the same volume.

⚡ Fractions first, then average. Ratio → milk fraction per vessel → weighted average.

⚡ Alligate the fractions. The target sits between the two vessel fractions; the distances give the volumes.

⚡ Weight by volume when sizes differ. Multiply each vessel's fraction by its volume before adding.

Time & Work

Work Rates & the LCM Method

One-day work

\frac{1}{T}

Combined time (two workers)

T = \frac{ab}{a + b}

Combined time (three workers)

T = \frac{abc}{ab + bc + ca}

Work done and left

\text{done} = \frac{t}{T}, \quad \text{left} = 1 - \frac{t}{T}

⚡ LCM units, not fractions. Set the job to the LCM of the individual times; every rate becomes a small whole number.

⚡ The ab over a+b reflex. For exactly two workers, one formula — no fraction addition at all.

⚡ Three workers through the LCM. Same method with three rates added in one line.

Efficiency, 'Twice as Good' & Ratio Cases

Efficiency and time

\frac{E_A}{E_B} = \frac{T_B}{T_A}

k-times worker

T_B = (k+1)T, \quad T_A = \frac{(k+1)T}{k}

Times from an efficiency ratio

E_A : E_B = a : b \Rightarrow T_A : T_B = b : a

Per cent more efficient

a\% \text{ more} \Rightarrow E_A : E_B = (100 + a) : 100

⚡ Rate units from the efficiency ratio. Let B = 1 unit/day, A = k units/day; together (k+1) per day prices the job.

⚡ Invert, never scale. 'A is 3 times as good' means A's days = B's days ÷ 3.

⚡ Days-difference cases. 'B takes 24 days more than A' plus an efficiency ratio pins both times.

Pipes & Cisterns

Net speed

\text{net} = (\text{inlets}) - (\text{outlets})

Speeds in units per hour, with tank = LCM of the times.

Time to fill

T = \frac{\text{capacity}}{\text{net speed}}

Two inlets

T = \frac{a b}{a + b}

Both pipes fill.

One inlet, one outlet

T = \frac{a b}{b - a}

a = filling time, b = emptying time, b > a.

Leak time from two fill times

T_{\text{leak}} = \frac{t_1 t_2}{t_2 - t_1}

t₁ = normal time, t₂ = time with the leak.

Stage method

t_2 = \frac{\text{capacity} - \text{speed}_1 \times t_1}{\text{speed}_2}

⚡ One inlet, one outlet: multiply over subtract. For one filling pipe and one emptying pipe, time = product ÷ difference. No LCM needed.

⚡ Leak from two fill times. The leak is the only thing that changed, so the drop in speed belongs to the leak. Use product ÷ difference of the two fill times.

⚡ Check the options before finishing. Filling pipes together are always faster than the fastest pipe alone. Adding an outlet always makes it slower than the inlets alone. Use this to cut two options in a few seconds.

Men–Days–Hours Chain & Provisions

MDH chain

M_1 D_1 H_1 E_1 = M_2 D_2 H_2 E_2 \quad (W_1 = W_2)

Men and days constant

M_1 D_1 = M_2 D_2

Provisions remaining

\text{days} = \frac{M_1 (D_{total} - t)}{M_2}

Work scaling

W_2 = W_1 \times \frac{M_2}{M_1} \times \frac{D_2}{D_1}

⚡ Multiply resources, equate products. Everything that grows the work sits beside M; the missing quantity lands alone.

⚡ Provisions after reinforcement. Stock left = original men × days left; then divide by the new headcount.

⚡ Work left after a share is done. First find the remaining man-days, then apply them to the new team.

Joining / Leaving Mid-work, Alternate Days & Wages

Work done by A in t days

\frac{t}{T_A}

Leaves t days before the end

\frac{T - t}{T_A} + \frac{T}{T_B} = 1

Two-day cycle

\frac{1}{T_A} + \frac{1}{T_B} \text{ per 2 days}

Wage split

w_A = \text{total} \times \frac{1/T_A}{1/T_A + 1/T_B}

⚡ Subtract the finished part. Whoever continues inherits only the remainder.

⚡ Count full cycles, then the tail. Alternate days: measure progress per 2-day cycle.

⚡ Wages follow work, not days alone. Rate ratio × equal days = work ratio.

Time, Speed & Distance

Speed, Distance, Time & Unit Conversion

Basic relation

S = \frac{D}{T}, \quad D = S \times T

km/h to m/s

\text{km/h} \times \frac{5}{18} = \text{m/s}

Equal-distance average

\bar{S} = \frac{2ab}{a + b}

General average speed

\bar{S} = \frac{D_1 + D_2}{T_1 + T_2}

⚡ Convert first, always. Train lengths are in metres, times in seconds — reach m/s before anything else.

⚡ Harmonic mean for round trips. Same distance out and back → 2ab/(a+b) in one line.

⚡ Early and late gaps are time equations. Both runs cover the SAME distance — equate or subtract the two time expressions.

Relative Speed

Relative speed

S_{rel} = S_1 \pm S_2

Meeting time

t = \frac{\text{initial gap}}{S_1 + S_2}

Overtake time

t = \frac{\text{gap or combined length}}{S_1 - S_2}

Gap between two movers

d = (S_1 \pm S_2) \times t

⚡ Subtract for the same direction. Overtaking uses the difference; for two trains the distance is both lengths together.

⚡ Add for opposite directions. The closing speed is the sum even when one side is just a walking man.

⚡ Distance flown till meeting. Find the meeting time first; any third object's distance = its speed × that time.

Trains Crossing Poles, Platforms & Trains

Pole / man

L_{train} = S \times t

Platform / bridge

L_{train} + L_{platform} = S \times t

Train vs train

L_1 + L_2 = S_{rel} \times t

Two-equation extraction

S = \frac{P}{t_{platform} - t_{pole}}

⚡ Pole first: it is the train's own length. The pole crossing IS the train's length in motion.

⚡ Subtract the pole equation. Platform time minus pole time covers exactly the platform.

⚡ Platform crossing from rest. No pole given: just add both lengths and divide by the speed.

Boats & Streams

Effective speeds

u = b + s, \quad v = b - s

Boat and stream from legs

b = \frac{u + v}{2}, \quad s = \frac{u - v}{2}

Time for two legs

t = \frac{d_1}{b + s} + \frac{d_2}{b - s}

Round-trip average

\bar{S} = \frac{2uv}{u+v} = \frac{b^2 - s^2}{b}

Drift

\text{drift} = s \times \text{time}

⚡ Halve the sum, halve the difference. Down and up speeds give the boat and the stream in one line each.

⚡ Extract u and v from trip times. Each trip is one equation; the pair is linear in the reciprocals.

⚡ Two double-trip equations. Two journeys pin down both leg speeds exactly.

Races & Handicaps

Beating margin in metres

\frac{S_A}{S_B} = \frac{D}{D - x}

Beating margin in time

t_B = t_A + t; \quad S_B = \frac{x}{t}

Start handicap

B \text{ runs } D - \text{start} \ ( - \text{win margin})

Circular track meeting

t = \frac{L}{S_1 \mp S_2}

⚡ Translate beats by x m into a speed ratio. Same finishing time — distances are in the speed ratio.

⚡ By x metres or t seconds reveals both speeds. B's last x metres took t seconds — that is B's speed for free.

⚡ Handle the handicap. A start shortens one runner's distance — recompute the ratio.

Algebra

Basic algebraic identities

Square of sum or difference

(a \pm b)^2 = a^2 \pm 2ab + b^2

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Squares combine

(a+b)^2 + (a-b)^2 = 2(a^2+b^2)

Squares subtract

(a+b)^2 - (a-b)^2 = 4ab

Square of a trinomial

(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)

Fourth powers from the ladder

a^4 + b^4 = (a^2+b^2)^2 - 2a^2b^2

⚡ Build higher powers from a sum and a product. Chain the rungs: square-sum first, then cube-sum, then fourth powers. The individual letters are never needed.

⚡ Recognise the identity inside decimals. Cubed decimals over a trinomial mean a cube identity. The value is the sum or difference of the bases.

⚡ Value-putting among expression options. Put small numbers into the question and each option. Wrong options die in one round; ties die in a second.

$x+\frac{1}{x}$ type expressions

Square rung

x^2+\frac{1}{x^2} = \left(x+\frac1x\right)^2-2 = \left(x-\frac1x\right)^2+2

Cube rung, sum

x^3+\frac{1}{x^3} = \left(x+\frac1x\right)^3 - 3\left(x+\frac1x\right)

Cube rung, difference

x^3-\frac{1}{x^3} = \left(x-\frac1x\right)^3 + 3\left(x-\frac1x\right)

Fourth rung

x^4+\frac{1}{x^4} = \left(x^2+\frac{1}{x^2}\right)^2 - 2

Fifth rung

x^5+\frac{1}{x^5} = \left(x^2+\tfrac1{x^2}\right)\left(x^3+\tfrac1{x^3}\right)-\left(x+\tfrac1x\right)

Ladders link

\left(x+\frac1x\right)^2 - \left(x-\frac1x\right)^2 = 4

Equal-end quadratic

ax^2 - bx + a = 0 \Rightarrow x+\frac1x = \frac ba

Mixed form

\left(px+\frac{1}{qx}\right)^2 = p^2x^2+\frac{1}{q^2x^2}+\frac{2p}{q}

⚡ Run the ladder in order. Square rung first, then cube rung, then reuse them for higher rungs. Never restart from k for each question part.

⚡ Divide the quadratic by x. Equal first and last coefficients mean the quadratic is a k-value in disguise. Divide by x and read it off.

⚡ Switch ladders with one square. The two ladders differ by 4 under a square: k squared equals m squared plus 4. Convert once, then stay on the new ladder.

$a^3+b^3+c^3-3abc$ and conditional identities

Master identity

a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Sum form

a^3+b^3+c^3-3abc = s\left(s^2-3P\right)

Half form

a^3+b^3+c^3-3abc = \tfrac12\,s\left[(a-b)^2+(b-c)^2+(c-a)^2\right]

Zero-sum case

a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc

Equal case

a^2+b^2+c^2 = ab+bc+ca \Rightarrow a=b=c

Power-sum expansion

a^3+b^3+c^3 = s^3 - 3sP + 3R

Pair products

(a+b)(b+c)(c+a) = sP - R

Pairwise from squares

ab+bc+ca = \dfrac{(a+b+c)^2-(a^2+b^2+c^2)}{2}

⚡ Hunt for a hidden zero sum. Brackets like x minus y, y minus z, z minus x always sum to zero. The cube-sum is three times the product.

⚡ Close numbers: use the half form. When the three numbers differ by little, the squared differences are tiny, and half of s times their sum is quick arithmetic.

⚡ Value-putting under a condition. If a condition like a+b+c=0 is given, pick simple numbers that satisfy it and evaluate the expression.

Surds: rationalisation and square roots of surds

Rationalisation

\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a-b}

Conjugate product

(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b

Root of a surd, plus

\sqrt{a+2\sqrt{b}} = \sqrt{m}+\sqrt{n},\ m+n = a,\ mn = b

Root of a surd, minus

\sqrt{a-2\sqrt{b}} = \sqrt{m}-\sqrt{n}\ \ (m > n)

Product-one pair

x = p+\sqrt{q},\ p^2-q = 1 \Rightarrow \tfrac1x = p-\sqrt{q},\ x+\tfrac1x = 2p

Telescoping sum

\sum \frac{1}{\sqrt{n}+\sqrt{n+1}} = \sqrt{\text{last}} - \sqrt{\text{first}}

Difference of roots

\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}

⚡ Denest by sum and product. Rewrite the inner coefficient as 2 root something, then find two numbers with the given sum and product.

⚡ Spot the product-one conjugate. For x = p + sqrt(q), test p squared minus q. If it is 1, the reciprocal is the conjugate and the ladder runs.

⚡ Telescoping rationalisation. Each fraction over consecutive roots becomes a difference; middle terms cancel, leaving last root minus first.

Linear equations, graphs and polynomials

Unique solution

\frac{a_1}{a_2} \ne \frac{b_1}{b_2}

lines cross once

No solution

\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}

parallel lines

Infinite solutions

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

same line

Area with the axes

\text{Area} = \frac{1}{2}\cdot\left|\frac{c}{a}\right|\cdot\left|\frac{c}{b}\right|

Remainder theorem

p(x) \div (x-a) \Rightarrow R = p(a)

for px - q, substitute q/p

Roots of a quadratic

\alpha+\beta = -\frac ba,\quad \alpha\beta = \frac ca

Discriminant

b^2 - 4ac \gtrless 0

decides root nature

⚡ Area from intercepts. Put y = 0 for the x-intercept and x = 0 for the y-intercept, then halve the product of the absolute values.

⚡ Remainder equals substitution at the zero. No long division: substitute the zero of the divisor into the polynomial.

⚡ Test the options for the intersection point. For a meeting-point question, plug each option into both equations instead of solving the pair.

Maxima and minima (AM ≥ GM, quadratics)

AM-GM

\frac{x+y}{2} \ge \sqrt{xy}

equality when x = y

Plus-form floor

ax + \frac{b}{x} \ge 2\sqrt{ab}\ \ (x>0)

at x = sqrt(b/a)

Vertex location

x = -\frac{b}{2a}

Extreme value

\frac{4ac-b^2}{4a}

max for a < 0, min for a > 0

Fixed sum

x+y = S \Rightarrow xy \le \frac{S^2}{4}

Fixed product

xy = P \Rightarrow x+y \ge 2\sqrt{P}

Always positive

ax^2+bx+c > 0 \iff a > 0,\ b^2 < 4ac

strict inequality, strict discriminant

⚡ Equal split for weighted sums. To maximise a product under a weighted sum, set the weighted pieces equal, then square.

⚡ Complete the square, read the floor. For a > 0, the completed square shows the least value directly as the loose constant.

⚡ Equal roots sit on the boundary. Equal roots mean the discriminant is exactly zero; solve the resulting equation for the unknown.

Geometry

Lines and angles

Angles on a line / around a point

180^\circ,\ 360^\circ

A straight line totals 180 degrees; one full turn totals 360.

Co-interior angles

a+b=180^\circ

The two inside angles on one side of a transversal, between parallel lines.

Complement / supplement

\text{supplement}-\text{complement}=90^\circ

Complement = 90 − x, supplement = 180 − x.

Equal pairs at parallels

\text{corresponding}=\text{alternate}=\text{vertically opposite}

Each of these pairs is equal.

⚡ Say the shape, then the rule. Trace the two angles with your finger. An F or Z shape means they are equal. A C shape means they add to 180^\circ.

⚡ Supplement minus complement is always 90. Whatever the angle, its supplement and its complement differ by exactly 90^\circ. Use it as a free check, or to build the equation.

Triangles and their centres

Angle sum and exterior angle

A+B+C=180^\circ,\quad \text{ext at }A=B+C

Exterior angle = sum of the two remote (far) interior angles.

Centroid division

AG:GD=2:1

G is the centroid on median AD; the vertex piece is twice the base piece.

Incentre angle

\angle BIC=90^\circ+\dfrac{A}{2}

I = incentre, where the angle bisectors meet.

Circumcentre angle

\angle BOC=2A

O = circumcentre; the angle at O stands on the same arc BC as angle A.

Orthocentre angle

\angle BHC=180^\circ-A

H = orthocentre, where the altitudes meet.

Apollonius (median length)

m_a^2=\dfrac{2b^2+2c^2-a^2}{4}

Median to side a of a triangle with sides a, b, c.

Isosceles median to base

m=\sqrt{a^2-\left(\dfrac{b}{2}\right)^2}

a = equal side, b = base.

Heron's area

K=\sqrt{s(s-a)(s-b)(s-c)},\ s=\dfrac{a+b+c}{2}

s = semi-perimeter (half the perimeter).

⚡ Centre angles from one input. Only \angle A is needed. Read which centre the question names, then use its formula.

⚡ Centroid cut in the ratio 2 : 1. Call the median 3 parts. The centroid gives 2 parts on the vertex side and 1 part on the base side.

⚡ Median in an isosceles triangle. Skip Apollonius when two sides are equal. The median to the base is \sqrt{a^2-(b/2)^2} with a the equal side and b the base.

⚡ Heron families worth memorising. (3,4,5) gives 6, (5,12,13) gives 30, (13,14,15) gives 84, (7,24,25) gives 84, (9,12,15) gives 54, (10,24,26) gives 120.

Congruence, similarity and BPT

Similarity ratios

\frac{a_1}{a_2}=k,\quad \frac{P_1}{P_2}=k,\quad \frac{K_1}{K_2}=k^2

a = side, P = perimeter, K = area; k = scale factor.

Areas from perimeters

\frac{K_1}{K_2}=\left(\frac{P_1}{P_2}\right)^2

Square a length ratio to get the area ratio; take a root to go back.

BPT (Thales)

DE\parallel BC\Rightarrow\frac{AD}{DB}=\frac{AE}{EC}

A line parallel to one side cuts the other two sides in the same ratio.

Midpoint theorem

D,E\ \text{midpoints}\Rightarrow DE=\frac{BC}{2}

The join of two midpoints is half the third side and parallel to it.

Angle bisector theorem

\frac{BD}{DC}=\frac{AB}{AC}

The bisector of angle A splits BC in the ratio of the sides AB and AC.

⚡ Square the perimeter ratio for areas. Perimeters in ratio p:q mean areas in ratio p^2:q^2. Going back, take the square root.

⚡ Spot the word midpoints. 'Midpoints of two sides' means the joining segment is half the third side, and parallel to it.

⚡ BPT: write the ratio directly. A line parallel to a side cuts equal ratios on both other sides. Write the proportion, substitute, solve.

Pythagoras theorem and triplets

Pythagoras

h^2=p^2+b^2

h = hypotenuse (longest side, opposite the right angle).

Missing leg

p=\sqrt{h^2-b^2}

Subtract when a leg is missing.

Median to hypotenuse

m=\dfrac{h}{2}

The median drawn from the right-angle corner.

Rectangle diagonal

d=\sqrt{l^2+b^2}

The corner angles of a rectangle are right angles.

Triangle type test

a^2\lessgtr b^2+c^2

a = longest side; equal means right, greater means obtuse, smaller means acute.

⚡ See two numbers, recall the third. 12 with 13 gives 5. 24 with 25 gives 7. 15 with 17 gives 8. Multiples scale: 10-24-26 is 5-12-13 times 2.

⚡ Median to the hypotenuse is half of it. In any right triangle, the median from the right angle equals half the hypotenuse.

⚡ Classify: square the longest side only. To name the triangle type, compare the longest side's square with the sum of the other two squares.

Quadrilaterals and polygons

Quadrilateral angle sum

A+B+C+D=360^\circ

Any four-sided figure.

Parallelogram angles

A+B=180^\circ,\quad A=C

Adjacent angles supplement; opposite angles equal.

Cyclic quadrilateral

A+C=180^\circ,\quad B+D=180^\circ

Opposite corners on one circle.

Rhombus

K=\frac{1}{2}d_1d_2,\quad a=\sqrt{\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2}

d1, d2 = diagonals; they cross at right angles.

Trapezium

K=\frac{1}{2}(a+b)h

a and b are the two parallel sides; h is the gap between them.

Parallelogram

K=bh

Height is measured perpendicular to the base.

Regular polygon

\text{ext}=\frac{360^\circ}{n},\quad \text{int}=180^\circ-\text{ext},\quad \text{diagonals}=\frac{n(n-3)}{2}

n = number of sides.

⚡ Exterior angle to number of sides. Exterior angle =180^\circ- interior, and n=\dfrac{360^\circ}{\text{exterior}}.

⚡ Rhombus side from half-diagonals. Halve both diagonals. They are the legs of a right triangle whose hypotenuse is the side.

⚡ Count diagonals in one line. Each of the n corners joins n-3 others, and every diagonal is counted twice, giving \dfrac{n(n-3)}{2}.

Circles: chords, tangents, secants and cyclic angles

Chord from distance

\ell=2\sqrt{r^2-d^2}

d = distance from centre to chord.

Equal chords

\ell_1=\ell_2\Rightarrow d_1=d_2

Equal chords sit equally far from the centre.

Centre vs circumference angle

\angle BOC=2\angle BAC

Both angles stand on chord BC.

Tangent length

PT=\sqrt{d^2-r^2}

P is d from the centre of a circle of radius r.

Tangent-secant

PT^2=PA\cdot PB

Tangent squared = outside part times whole secant.

Intersecting chords

PA\cdot PB=PC\cdot PD

Two chords crossing inside the circle.

Alternate segment

\angle(\text{tangent},\ \text{chord})=\angle\text{ in alternate segment}

The angle between a tangent and a chord equals the angle the chord makes on the far side.

Common tangents (transverse / direct)

L_T=\sqrt{d^2-(r_1+r_2)^2},\quad L_D=\sqrt{d^2-(r_1-r_2)^2}

d = distance between centres.

⚡ The triplet inside the circle. Radius, distance and half-chord are the sides of a right triangle. Radius 13 and distance 5 give half-chord 12.

⚡ Tangent-secant: multiply the pieces. Tangent squared equals outside part times the whole secant. Whole = outside + inside.

⚡ Count common tangents from d. Compare the centre distance with r_1+r_2 and r_1-r_2 and read off 4, 3, 2, 1 or 0.

⚡ Chord of the outer circle touching the inner one. Concentric circles: a chord of the big circle that just touches the small one is 2\sqrt{R^2-r^2}. The small radius is its distance from the centre.

Mensuration (2D)

Areas of triangles

Triangle area

K=\frac{1}{2}bh

b = any side, h = perpendicular height on it.

Equilateral triangle

K=\frac{\sqrt3}{4}a^2,\quad h=\frac{\sqrt3}{2}a

a = side; height is root-three over two of the side.

Heron's formula

K=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}

s = half the perimeter.

Altitude to hypotenuse

h=\frac{ab}{c}=\frac{2K}{c}

a, b legs, c hypotenuse; from equating two areas.

Inradius / circumradius

r=\frac{K}{s},\quad R=\frac{abc}{4K}

K = area, s = half-perimeter.

Median split

\text{median}\Rightarrow\text{two equal areas}

Each median halves the area of a triangle.

⚡ Triplet beats Heron. Before Heron, check for a triplet. A right triangle needs only half the product of its legs.

⚡ Isosceles: half it first. The median to the base is the height. Halve the base, then use Pythagoras with an equal side.

⚡ Two radii from Heron. After Heron gives the area, both radii are one division away: r divides by s, R uses abc over 4K.

Areas of quadrilaterals

Rectangle

K=lb,\quad P=2(l+b),\quad d=\sqrt{l^2+b^2}

Three linked facts; any two fix the third.

Square

K=a^2,\quad d=a\sqrt2

Diagonal is root-two times the side.

Parallelogram

K=bh=ab\sin\theta

theta = angle between the two given sides.

Rhombus

K=\frac{1}{2}d_1d_2

Diagonals cross at right angles and halve each other.

Trapezium

K=\frac{1}{2}(a+b)h

a, b = the two parallel sides.

Any quadrilateral

K=\frac{1}{2}d(h_1+h_2)

d = a diagonal; h1, h2 = perpendiculars onto it.

⚡ Rectangle identities, not quadratics. With perimeter and diagonal, jump straight to (l+b)^2=l^2+b^2+2lb and read off the area.

⚡ Rhombus ratio to perimeter. Ratio plus area fixes both diagonals. Halve them, spot the triplet, read the side.

⚡ Diagonal splits any quadrilateral. No special shape? Draw one diagonal. The area is half the diagonal times the sum of the two heights.

Circles, sectors and rings

Circle

K=\pi r^2,\quad C=2\pi r

Take pi as 22/7 unless stated.

Arc and sector

\text{arc}=\frac{\theta}{360}2\pi r,\quad \text{sector}=\frac{\theta}{360}\pi r^2

Same fraction of circumference and area.

Ring

K=\pi(R^2-r^2)

R = outer radius, r = inner radius.

Wheel revolutions

N=\frac{D}{C}=\frac{\text{distance}}{2\pi r}

Count turns by dividing distance by one circumference.

Quadrant / semicircle

\text{quad}=\frac{\pi r^2}{4},\quad \text{semi}=\frac{\pi r^2}{2}

Quarter and half of the circle area.

⚡ Radius first, always. Circumference, diameter or area given: convert to the radius before anything else. Every formula lives on r.

⚡ Simplify the sector fraction. Reduce theta over 360 first: 72/360 is 1/5, 90/360 is 1/4. Then one multiplication finishes.

⚡ Wheels: count the turns. One turn covers one circumference. Divide total distance by it, in the same units.

Regular polygons and inscribed figures

Regular hexagon

K=\frac{3\sqrt3}{2}a^2

Six equilateral triangles of side a.

Polygon from apothem

K=\frac{1}{2}\times P\times a

P = perimeter, a = apothem (centre to a side).

Exterior angle

\text{ext}=\frac{360^\circ}{n}

Equal turns around the boundary.

Interior angle

\text{int}=180^\circ-\text{ext},\quad \text{sum}=(n-2)180^\circ

One angle plus the total for all n.

Diagonals

d=\frac{n(n-3)}{2}

n sides give this many diagonals.

⚡ Hexagon = 6 equilateral triangles. Six triangles of side a. Use the equilateral area times six; the root-three factor is already familiar.

⚡ Exterior angle finds n. Divide 360 by the exterior angle to get the side count; the interior angle is its partner to 180.

⚡ Diagonal equation factors. Set n(n-3)/2 equal to the given count, then factor the quadratic. Exam answers are whole numbers.

Percentage change, similarity and re-bent shapes

Similar figures

\frac{a_1}{a_2}=k\Rightarrow\frac{K_1}{K_2}=k^2

Lengths take k once; areas take k squared.

Successive percentage change

\text{net}=a+b+\frac{ab}{100}

Works for two changes in a row, like both dimensions.

Reverse percentage area change

1+\frac{x}{100}=\left(1+\frac{y}{100}\right)^2

Area factor is the side factor squared.

Map areas

\text{true area}=\text{map area}\times(\text{scale})^2

Square the linear scale before converting units.

⚡ a + b + ab/100 in one line. Both dimensions change by the same percent: plug once, no quadratics, no decimals.

⚡ Reverse the square root. Area up 21%? Factor 1.21=1.1^2, so the side rose 10%. Recognise perfect squares of decimals.

⚡ Wire into shapes: circle wins. Same perimeter, compare areas. Circle beats square beats any rectangle, so guess before computing.

Mensuration (3D)

Cube and cuboid

Cuboid

V=lbh,\quad \text{LSA}=2h(l+b),\quad \text{TSA}=2(lb+bh+hl)

l, b, h are the three dimensions.

Cube

V=a^3,\quad \text{LSA}=4a^2,\quad \text{TSA}=6a^2

a = edge.

Diagonals

d=\sqrt{l^2+b^2+h^2},\quad d_{\text{cube}}=a\sqrt3

Corner to opposite corner.

Capacity

1\text{ m}^3=1000\text{ L},\quad 1\text{ L}=1000\text{ cm}^3

Convert once, at the end.

Painted cube counts

8,\ 12(n-2),\ 6(n-2)^2,\ (n-2)^3

3, 2, 1, 0 painted faces for n by n by n.

⚡ Cubes table, not cube roots. Know the cubes to 12 by heart. Reverse questions become look-ups: 343 is 7 cubed.

⚡ Capacity: metres to litres. Compute the volume in cubic metres, then multiply by 1000 for litres. Never by 100.

⚡ Edge scaling in powers. Edge times k: surface times k squared, volume times k cubed. Doubling is 4 and 8.

Cylinder

V=\pi r^2h,\quad \text{CSA}=2\pi rh,\quad \text{TSA}=2\pi r(h+r)

r = radius, h = height.

Missing height

h=\frac{V}{\pi r^2}

Reverse of the volume formula.

Capacity

\text{litres}=\text{m}^3\times1000

Same conversion as tanks.

Dimension change

V\propto r^2h,\quad \text{CSA}\propto rh

Scale each symbol by its own factor.

⚡ 154 and friends. pi r squared for r = 7, 14, 21 is 154, 616, 1386. Reverse questions then divide by a friendly number.

⚡ Scale the symbols, not the shape. Radius doubled, height halved: volume factor is 4 times 1/2 = 2. Track r squared and h separately.

⚡ Melt: equate volumes. Melting conserves volume. Write both volume formulas equal, cancel, solve for the new length.

Cone

Slant height

\ell=\sqrt{r^2+h^2}

Radius, height, slant: a right triangle.

Cone volume

V=\frac{1}{3}\pi r^2h

One-third of the same-base cylinder.

Cone surfaces

\text{CSA}=\pi r\ell,\quad \text{TSA}=\pi r(\ell+r)

Skirt alone, or skirt plus base.

Same base ratios

V_{\text{cone}}=\frac{V_{\text{cyl}}}{3}

Equal base and height.

⚡ Hunt the triplet. 7-24-25, 3-4-5 and their multiples cover nearly every cone. Two lengths known, read the third.

⚡ One-third both ways. Cone to cylinder: divide by 3. Cone volume given: multiply by 3 before dividing by the base area.

⚡ Tent = curved surface. Canvas touches only the slanted side, so use pi r l. Add the base circle only for a solid cone.

Sphere and hemisphere

Sphere

V=\frac{4}{3}\pi r^3,\quad S=4\pi r^2

One radius drives both.

Hemisphere

V=\frac{2}{3}\pi r^3,\quad \text{curved}=2\pi r^2,\quad \text{total}=3\pi r^2

Total adds the flat circle.

Melting into n parts

r_{\text{small}}^3=\frac{R^3}{n}

Divide the cubed length, then cube-root.

Radius scaling

V\to k^3V,\quad S\to k^2S

k = radius scale factor.

⚡ Total hemisphere = 3 circles. Curved shell 2 pi r squared plus flat face pi r squared equals 3 pi r squared. Never 2.

⚡ Cube-root the split. One sphere into 8 equal spheres: each cubed radius is one-eighth, so each radius is half.

⚡ Ratio: cube it or square it. Radii 3:4 mean volumes 27:64 and surfaces 9:16. Cancel pi, apply the right power.

Prisms, pyramids and painted cubes

Prism

V=\text{base area}\times\text{length}

Base can be any polygon.

Pyramid

V=\frac{1}{3}\times\text{base area}\times h

h = perpendicular height.

Frustum

V=\frac{\pi h}{3}(R^2+r^2+Rr)

R, r = the two end radii.

Lateral surfaces

\text{prism}=P\times L,\quad \text{pyramid}=\frac12 P\times\ell

P = base perimeter; L = length; slant for pyramid.

⚡ One-third rule of thumb. Same base and height: pyramid = one third of the prism. Use it to sanity-check any answer.

⚡ Add volumes, then root. Melting several solids: add the volumes, then take the cube root for a cube's edge.

⚡ Frustum: three terms. R squared, r squared, Rr. Radii 5 and 3 give 25 + 9 + 15 = 49, and the numbers turn friendly.

Trigonometry

Ratios and standard values

Primary ratios

\sin\theta=\frac{o}{h},\quad \cos\theta=\frac{a}{h},\quad \tan\theta=\frac{o}{a}

o = opposite, a = adjacent, h = hypotenuse.

Reciprocals

\text{cosec}=\frac{1}{\sin},\quad \sec=\frac{1}{\cos},\quad \cot=\frac{1}{\tan}

Flip the fraction.

Standard values

\sin\theta=\frac{\sqrt{k}}{2},\ k=0,1,2,3,4

For 0, 30, 45, 60, 90 degrees; cos runs the row backwards.

One ratio to all

\sin\theta=\frac{3}{5}\Rightarrow\cos=\frac45,\ \tan=\frac34

Draw the 3-4-5 triangle and read every ratio off it.

⚡ The root-k-over-2 row. sin at 0, 30, 45, 60, 90 is root-0, root-1, root-2, root-3, root-4, all over 2. Cos is the same row reversed.

⚡ Triplet finishes the ratios. Given sin = 3/5, place 3 and 5 in the triangle; the third side 4 completes 3-4-5 and every ratio follows.

⚡ Angle from a value. Isolate the ratio, then match the table entry. 2 sin = root 3 means sin = root-3 over 2, the 60-degree slot.

Fundamental identities

Pythagorean identities

\sin^2+\cos^2=1,\quad 1+\tan^2=\sec^2,\quad 1+\cot^2=\cosec^2

Three engines from one identity.

Conjugate pairs

(\sec+\tan)(\sec-\tan)=1,\quad (\cosec+\cot)(\cosec-\cot)=1

Sum and difference are reciprocals.

Squares of sums

(a+b)^2+(a-b)^2=2(a^2+b^2)

Cross terms cancel in pairs.

Reciprocal products

\sin\cdot\cosec=\cos\cdot\sec=\tan\cdot\cot=1

The constant that kills cross terms.

⚡ Sum given, difference taken. sec + tan = 5 means sec - tan = 1/5, because their product is 1. Then add or subtract the pair.

⚡ Divide through by cosine. A sin-cos fraction becomes a tan fraction when every term is divided by cosine. Substitute tan and finish.

⚡ Square and subtract two. tan + cot = 5: square it, use tan times cot = 1, and the squared sum is 25 - 2.

Complementary angles

Complementary swaps

\sin(90^\circ-\theta)=\cos\theta,\ \tan(90^\circ-\theta)=\cot\theta,\ \sec(90^\circ-\theta)=\cosec\theta

Drop the co- or add it.

Right triangle angles

A+B=90^\circ\Rightarrow\sin A=\cos B

The two acute angles are partners.

Pairing to one

\tan\theta\cdot\tan(90^\circ-\theta)=1

Complementary tangents multiply to 1.

⚡ Sum the angles first. Before computing anything, add the two angles. Ninety degrees means the terms are twins.

⚡ Chains pair from the ends. tan 5 with tan 85, tan 25 with tan 65: each pair multiplies to 1, and the lone tan 45 is 1.

⚡ Match partners to find the angle. sin of something = cos of something: the two somethings must add to 90. That gives a linear equation.

Value-putting and given-ratio questions

Square of sin+cos

(\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta

The bridge from a sum to a product.

Divide by cosine

\frac{a\sin+b\cos}{c\sin+d\cos}=\frac{a\tan+b}{c\tan+d}

After dividing every term by cosine.

Cot fraction to triangle

\cot\theta=\frac{21}{20}\Rightarrow\text{hyp}=29

Two sides given, Pythagoras gives the third.

Reciprocal pair sum

x+\frac{1}{x}\ \text{from}\ x\cdot\frac{1}{x}=1

Conjugates of cosec plus cot.

⚡ Symmetry cancels first. cos squared 30 and sin squared 60 are the same number; scan for such twins before computing.

⚡ Square the sum. sin + cos given: square it to reach 1 + 2 sin cos, then read off the product.

⚡ Condition straight into the fraction. 5 tan = 4: divide the fraction by cosine, substitute, one line of arithmetic.

Maximum and minimum values

Amplitude of a sin + b cos

\max=\sqrt{a^2+b^2},\quad \min=-\sqrt{a^2+b^2}

General angles; on 0 to 90 check the endpoints too.

sin times cos

\sin\theta\cos\theta=\frac{\sin2\theta}{2}\le\frac12

Peak at 45 degrees.

AM-GM floor

x+\frac{1}{x}\ge2

For positive x; equality when x = 1.

Weighted squares

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

Rewrite as one constant plus one square.

⚡ Square, add, root. The maximum of a sin + b cos is the hypotenuse of the a-b right triangle. 4 and 3 give 5.

⚡ AM-GM floor of two. Anything plus its own reciprocal bottoms at 2: tan + cot, sec + cosec, all the same.

⚡ Weighted squares: constant plus square. Rewrite 5 sin squared + 12 cos squared as 5 + 7 cos squared. The range is then obvious.

Heights and Distances

Angles of elevation and depression

Tangent rule

\tan\theta = \frac{\text{height above eye}}{\text{horizontal distance}}

The angle sits at the observer. Height is opposite, distance is next to the angle.

Height and distance

h = d\tan\theta,\qquad d = h\cot\theta

Slanting length (thread, wire, ladder)

h = L\sin\theta,\qquad d = L\cos\theta

L is the slanting line of sight, the hypotenuse of the triangle.

Depression to elevation

\text{depression from top} = \text{elevation from bottom}

The two horizontal lines are parallel, so the angles are equal.

⚡ Swap depression for elevation first. Never work with a downward angle directly. Redraw it at the bottom of the tower and solve an ordinary elevation question.

⚡ Shadows without trigonometry. Same sun, same time, similar triangles. Set up the stick ratio and multiply; no tangent needed.

⚡ Forty-five degrees means equal legs. Whenever the angle is 45^\circ, the height above the eye equals the horizontal distance. Write the equal pair without any tangent.

Standard angles: 30°, 45°, 60°

Tangent values

\tan 30^\circ = \frac{1}{\sqrt{3}},\quad \tan 45^\circ = 1,\quad \tan 60^\circ = \sqrt{3}

Distance from height

d = h\cot\theta

cot 30 = sqrt3, cot 45 = 1, cot 60 = 1/sqrt3.

Ladder on a wall

h = L\sin\theta,\quad d = L\cos\theta

Theta is the ladder's angle with the ground.

Fifteen and seventy-five

\tan 15^\circ = 2-\sqrt{3},\qquad \tan 75^\circ = 2+\sqrt{3}

⚡ Read distance straight off the cot column. With a standard angle, distance = height \times the cot value. No division, no fraction juggling.

⚡ Fifteen and seventy-five multiply to one. Angles that add to 90^\circ have tangents that multiply to 1. Use this to check answers or flip a division into a multiplication.

Two observation points (two angles)

Same side (walk towards)

h = \frac{d}{\cot\alpha - \cot\beta}

d is the distance walked; beta is the nearer, bigger angle.

Opposite sides

h = \frac{d}{\cot\alpha + \cot\beta}

d is the full distance between the two observers.

Gap between two objects from a height

\text{gap} = h(\cot\alpha - \cot\beta)

Tower seen from foot and roof of a building

d = \frac{b}{\tan\beta - \tan\alpha},\quad H = d\tan\beta

b = building height; beta from the foot, alpha from the roof.

⚡ Thirty-sixty fast numbers. For the 30 and 60 pair: same side h = d\times\dfrac{\sqrt{3}}{2}; opposite sides h = d\times\dfrac{\sqrt{3}}{4}. Both come from \cot 30^\circ - \cot 60^\circ = \dfrac{2}{\sqrt{3}}.

⚡ Write the cot pair before the numbers. Always simplify \cot\alpha - \cot\beta (or the sum) first. Substituting numbers into unsimplified surds is where errors creep in.

Moving observers: speed and time

Moving observer chain

v\,t = h(\cot\alpha - \cot\beta)

alpha = first (farther) angle, beta = second (nearer) angle.

Time to reach the foot

t = \frac{h\cot\beta}{v}

Use the angle at the car's current position.

Vertical rise

\text{rise} = d(\tan\beta - \tan\alpha)

d = fixed horizontal distance; angles of depression shrink as the balloon rises.

Speed conversion

1\ \text{km/h} = \frac{5}{18}\ \text{m/s}

⚡ One chain, any unknown. v t = h(\cot\alpha - \cot\beta) contains every moving-observer question. Cover the unknown and solve.

⚡ Check the units before the triangle. Convert km/h to m/s with 5/18 before anything else. A speed in the wrong unit spoils an otherwise perfect triangle.

Compound figures: buildings, pedestals, broken objects

Stacked object (statue on pedestal)

s = d(\tan\beta - \tan\alpha)

d comes from the lower triangle: d = pedestal height x cot alpha.

Tower on a building

t = d(\tan\beta - \tan\alpha),\quad d = b\cot\alpha

From a roof: depression and elevation

d = b\cot\alpha,\qquad H = b + d\tan\beta

Broken tree

\text{stump} = x\tan\theta,\quad \text{broken} = \frac{x}{\cos\theta}

x = distance from the foot to where the top touches.

⚡ Subtract the tans, multiply once. For any stacked object, extra height = d(\tan\beta - \tan\alpha). Compute the tangent difference first, then one multiplication.

⚡ Broken tree: tan plus sec. Total height = x(\tan\theta + \sec\theta) where x is the ground distance to the touching point. At 30^\circ that is x\left(\dfrac{1}{\sqrt{3}} + \dfrac{2}{\sqrt{3}}\right) = x\sqrt{3}.

Statistics

Mean, weighted mean and combined mean

Average

\bar{x} = \frac{\text{sum of values}}{\text{count}}

Total

\text{total} = \bar{x} \times n

Combined average

\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}

Missing value

x = n\bar{x} - \sum(\text{known values})

New member

w = (n+1)\bar{x}_{new} - n\bar{x}_{old}

Use the same pattern for a leaving member with n-1.

⚡ Deviations from a round base. Pick a round number near the values. Add the small differences and divide by the count. The base can be anything.

⚡ Watch where the average is pulled. The combined average always lies between the two group averages, closer to the bigger group. Use this to reject impossible options in seconds.

Median and mode of raw data

Median (odd n)

\text{value at } \frac{n+1}{2}\text{th place}

Position in the sorted list.

Median (even n)

\frac{\text{(n/2)th} + \text{(n/2+1)th}}{2}

Empirical relation

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Rough rule for mildly skewed data; rearrange for any missing one.

Median from mode and mean

\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3}

Transform

y = kx + c \Rightarrow \text{Med}_y = k\,\text{Med}_x + c

⚡ Count positions, do not hunt. For odd n, the median sits at position (n+1)/2 of the sorted list. Count to that position instead of scanning for the middle by eye.

⚡ Median without sorting everything. You only need the middle order statistics. In a long list, quickly bucket values as low or high; full sorting wastes time.

Median and mode of grouped data

Grouped mean

\bar{x} = \frac{\sum f x}{\sum f}

x is the midpoint of each class.

Grouped median

\text{Med} = L + \frac{\frac{n}{2} - c}{f} \times h

L: lower limit of median class; c: cf before it; f: its frequency; h: width.

Grouped mode

\text{Mode} = L + \frac{f_m - f_1}{2f_m - f_1 - f_2} \times h

f_m: modal class frequency; f_1, f_2: neighbouring frequencies.

⚡ Build the cf column once, use it thrice. Median, quartiles and 'how many below a value' questions all read the same cumulative column. Write it before touching any formula.

⚡ Cross-check the median class fast. Half of n must fall inside the median class. Glance: the cf before it is below n/2, the cf through it is at or above n/2.

Range, variance and standard deviation

Range

R = \text{max} - \text{min}

Variance

\sigma^2 = \frac{\sum (x-\bar{x})^2}{n}

Average of squared distances from the mean.

Standard deviation

\sigma = \sqrt{\sigma^2}

Shift and scale

\text{SD}(kx + c) = |k|\,\text{SD}(x)

Adding c changes nothing; multiplying scales the SD.

First n naturals

\sigma^2 = \frac{n^2-1}{12}

Two values

\text{SD} = \frac{|p-q|}{2}

Coefficient of variation

CV = \frac{\sigma}{\bar{x}} \times 100\%

Sum of squares

\sum x^2 = n(\bar{x}^2 + \sigma^2)

⚡ Skip the squares for symmetric lists. Values spaced evenly around their mean cancel in pairs: 2,4,6,8,10 gives squared deviations 16,4,0,4,16. Write only the distinct squares.

⚡ AP spread without listing. For k, 2k, 3k, ..., nk use variance k²(n²−1)/12. No squaring of long lists.

Averages of special series

Sum of first n naturals

\sum k = \frac{n(n+1)}{2}

Sum of squares

\sum k^2 = \frac{n(n+1)(2n+1)}{6}

Sum of cubes

\sum k^3 = \left[\frac{n(n+1)}{2}\right]^2

The square of the sum of the first n naturals.

Mean of first n naturals

\frac{n+1}{2}

Mean of first n odds

Mean of first n evens

n+1

Multiples of k

\text{sum} = \frac{kn(n+1)}{2},\; \text{mean} = \frac{k(n+1)}{2}

⚡ The middle is the mean. For any equally spaced list the mean is the middle term. Use it forwards (find the mean) and backwards (rebuild the list).

⚡ Odds add to squares. 1 + 3 + ... up to n odd numbers is exactly n². Use it to test counts fast.

Data Interpretation

Reading tables: totals, differences, ratios

Row total

\text{total} = \sum \text{row cells}

Row average

\bar{x} = \frac{\text{row total}}{\text{number of columns}}

Cell ratio

\text{ratio} = a : b \text{ (reduced)}

Rate from counts

\text{rate} = \frac{\text{passed}}{\text{appeared}} \times 100\%

⚡ Sweep the row with tick marks. For count-type questions (how many years above a value), tick the qualifying cells while reading once. No rewriting, no calculator.

⚡ Column-first for year questions. Any question about one year (best year, total of a year) lives in a single column. Add that column alone; ignore the rest of the table.

Percentage change and comparisons

Percentage change

\frac{\text{new}-\text{old}}{\text{old}} \times 100\%

Base = the old / original value.

\frac{\text{part}}{\text{whole}} \times 100\%

Points vs percent

\text{points} = a - b; \quad \%\text{age} = \frac{a-b}{b} \times 100

⚡ The 1% probe. Find 1% of the base first; then any percentage is that times n. 1% of 250 is 2.5, so 70 units is 28%.

⚡ Anchor on the double. A value that doubles is +100%; halves is −50%. Spot these before computing: they frame every other answer.

Pie charts: degrees, shares and totals

Slice value

\text{value} = \frac{\theta}{360} \times \text{total}

theta is the slice angle in degrees.

Angle to percent

\% = \frac{\theta}{3.6}

Percent to angle

\theta = \% \times 3.6

Part to angle

\theta = \frac{\text{part}}{\text{total}} \times 360

Per-degree value

1° = \frac{\text{total}}{360}

⚡ Divide by 36 for percent. Angle over 3.6 gives percent; angle over 36 gives percent over 10. 108° -> 30%, 54° -> 15%, 90° -> 25%.

⚡ Per-degree shortcut. Given one slice's value, find the value of 1 degree once, then price every other slice (and the whole pie) with one multiplication each.

Averages from data

Simple average

\bar{x} = \frac{\sum x}{n}

Weighted average

\bar{x} = \frac{\sum n_i x_i}{\sum n_i}

n_i is the count in each band or row.

Entry to hit a target average

x = (n+1)\bar{x}_{new} - n\bar{x}_{old}

⚡ Deviations from a round base. Table values near a round number: average = base + (sum of differences ÷ count). 480, 520, 460, 540, 500, 500 around 500: deviations sum to 0, average exactly 500.

⚡ Balance around the new average. If an added value lowers the average, each old entry gains (old avg − new avg); the new entry supplies all of it.

Growth rates and successive changes

Growth multiplier

\text{new} = \text{old}\left(1+\frac{r}{100}\right)

Successive growth

r_{total} = \left(1+\frac{a}{100}\right)\left(1+\frac{b}{100}\right)-1

Reverse to original

\text{old} = \frac{\text{new}}{1+\frac{r}{100}}

Total multiplier

\frac{\text{end}}{\text{start}} = 1 + \frac{r_{total}}{100}

⚡ Multipliers beat repeated percenting. Chain growth as multipliers: +10% then +10% is 1.1 x 1.1 = 1.21. One multiplication, no compounding errors.

⚡ End over start. The total growth multiplier over any span is just the last value divided by the first. 144/80 = 1.8 → +80%.

General Intelligence & Reasoning

Analogy

The idea of analogy

⚡ Say the link first, read the options second. Turn the full pair into a sentence before you look at the options. Then test each option with that same sentence. Options that are only from the same field fail the sentence at once.

Letter / letter-cluster analogy

Shift rule

\text{new position} = \text{old position} + k

k is the shift. Above 26, subtract 26; below 1, add 26.

Opposite letter

\text{opposite} = 27 - p

p is the position. A and Z, or M and N, always add to 27.

Rising shift

\text{shift of slot } i = i

Slot 1 moves +1, slot 2 moves +2 and so on, e.g. ABCD → BDFH.

⚡ Subtract once, then copy the pattern. Write one row of subtractions for the model pair. The pattern you get is the rule. Apply the same pattern to the third group and read the letters with EJOTY.

⚡ Spot opposite letters by the 27 total. If matching letters of the model pair add up to 27, the rule is opposite letters. Then write the opposite of each letter. No counting is needed.

⚡ Read the second group backwards. Before any counting, read the second group from right to left. If it gives the first group, the rule is plain reversal.

Word analogy

⚡ Two ticks on every option. Give each option a tick for "same kind of link" and a tick for "same exact link, same direction". The option with two ticks is the answer.

⚡ Tag animal pairs before reading options. For an animal pair, first tag it: adult : young, male : female, or animal : home, sound or group. Options often mix a young one with a female.

Number analogy

Multiply-and-add rule

b = k a + r

a is the first number, b the second. Find k and r from the model pair, then check.

Square family

b = a^2 \pm r,\ a(a+1),\ (a+1)^2

Try these when b is close to the square of a.

Digit rules

b = \text{digit sum},\ \text{digit product},\ (\text{digit sum})^2

Not allowed when the whole-number note is given.

⚡ The size test picks the family. Compare b with a before anything else. A few times a: multiply and add. Near a squared: square family. Very large: cubes. Smaller: roots or digit rules.

⚡ Check the rule on the model pair. Whatever rule you guess, make sure it gives b from a exactly. The check takes two seconds and prevents most wrong answers.

Number sets (triads) analogy

Chain rule

b = ka + r,\quad c = kb + r

a, b, c are the three numbers of a set. One step is used twice.

Third from first two

c = ab,\ a^2 + b^2,\ k(a+b),\ k(a-b)

Test on every model set.

Power set

(a,\ a^2 \pm r,\ a^3 \pm r)

Near-squares and near-cubes of the first number.

⚡ Compute only the last number. Once you have the rule, work out what the last number of each option should be. Traps nearly always change only that number, so one pass finds the answer.

Mixed clusters and same-relation selection

Word to number

\text{code} = k \times (\text{sum of positions}) + r

Often k = 1 and r = 0: the plain sum of letter positions.

Reverse position

\text{reverse value} = 27 - p

A = 26, B = 25 ... Z = 1.

Reverse sum shortcut

\text{reverse sum} = 27n - \text{plain sum}

n is the number of letters in the word.

⚡ Two columns: letters and numbers. Make one column for the letters and one for the numbers. Solve each column on its own. An error then stays inside one column.

⚡ One sentence for all four pairs. In same-relation questions, make the link sentence once from the model pair. Test every option against that same sentence, in the same order.

Classification (Odd One Out)

The idea of odd one out

⚡ Write the majority rule, not the exception. Instead of hunting the odd item, write the rule the other three follow. Confirm each of the three obeys it and the last one breaks it. This stops you marking an item that is merely unusual.

Word odd one out

⚡ Category first, then a binary property. First find the broad category shared by three words. If the fourth also fits that category, switch to a binary property: salt or fresh, input or output, element or alloy, male or female. One word will stand alone on the …

Number odd one out

Divisibility by 11

|S_{odd} - S_{even}| \equiv 0 \pmod{11}

S = sums of alternate digits from the left; e.g. 2728: (2+2) − (7+8) = −11, divisible.

Perfect square endings

n^2 \in \{0,1,4,5,6,9\}

A square never ends in 2, 3, 7 or 8 — instant elimination.

Digit sum rule for 9

9 \mid n \iff S(n) \equiv 0 \pmod 9

S(n) is the sum of the digits of n.

⚡ Last-digit scan for squares. Squares end only in 0, 1, 4, 5, 6 or 9. An option ending in 2, 3, 7 or 8 cannot be a square. For the rest, place the number between two known squares.

⚡ Prime check by small divisors. To test a number under 200 for prime, divide only by 2, 3, 5, 7, 11 and 13. Memorise the fake-prime list: 51, 57, 87, 91, 119, 133, 143, 161.

Letter-cluster odd one out

Step of a cluster

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Q is the alphabet position. Compute steps for every option; three must match as tuples.

Opposite-pair sum

Q(X) + Q(Y) = 27

Letter pairs whose positions add to 27: AZ, BY, ..., MN.

⚡ Step tuples in one line. Under each cluster write its steps: ACE → (+2, +2), BDF → (+2, +2), GIK → (+2, +2), MOP → (+2, +1). The mismatched tuple is the answer. No vowel counting is needed.

⚡ Vowel count as tiebreaker. If the step tuples agree, or the clusters look irregular, count the vowels (A, E, I, O, U). Three clusters with no vowel and one with a vowel (or the reverse) is a complete exam pattern.

Pair odd one out

⚡ Relation sentence, then four ticks. Form the relation sentence from the three pairs that clearly agree, then tick or cross all four options. In number pairs, compute the rule for each pair; three will match.

Number pairs and sets odd one out

Pair rule

b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}

Find f from one pair, then check the other three pairs.

Chain triad

(a,\ f(a),\ f(f(a)))

The same operation applied twice, e.g. ×3 gives (3, 9, 27).

Outer-to-middle triad

(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2

The middle number is made from the two outer numbers.

First-two-to-third triad

(a,\ b,\ h(a, b)),\; h = k(a+b) \text{ or } ab \pm r

The third number is made from the first two.

⚡ Size tells the operation. Compare sizes first. If the second number is near the square of the first, think square. If the middle number is large and the outer ones small, think product of the outer numbers.

⚡ Test the rule on two options before trusting it. A rule found from one option can be a coincidence. Confirm it on a second option, then check the rest. The option that fails is the answer.

Series (Number & Letter)

How every series is cracked

Gap between terms

d_n = a_{n+1} - a_n

Write these gaps under the series. If they are not clear, take gaps of the gaps.

Same number added

a_n = a_1 + (n - 1)d

d is the equal gap. Use it when the first row of gaps is constant.

⚡ Two rows of gaps in ten seconds. Write the gaps under the series. If they are not equal, write the gaps of the gaps. Stop when a row is clear. Then work back up.

⚡ Say the rule aloud in words. Before you calculate, say the rule in one plain sentence, like 'add 3, then add 6, then add 9'. If you cannot say it in a sentence, the rule is wrong.

Number series

Same gap (AP)

a_n = a_1 + (n - 1)d

d is the equal gap from the ladder.

Multiply then adjust

a_{n+1} = k \cdot a_n \pm r

Try k = 2 and 3 first, with r = 1 or 2.

Squares family

a_n = n^2 \pm k \ \text{ or } \ n^2 + n

If the second gaps are all 2, the series is built on n squared.

Same ratio (GP)

a_n = a_1 \cdot r^{\,n-1}

r is the fixed ratio. Test ×2, ×3 and ×1.5.

⚡ Ratio test before anything fancy. Divide each term by the one before it. If the result is close to 2 or 3, try ×k ± r. Find r from what is left over.

⚡ Split a jumpy series. If the terms go up and down, write the odd places on one line and the even places on another. Solve each line on its own. Then find which line the asked place belongs to.

⚡ Cubes hiding in the gaps. If the gaps grow very fast, check for cubes (1, 8, 27, 64, 125) or powers of 2 (1, 2, 4, 8, 16).

Letter series

Opposite letter

\text{place of opposite} = 27 - \text{place}

A and Z, B and Y, and so on. Each pair adds up to 27.

Wrap around

\text{place} > 26 \Rightarrow \text{place} - 26,\ \ \text{place} < 1 \Rightarrow \text{place} + 26

Use this whenever the rule takes you past Z or before A.

⚡ A step row under the letters. Write the place under every letter, then the gap between places. Work on the numbers only. Convert to a letter once, at the end.

⚡ Wrap around with the number 26. Never count letters on your fingers past Z. Add or subtract 26 on the number, then convert.

Alphanumeric series

⚡ Two columns, two answers, one join. Write letters in one column and numbers in the other. Solve each column on its own. Join only at the end, in the format of the question.

⚡ Test the letter-place link first. Check whether the number equals the place of the letter, or its square. If yes, you only need the next letter.

Wrong number in the series

⚡ Ladder with one broken rung. Write the gaps. Find two bad gaps side by side. The term between them is the wrong term.

Repeating letter series — fill the blanks

⚡ Write the blocks one below another. Cut the line into equal rows. Each column must hold one letter. Blanks take the letter of their column.

⚡ A double letter marks a mirror turn. If the line has a doubled letter like rr or nn, the block probably reads forward and then backward. The block is twice the length of the first half.

Coding - Decoding

The coding-decoding family

⚡ Find the difference list first. Write the sample word above its code. Subtract the positions slot by slot. The list of differences is the rule. Apply the same list to the new word.

Letter shift coding

Shift list

\text{code}_i = \text{letter}_i + d_i \pmod{26}

d is the shift for place i, found from the sample. Above 26, subtract 26.

Decoding

\text{letter}_i = \text{code}_i - d_i \pmod{26}

To decode, subtract the same shifts. At 0 or below, add 26.

⚡ Hop from an anchor, do not count. To shift a letter, jump from the nearest EJOTY anchor. W is Y − 2, so W + 4 = Y + 2 = A. Two small hops are faster than counting W, X, Y, Z, A.

⚡ Decoding: run the shifts backwards. When the code is given and the word is asked, subtract the shift instead of adding it.

Reversal and reverse-plus-shift coding

⚡ Read the code backwards first. Read the code from right to left. If you see the word, just reverse new words. If every letter is off by the same k, reverse and shift by k.

Number coding

Position sum

\text{code} = \sum p_i

p = position of a letter, A = 1 … Z = 26.

Sum with an extra step

\text{code} = k \cdot \sum p_i + r

Find k (multiply) or r (add) from the sample. Check with a second sample if one is given.

Weighted sum

\text{code} = \sum i \cdot p_i

i = place of the letter in the word (1st, 2nd, 3rd …).

Reverse positions

p^{\text{rev}} = 27 - p

Z counts as 1 and A as 26.

⚡ Plain sum first. Work out the position sum of the sample. If it equals the code, you are done. If not, compare: code − sum, then code ÷ sum. Only then try weighted or side-by-side codes.

⚡ Anchors make sums quick. Letters near E, J, O, T and Y are small hops from 5, 10, 15, 20 and 25. Group them to add fast.

Symbol and digit coding

⚡ Common letters match common symbols. When two coded words share letters, circle the shared letters and the shared symbols. They belong together. The letters left over take the symbols left over.

⚡ Conditions before the table. Read the conditions and test them on the word before you touch the table. The table gives the symbols; the condition only changes their order.

Substitution and sentence coding

⚡ Line up the sentences. Write each sentence above its code. Tick the words that repeat and the code words that repeat. The word you need is either matched directly or is the one item left over.

Language (sentence) coding with common words

⚡ Intersect, then subtract. For a word, take the code words shared by all sentences that contain it. Remove codes already fixed for other words. One left is the answer; two left means cannot be determined.

Mathematical Operations

Operator puzzles and BODMAS

BODMAS order

B \to O \to D/M \to A/S

D and M share a rank and go left to right; so do A and S.

Missing number

? = \dfrac{\text{RHS} - \text{loose terms}}{\text{factor}}

Undo + and − first, then the multiply or divide around the ?.

⚡ Circle the × and ÷ first. Read the line once and circle every × and ÷. Work those pieces first. Then read the line again for + and −. Your eye lands on the high-rank work before any arithmetic.

Sign substitution

⚡ Compute the trap value too. After the correct value, evaluate the original expression once. If that number sits among the options, the examiner planted it. Seeing it confirms your substitution is the different one.

Interchanging signs and numbers

⚡ One rewrite for every swap. Write the swaps on the left, like + ↔ × and 2 ↔ 8. Then rewrite the expression once with all swaps applied. Evaluate. Piececemeal swapping invites double swaps.

⚡ Fixed-order sweep for fix-the-equation. Sweep number pairs left to right: (n1,n2), (n1,n3), (n1,n4), (n2,n3), (n2,n4), (n3,n4). One written value per row. The row matching the right side is the answer.

Balancing and operator-filling

⚡ Value column for pick-the-equation. List the four options in a column. Evaluate each left side in one written line. Exactly one row matches its right side. The column doubles as your final check.

Hidden-rule and defined operations

Sum of squares

a \# b = a^2 + b^2

A very common hidden rule: 5 # 3 = 34.

Difference of squares

a @ b = a^2 - b^2

Also equals (a+b)(a−b): 8 @ 2 = 60.

Product plus sum

a \diamond b = ab + a + b

Multiply, then add both numbers: 4 ◇ 3 = 19.

⚡ Rule ladder for hidden operations. Test in this order: a + b, a × b, a² + b², a² − b², (a+b)², (a−b)², ab + a + b. Stop at the first rule that fits ALL examples.

Missing Number

How missing-number puzzles work

Rule check

f(a_1, b_1) = c_1 \text{ and } f(a_2, b_2) = c_2 \Rightarrow \text{use } f

One row fitting proves nothing. Two rows fitting is strong proof.

Common families

a + b,\ a - b,\ ab,\ ab \pm k,\ (a + b)k,\ a^2 \pm b

Sums and products cover most puzzles. Try them first.

⚡ Test the spare row. Write your guessed rule as a formula. Test it on the second complete row. Only then use it on the row with the ?. This takes ten seconds and stops the most common wrong answer.

⚡ Build the biggest number first. Pick the biggest number in the row. Try to make it from the smaller ones. This finds the rule faster than reading left to right.

Grids with row (or column) rules

Sum times a number

c = (a + b) \times k

Rows (4, 7, 22) and (6, 3, 18) give k = 2.

Square minus the second

c = a^2 - b

Rows (8, 15, 49) and (7, 10, 39) fit.

Sum of two squares

c = a^2 + b^2

Rows (3, 4, 25) and (6, 8, 100) fit.

Average of two

c = \dfrac{a + b}{2}

Rows (12, 8, 10) and (20, 6, 13) fit.

⚡ Build the biggest cell first. Write the biggest cell of a row. Rebuild it from the other two. This shows the rule faster than reading left to right.

⚡ Copy the grid as three rows. Write the grid as three lines in your rough space, with the ? in place. Reading numbers off the figure while you calculate leads to mistakes.

Digit-sum and digit-reversal rules

Digit sum

S(47) = 4 + 7 = 11

Add the digits of the number.

Reversal

R(43) = 34

Write the digits in the opposite order.

Digit product

P(47) = 4 \times 7 = 28

Multiply the digits of the number.

⚡ Switch to digits when standard rules fail. Give ordinary rules about 30 seconds. If both complete rows refuse them, test these in order: digit sums, reverse of the sum, digit sums plus a number.

⚡ Add first, then flip. A reversal answer must read backwards cleanly. Add the two numbers first, then reverse the sum. Confirm on two rows.

Blood Relations

How to solve blood-relation questions

Mirror pairs

\text{father} \leftrightarrow \text{son},\ \text{uncle} \leftrightarrow \text{nephew}

Every relation has a mirror. If A is B's uncle, B is A's nephew or niece.

Marriage words

\text{child's spouse} \to \text{son/daughter-in-law}

Spouse's parent = father/mother-in-law. Sibling's spouse or spouse's sibling = brother/sister-in-law.

Cousin

\text{cousin} = \text{child of a parent's sibling}

An uncle's or aunt's child is always a cousin, never a nephew.

⚡ Write each sentence as an arrow. In the margin, turn every sentence into an arrow: 'Suresh's daughter is Pooja' becomes Suresh to Pooja (F). When every sentence is an arrow, the tree draws itself.

⚡ Flip the word, not the tree. One tree answers both directions. Keep the drawing, swap to the mirror word: father goes to son, uncle to nephew, grandfather to grandson.

Multi-statement tree puzzles

⚡ Start from the root. The root is the person nobody calls a child. Build downward from there. Every statement then clicks into an empty slot.

⚡ Count arrows, then gender. The arrow count picks the family of the word (father, grandfather, great-grandfather). Only then does gender pick inside it. Never guess the word first and draw later.

Pointing and introducing

⚡ One layer per line. Underline the innermost 'my' and work outward. Each line of rough work resolves one 'of'. Then place the speaker and the target in a two-person sketch.

⚡ Scan for the fold-back. Before peeling anything, scan the phrase for 'my mother's only child', 'my father's only son', 'my mother's only daughter'. If one is there, test the speaker's gender against it first.

Coded (symbol) relations

⚡ Write the sentences under the string. For each symbol, write its one-line sentence under the string, then draw. Six symbols become three short sentences, and the drawing finishes the job.

⚡ Audit genders before answering. Before marking, list every letter with the gender its symbols fixed. If the asked word needs a gender that stayed blank, the answer is the neutral word or 'Cannot be determined'.

Direction & Distance

Direction conventions and displacement

Shortest distance

d = \sqrt{x^2 + y^2}

x = net East-West move, y = net North-South move, after cancelling.

Number triples

3\text{-}4\text{-}5,\ 5\text{-}12\text{-}13,\ 8\text{-}15\text{-}17,\ 7\text{-}24\text{-}25

Doubles and triples of these also work: 6-8-10, 9-12-15.

Diagonal leg

k\sqrt{2}\ \text{along NE} = k\ \text{North} + k\ \text{East}

The same rule holds for NW, SE and SW.

⚡ Two-line tally. Keep one running total for East-West and one for North-South. Write West and South as minus. The two totals are the whole answer, with no drawing needed.

⚡ Spot the triple. If the two net moves are 3 and 4, or 5 and 12, or 8 and 15, or 7 and 24 (or their multiples), the answer is the third number. No square root is needed.

Direction faced and net direction

Clockwise cycle

N \to E \to S \to W \to N

Every right turn moves one step along this circle.

Anticlockwise cycle

N \to W \to S \to E \to N

Every left turn moves one step along this circle.

Reading the end point

(x>0,\ y>0) \Rightarrow \text{North-East}

x is the net East move and y is the net North move. Their signs give the compass word.

⚡ Write the direction after every turn. After each turn, write one letter: N, E, S or W. Overwrite the old letter. Three written letters are safer than one long chain in your head.

⚡ Reverse the pair, flip the word. If B is North-East of A, then A is South-West of B. The mixed words flip together.

Shadow questions

⚡ Change the shadow into a facing first. Before any other step, rewrite the sentence as a facing sentence. Morning and shadow behind him means he faces East. Evening and shadow in front means he faces East.

Position of one point from another

Distance between two points

d = \sqrt{(\Delta E)^2 + (\Delta N)^2}

Delta E is the East gap and Delta N is the North gap between the two people.

Distance from speed and time

\text{distance} = \text{speed} \times \text{time}

Use it first when two people walk at different speeds.

⚡ Start from the person named after 'from'. For "direction of C from B", draw B first at the middle. The chain then hangs off B. The answer is simply where C lands.

Order & Ranking

How ranking questions work

Rank from the other end

p' = n + 1 - p

n = total people, p = rank from one end.

Total from two ranks

n = a + b - 1

The same person is a-th from one end and b-th from the other end.

Between count (same end)

\text{between} = |a - b| - 1

Both ranks are from the same end.

Ahead and behind

\text{ahead} = r - 1,\quad \text{behind} = n - r

r = the person's rank from the front.

⚡ Check that the two ranks add to total plus 1. For one person, the two ranks from opposite ends always add up to the total plus 1. If they do not, you have misread a number.

⚡ Draw the end zones. Draw a short row of boxes. 'a-th from the left' means a - 1 people stand before the person. Counting people at each end is often faster than any formula.

Ranks and positions

Rank from the other end

p' = n + 1 - p

n = class size or row size.

Total from two ranks

n = a + b - 1

The same person, ranked from both ends.

People joining or leaving ahead

r_{\text{new}} = r_{\text{old}} \pm k

Plus for k joiners ahead. Minus for k leavers who were ahead.

⚡ Check that the sum of the two ranks is total plus 1. For one person, the two ranks add up to the total plus 1. Test it before you mark the answer.

People between two positions

Between, same end

\text{between} = |a - b| - 1

Both ranks are counted from the same end.

Between, opposite ends

\text{between} = n - a - b

n = total people, a and b = ranks from opposite ends.

Middle person

m = \dfrac{a + b}{2}

a and b from the same end. The result must be a whole number.

Rank from the gap

b = a + k + 1

k people are between a and b, counted from the same end.

⚡ Convert first, then count. Never subtract ranks from different ends. Convert one rank with n + 1 - p. Then use the same-end formula.

Ordering and comparison chains

⚡ Write two symbols for each clue. Turn each sentence into two letters with a sign, such as S < V. Keep the sign pointing the same way all through. The reversed words then become easy to see.

⚡ Start from the person named most often. The person who appears in the most clues is the best starting point. Build the chain outwards from that person and it forms in one pass.

Seating Arrangement

How seating puzzles are set

Circular neighbour rule

L(i) = (i+1) \bmod n,\quad R(i) = (i-1) \bmod n

Seats numbered clockwise, everyone facing the centre. Both swap if facing outward.

Opposite seat

i \leftrightarrow (i + n/2) \bmod n

Only when n is even. An odd table has no opposite seat.

People between two seats

\text{between} = |a - b| - 1

a and b are seat numbers in the same row.

Viewer versus person

\text{person faces south} \Rightarrow \text{their left} = \text{your right}

Applies to the person's own left and right, never to the row ends.

⚡ Pin the absolutes first. Ends, the middle and opposite seats fix a person outright. Fill them before you touch any chain. The remaining people usually fall into one or two slots.

⚡ Test one clue both ways. If you are unsure of the facing, place one clue for north and for south. The two seats differ, and this shows you which one you must use.

Single-row arrangements

⚡ Chain links, do not guess gaps. 'Second to the right of A' pins B exactly two seats from A. Mark it with an arrow as soon as you read it. Most puzzles have four such arrows, and the picture then completes itself.

⚡ Turn the row to face up. If the row faces south, redraw it from east to west. Now every person 'faces up' and left is your left. Use this drawing for the whole puzzle.

Round-table arrangements

⚡ Walk the circle in one direction. Take the longest chain of 'immediate left' clues and lay it out clockwise. Close the loop with the last clue.

⚡ Convert left to right around a circle. In a circle of n, the k-th to the left is the same person as the (n - k)-th to the right. Use the smaller number to save steps.

Exam strategy for puzzles

⚡ Two-seating bail-out. When exactly two seatings survive, check what the question asks. If both give the same answer, mark it and move on.

⚡ Verify with one closed loop. In a circle, walk the left-neighbour links around your final drawing. If the string does not close, one arrow was drawn backwards.

Syllogism

What a syllogism is and how to test it

Transitivity of All

A \subset B \wedge B \subset C \Rightarrow A \subset C

'All A are B + All B are C → All A are C' is the only freely chained rule.

Conversion of All

\text{All } A \subset B \Rightarrow \text{Some } B \subset A

Valid (classes are assumed non-empty): 'Some B are A' follows.

Conversion of No / Some

\text{No } A \cap B \Leftrightarrow \text{No } B \cap A;\quad \text{Some } A \cap B \Leftrightarrow \text{Some } B \cap A

Both convert symmetrically.

⚡ Find the shared term. The conclusion must join the two outer terms. Find the term that appears in both statements, and ask what travels through it.

⚡ One counter-picture kills it. You do not need to prove a conclusion. You need only one picture that fits the statements and breaks it.

Possibility conclusions

⚡ Defence lawyer thinking. For a possibility, flip your mindset. Build one picture where it holds. Done.

Either-or (complementary pairs)

⚡ Spot the opposite pair. Look for (All, Some-not) or (Some, No) on the same two terms. Then test each alone.

Only-a-few and definite-case rulings

⚡ Split on sight. Rewrite every 'Only a few A are B' as two lines: Some A are B, and Some A are not B. Then solve as normal.

Venn Diagrams

What a Venn diagram encodes

Two-set inclusion-exclusion

|A \cup B| = |A| + |B| - |A \cap B|

The intersection is counted twice on the right, so subtract once.

Only-regions

|A \setminus B| = |A| - |A \cap B|

'Only A' is A minus the overlap.

⚡ Label regions, don’t imagine them. Write the numbers INTO the regions: |A∩B| in the lens, only-A and only-B in the moons, neither outside. Sums become single looks instead of formula recalls.

Two-circle counting

Two-set inclusion-exclusion

|A \cup B| = |A| + |B| - |A \cap B|

The intersection is counted twice on the right, so subtract once.

Only-regions

|A \setminus B| = |A| - |A \cap B|

'Only A' is A minus the overlap.

Neither

\text{neither} = \text{total} - |A \cup B|

Everyone outside both circles.

⚡ Label regions, don’t imagine them. Write the numbers INTO the regions: the lens, the two moons, and outside. Sums become single looks instead of formula recalls.

Three-circle counting

Three-set union

|A \cup B \cup C| = \sum|A| - \sum|A \cap B| + |A \cap B \cap C|

Add singles, subtract pairs, re-add the triple.

Exactly two sets

\sum (|A \cap B|) - 3|A \cap B \cap C|

Sum of the three pairwise figures minus three times the triple.

Exactly one set

|A \cup B \cup C| - \text{exactly two} - \text{all three}

Strip the multi-set members from the union.

⚡ Triple-strip subtraction. Whenever a question says 'exactly two', your reflex is: pairwise sums overcount the core by 3. Compute pairwise sums, subtract 3×(all three), done.

Choosing the correct diagram

⚡ Test one pair at a time. Never judge the whole picture at once. Run necessity/impossibility on each pair; the correct option is the only one agreeing on every pair.

Dictionary Order & Alphabet

Dictionary order rules

⚡ Find the first split point. Write the words one under another and scan the columns from the left. The first column with different letters decides the whole comparison. Ignore every letter after it.

Arranging words and picking slots

⚡ Sort once, count slots. Fix one numbered order for the whole question, then answer any slot lookup from it. Re-sorting for each option is where errors and lost time come from.

Letter positions and shifts

Position from the right

p_{\text{right}} = 27 - p_{\text{left}}

The two positions of the same letter add to 27.

Shifted position

p' = p \pm k

Move right (+) or left (−) by k letters, staying inside 1 to 26.

⚡ EJOTY plus a tiny shift. Anchor at the nearest of E, J, O, T, Y and step from there. Position 18 is T (20) minus 2, which is R. Two seconds, no counting.

⚡ The 27 mirror. Any from-the-right count becomes from-the-left by 27 − k. Convert first, then do everything else from the left.

Dictionary rank and word surgery

Rank of a word

\text{rank} = 1 + \sum_i c_i \times (r_i)!

c_i = unused letters smaller than the letter at position i; r_i = letters remaining after position i.

⚡ Count smaller unused letters. Freeze the sorted letter list. For each letter of the word, left to right, count how many still-unused letters are smaller. Multiply by the factorial of what remains, sum, add one.

⚡ Three letters: write all six. For a three-letter word the whole list has just 6 entries. Writing them beats any formula and never miscounts.

Statement & Conclusion

Statement and conclusions

⚡ The extreme-word filter. Circle words like only, all, always, never, surely, best, entire and must. If the conclusion has one and the statement does not, it almost always fails.

⚡ Needed is not the same as enough. 'Only X can do Y' means X is required. It does not mean X is enough. Check which way the conclusion runs.

Statement and assumptions

⚡ The negation test. Put 'not' into the assumption. Does the speaker still have a reason to speak? If not, the assumption is implicit.

Courses of action

⚡ The proportion check. Ask: would a sensible official do this tomorrow? Inspecting, repairing, warning, supplying and treating usually pass. Banning, shutting forever and blaming everyone usually fail.

Counting Figures

Counting triangles

Apex lines

T = \frac{(k+1)(k+2)}{2}

k extra lines drawn from the apex to the base.

Apex lines + horizontal cuts

T = \frac{(k+1)(k+2)}{2}\times(h+1)

h lines parallel to the base, each crossing every apex line.

⚡ Square with diagonals: 8 or 16. A rectangle with both diagonals always has 8 triangles. Add both midlines and it becomes 16. Memorise both.

⚡ Count the lines, then choose two. In an apex figure, count the lines through the apex (extra lines plus both sides) and choose any two of them.

Counting squares and rectangles

Rectangles in an m × n grid

R = \binom{m+1}{2}\binom{n+1}{2}

Pick 2 of the m + 1 vertical lines and 2 of the n + 1 horizontal lines.

Squares in an m × n grid

S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)

An n × n grid gives 1 squared + 2 squared + ... + n squared.

⚡ Line-pair trick. A rectangle is decided by its two vertical sides and two horizontal sides. Count line pairs, never shapes.

⚡ Memorise the small grids. 2 × 2: 5 squares, 9 rectangles. 3 × 3: 14 squares, 36 rectangles. 4 × 4: 30 squares, 100 rectangles. They appear inside bigger questions.

Counting straight lines

Lines in a grid

L=(m+1)+(n+1)

m columns, n rows; add extra slanting lines separately.

⚡ Direction sweep. One pass for horizontals, one for verticals, one per slant direction. The running total is the answer.

⚡ Merge before counting. When cell diagonals sit end to end on one path, they are one line. Join them in your head first.

Mirror & Water Images

Mirror images (vertical mirror)

⚡ Check the ends first. The image starts with the last character of the word, flipped. Reject options by looking at the first character only.

⚡ Symmetric-letter shortcut. A word made only of A H I M O T U V W X Y that reads the same backwards looks unchanged in a vertical mirror.

Water images (horizontal mirror)

⚡ Order stays, letters go upside down. In the options, look for the one that keeps the original order and has only the unsymmetric letters turned upside down.

⚡ Name the image from two clues. Same order and upside down: water. Reversed order and flipped left to right: mirror. Reversed and upside down: half turn.

Mirror image of a clock

Mirror time

T_{\text{image}} = 11{:}60 - T_{\text{actual}}

Works in both directions. For 12:xx, use 23:60 minus the time.

⚡ 11:60 rule. Subtract the hours from 11 and the minutes from 60.

⚡ Check by adding. Add the given time and your answer. The sum must be 12:00.

Paper Folding & Cutting

Folding and punching: unfolding the pattern

⚡ Reflect, do not rotate. Each unfold is a mirror. The copy is at the same distance from the fold line, on the other side. A half turn of the flap is wrong.

⚡ Count first, then check positions. Holes = punches times layers under each punch. Use the count to remove wrong options, then check positions.

Counting layers and holes

Holes after full half-folds

H = p \times 2^{n}

p punches through n full half-folds.

Layers after full half-folds

L = 2^{n}

Each full fold doubles the layers.

⚡ Doubling chain. Write 1, 2, 4, 8 as you read each full fold. Multiply by the number of punches at the end.

⚡ Multiply the steps. Each step multiplies the layers: a half fold by 2, a three-way fold by 3.

Embedded Figures

Finding the hidden part in a figure

⚡ Reject by a missing direction. Look at which slants the big figure has. An option with a slant the figure lacks cannot be hidden in it.

⚡ Count strokes by type. Count the flat, upright and slanting strokes in the part. The big figure must have at least as many of each type.

⚡ Two finalists: compare at the anchor. If two options look alike, compare only the strokes that touch the anchor. The wrong one usually breaks there.

Which figure contains the given part

⚡ Reject by a missing direction. List the directions in the part: flat, upright, / and \. An option that lacks one of them is out at once.

⚡ Count the strokes first. The option must have at least as many strokes as the part. Fewer strokes means it cannot contain the part.

⚡ Two finalists: compare at the anchor. If two options survive, compare only the strokes that touch the anchor. The wrong one usually breaks a stroke there.

Cube & Dice

Dice: finding opposite faces from positions

Standard die

1+6 = 2+5 = 3+4 = 7

Opposite faces of a standard die add to 7.

Hidden faces of a standard die

\text{hidden} = 21 - \text{shown}

All six faces add to 21.

⚡ Strike out the neighbours. List every face seen with the asked face and cross them out. If one face is left, it is the answer.

⚡ Two common faces. Two pictures with two faces in common: the two other faces are opposite.

⚡ Hidden faces of a standard die. All six faces add to 21. Subtract the three faces you can see.

Cube nets: folding a sheet into a cube

⚡ Skip one. Along a row or column, skip one square. The squares on both sides of the skipped one are opposite.

⚡ Reject first. In a which-cube question, throw out every option that shows an opposite pair. Often only one option is left.

Painted cube cut into small cubes

Two faces painted

12(n-2)

Cubes on the edges, not the corners.

One face painted

6(n-2)^2

Cubes in the middle of each face.

No face painted

(n-2)^3

The hidden inner block.

Total check

8 + 12(n-2) + 6(n-2)^2 + (n-2)^3 = n^3

The four counts add up to all the small cubes.

⚡ Check the total. The four counts must add up to n cubed. Use it to catch mistakes.

⚡ At least one painted. Take all the small cubes and remove the core.

Figure Series

Movement and rotation

Position after k steps

p_{k} = (p_{0} + k\,s) \bmod 8

s = step (positive = clockwise) on the 8 border places.

⚡ One element, one line. Write one short row for each element: the places in every frame, then the step, then the next place. Any option that fails a row is out.

⚡ Arrows as clock hours. Think of the arrow as the hand of a clock. A 90-degree clockwise turn goes 12, 3, 6, 9.

⚡ Cross out by one element. Test the easiest element first, such as the arrow. Cross out the options that fail it before you look at anything else.

Adding, removing and changing elements

⚡ Count first. Count the elements in each frame. Only options with the right count survive. Then check positions or details.

⚡ Differences of differences. If the differences are not equal, write them as a list. A list like 1, 2, 3 grows by 1 each time.

⚡ Cross out by the detail. Test the small detail first, such as a shape or an arrow. It removes options quickly.

English Comprehension

Reading Comprehension

How to attempt an RC passage in SSC exams

⚡ Two-minute structure read. On the first read, do not hunt for facts. After each paragraph say its job in three words. Facts are cheap to re-find; structure is what saves you.

⚡ Vocab questions first, main idea last. Answer the replace-the-word question and line-specific details before the main-idea question — by then you will have re-touched half the passage and the theme is clear.

⚡ Line anchoring. Every detail question has a home line. Match a distinctive word in the question (a name, a number, an odd noun) to the passage, read two lines around it, and choose the option that paraphrases them.

Main idea, central theme and best-title questions

⚡ First and last sentence sketch. Read only the opening and closing sentences of the passage, plus the first sentence of each middle paragraph. That skeleton usually states or implies the theme — build your ten-word summary from it before looking at opti…

⚡ Count the mentions. The concept the author repeats in different words across paragraphs is the theme. If an option's core idea appears in only one paragraph, it is a detail, not the theme.

⚡ Title vs theme wording. Titles are noun phrases ('The quiet comeback of the public library'); theme options are full claims. Do not reject a title for lacking a verb — reject it for wrong scope.

Factual detail questions ('According to the passage…')

⚡ Paraphrase before you look. After locating the line, cover the options, answer aloud in your own words, then uncover. This one habit defeats most engineered wrong options.

⚡ Reversal scan. Before marking a detail option, check its direction words: only/always/all, may/often/some, can/cannot. Wrong options flip these while keeping the same topic words.

⚡ Option pairing. When two options say the same thing in different words, both are wrong — a fact has one meaning. Kill the pair, then judge the remaining two.

Inference and extrapolation questions

⚡ One-step rule. Correct inference = stated idea + one logical step. If you need two steps, extra facts, or a change of scale (one city → all countries), reject it.

⚡ Verbatim is a verdict. In an inference question, an option that repeats passage wording without adding the step is wrong by definition. Cross it out first.

⚡ Extremity filter. Inferences with *only, never, must, inevitably* are almost always one step too far. Moderate options (*may, tends to, is likely*) fit the must-be-true test better.

Vocabulary in context (replace-the-word / meaning questions)

⚡ Own-words first. Cover the options, replace the word with your own simple word, then find the option matching YOUR word. This kills the dictionary-meaning bait.

⚡ Tone match. The substitute must keep the sentence's attitude. In a critical sentence, choose the critical option; a neutral synonym that drains the criticism is wrong.

⚡ Two-line rule for phrases. For phrase-meaning questions, read one line before and after: figurative phrases get their meaning from the surrounding argument, not from the words themselves.

Tone, attitude and author's-opinion questions

⚡ Adjective audit. Sweep the passage for adjectives and adverbs with feeling (quietly, badly, remarkable, merely, confident). Three of them in the same direction fix the tone.

⚡ However counts double. The clause after *but/however/yet* carries the author's real position; the clause before it is the position being set aside. Tone and opinion questions key off the second half.

⚡ Degree matching. Match intensity, not just direction: 'sceptical' ≠ 'hostile', 'cautiously optimistic' ≠ 'confident'. Pick the option at the same temperature as the passage.

Cloze Test

What a cloze test is and how to attack it

⚡ Tense anchor first. Before touching any blank, underline the time expressions: 'Every morning...', 'Last month...', 'by evening'. Verb blanks and even prepositions obey these anchors.

⚡ Predict, then match. Cover the options with your hand, read the sentence, and whisper your own word. Options are written to include your word's exact meaning — anything far from it is usually wrong.

⚡ The read-aloud finish. After answering all five, read the passage silently as if it were written by someone else. Grammar errors and logic breaks become audible — re-check any blank where you stumble.

Grammar inside the passage: tense, agreement, articles

⚡ Strip-to-subject. Delete every phrase between the subject and the verb blank, then apply agreement to the bare skeleton.

⚡ Only-one-past trick. When a narrative blank offers just one past-tense option among present/gerund forms, the anchor already decided it — mark it and move on without re-reading the paragraph.

⚡ Say the article aloud. For a/an blanks, pronounce the *next word's first sound*: 'an hour' but 'a university'. Silent letters and 'yu' sounds decide it, not the first letter.

Vocabulary blanks: meaning, register and collocation

⚡ Name the missing idea. Before looking down, say what the sentence lacks in plain words: 'it needs the idea of CONSISTENCY here'. Then choose the option that says it — this defeats look-alike distractors.

⚡ Collocation pairs to bank. Memorise high-frequency pairs: gain momentum, pay attention, take measures, reach a decision, bear fruit, meet a deadline, break the news, strike a balance. Cloze nouns and verbs revolve around them.

⚡ Polarity check. Mark the sentence + or − before choosing. A praising sentence rejects negative words even when they are grammatically perfect.

Connector blanks: contrast, cause, result, addition

⚡ Opposite-direction test. Give each clause a + or − sign. Same signs → addition/result connector; opposite signs → contrast connector.

⚡ Partner hunt. Scan the sentence for the other half of a correlative pair. Seeing 'not only' fixes the blank to 'but also' — zero thinking needed.

⚡ Comma tell. 'clause, _ clause' with a single-word slot wants *yet/but/so/for/and*. Options like 'however/therefore' fit 'clause. _, clause' patterns instead.

Preposition and phrasal-verb blanks

⚡ Governor chant. Underline the word governing the blank and recite its pair: depend—on, afraid—of, good—at, solution—to. Ten pairs cover most paper prepositions.

⚡ Particle direction sense. Ask what the sentence does to the object: cancels it (*call off*), rejects it (*turn down*), erects it (*set up*). The particle encodes the action's direction.

⚡ Substitute the simple verb. Replace the phrasal option with a one-word verb in your head: look after = tend. The sentence must keep its meaning with the substitution; if not, that phrasal is wrong.

Error Spotting

The master checklist: SVA → tense → article → preposition

⚡ The sweep in one line. Verb first: find the subject and check its number. Then tense words. Then a / an / the by sound. Then the preposition partner. Only then pronouns, pairs and word choice.

⚡ Strip the middle. Delete the phrase between the subject and the verb, such as 'of my friends' or 'as well as the players'. Read the bare skeleton. Agreement errors become loud.

⚡ Earn the No error answer. Choose 'No error' only after the full sweep. About one in five of these items hides a quiet agreement or preposition fault.

Subject–verb agreement errors

⚡ Hunt the subject backwards. Start at the verb and walk backwards past every 'of ...' phrase. The first noun or pronoun with no preposition in front is the subject.

⚡ Nearness rule flash. See neither ... nor or either ... or with two subjects? The verb matches the second one. The item usually solves at once.

⚡ A quantity is singular. Money, distance, time and weight taken as one amount use a singular verb, even in plural form: 'Ten kilometres is a long walk.'

Tense and sequence-of-tenses errors

⚡ Circle every anchor. Circle yesterday, ago, since, for, by the time, when and next week before you judge any verb. Most tense items solve themselves.

⚡ The will check. Scan for will inside a when / if / until clause with future meaning. SSC plants this error again and again.

⚡ Since and for sorting. Since + starting point (since Monday, since 2019, since childhood). For + length (for ten years, for a week). The wrong partner is the answer.

Article and preposition errors

⚡ Say it aloud. Articles are sound decisions. Say the phrase. 'An university' fails the ear because of the 'yu' sound. 'A honest man' fails too.

⚡ Governor chant. Underline the word the preposition depends on. Recite its partner: insist on, angry with (person), prefer to, married to. A wrong partner is the answer.

⚡ The with superlatives. Superlatives and ordinals need the: 'the most intelligent girl', 'the first attempt'. A bare superlative hides an article error.

Pronoun and correlative-conjunction errors

⚡ Preposition plus I? Kill it. Between you and I, for he and I: all wrong. After a preposition, use the object form every time.

⚡ Did plus base form. After did, does or do, the verb keeps its base form: 'No sooner did the bell ring'. A past form after did is always an error.

⚡ Cover and match the pair. Cover everything except the pair words. If one half stands with the wrong partner (hardly ... than), that part is the error.

Comparison, redundancy and confusable-word errors

⚡ Scan for more + er. Look for more before a comparative (more bigger) or most before a superlative. That is an error at once.

⚡ Flinch at echo pairs. In pairs like return back, repeat again and revert back, the second word repeats the first. These are planted answers.

⚡ The that-of test. When two nouns' qualities are compared, the second noun needs that of or those of: 'The roads of Delhi are wider than those of Patna.'

Verb forms, question tags, conditionals and structure

⚡ Flip the tag. Hear the end of the sentence. A positive statement needs a negative tag. A negative word (never, rarely, hardly) needs a positive tag.

⚡ Unreal past conditional. After 'if' about the unreal past, use had + participle. 'Would have' in the if-part is the planted error.

⚡ Copy the first item. In a list, the first item sets the shape (-ing, to + verb or noun). Every other item must copy it.

Sentence Improvement

The elimination ladder: how to choose in 20 seconds

⚡ Find the anchor before you read the options. Look for the time word or helper verb near the bold part. Decide what the verb must look like. Then pick the option that matches your form.

⚡ Tag every option with its flaw. In your head, label each option: 'plural verb', 'wrong partner', 'breaks inversion'. If no option has a flaw, the answer is No substitution required.

⚡ Meaning guard. Two options are both grammatical? Choose the one that keeps the original meaning word for word. An option that changes who did what is wrong.

Tense and sequence fixes

⚡ The yesterday test. See a finished-time word? Cross out every perfect and continuous option at once. Usually one option is left.

⚡ Backshift plus straight order. In a reported question, two things change: the tense moves back and the order becomes a statement. Each wrong option usually breaks one of these. Check both.

⚡ Since and for health check. 'Since 2015' or 'for ten years' with a continuing action needs 'has / have been'. Options without it die at once.

Agreement and verb-form fixes

⚡ Cover the phrase and read the skeleton. Cover 'of the students' type phrases with a finger. Read what is left: 'One _ won'. The right helper is now easy.

⚡ Check the helper chain. did takes the base verb, have takes the participle, be takes -ing or the participle. If the bold part has a helper, check its partner form first.

⚡ Two subjects with and. Two subjects joined by 'and' take a plural verb. The exception is a pair that names one thing, like 'bread and butter is'.

Comparison and structural fixes

⚡ Insert that of or those of. Two 'the X of A' phrases are compared, but the second is bare. Add 'that of' for a singular noun or 'those of' for a plural noun.

⚡ Check for a double marker. Look for more + a word that already has -er, or most + a word that already has -est. If you find one, the repair is a subtraction.

⚡ Count first. Find the count phrase in the sentence. 'Of the two' means the comparative with 'the'. 'Of all' means the superlative.

Precision fixes: articles, prepositions, pronouns, word choice

⚡ Sweep the small words. Read only the small words in the bold part: a, an, the, in, on, at, me, myself. A wrong one usually stands alone. Fix it and finish.

⚡ The myself firewall. A reflexive like 'myself' never replaces a plain object pronoun. Any option that offers 'myself' where 'me' belongs is a decoy.

⚡ Redundancy flinch. Learn the fatal pairs: return back, repeat again, revert back, final conclusion. When one is bold, the shorter option is the key.

Structural fixes: inversion, dangling modifiers, parallelism

⚡ Opener triggers a flip. Negative word first? Expect helper before subject: had I reached, did he arrive. Then check the partner: than or when.

⚡ Who does the -ing?. Name the doer of the opening action. If the main subject is not that doer, pick the option that makes the doer the subject.

⚡ Shape match across pairs. Cover the pair words. Compare the shapes on both sides. If one side is a verb and the other a noun, that is the fault.

Fill in the Blanks

Collocation blanks: word partnerships that decide the answer

⚡ Governor-first reading. Read the sentence up to the blank, stop, and ask: 'What does this word usually take next?' Answer from memory, then look at the options — your answer is usually sitting there verbatim.

⚡ Topic veto. Collocation options in SSC are often same-suffix decoys (-tions, -ments). Kill them by topic first: a border dispute invites *talks/deliberations*, never *medications*.

⚡ Verb-object completeness. For verb blanks, complete the object in your head: 'attention is _' → paid. If your completing verb is an option, mark it and move on.

Phrasal-verb blanks: verb + particle as one unit

⚡ Name the action, then match the unit. Say the sentence's action in one plain word (cancelled, tolerated, abolished), then pick the phrasal unit that translates it. The wrong particles translate other actions.

⚡ Particle direction check. Ask what the sentence does to its object — kills it (off/out), raises it (up), lowers it (down), accepts it (in). Match the particle's direction to the verb's fate.

⚡ Simple-verb substitution. Replace each option with a one-word verb (call off = cancel). The option whose substitution keeps the sentence's meaning is the answer.

Verb-form and tense blanks

⚡ Preposition-to test. Check what owns the 'to'. Owned by a preposition-verb or noun → -ing; owned by an infinitive verb → base form. This one test answers the largest tense-form family.

⚡ By-the-time ladder. Two events with 'by the time' → earlier verb in past perfect, later in simple past. Fit the blank to its rung of the ladder.

⚡ Verb-category lists. Memorise the three mini-lists: -ing verbs (avoid, enjoy, mind, suggest, finish), to-verbs (decide, hope, refuse, plan), base-after (had better, would rather, let, make).

Connective and relative-pronoun blanks

⚡ Partner scan before choosing. Search the sentence for the other half of the pair. Seeing 'Hardly had' fixes the blank to *when*; seeing 'No sooner' fixes it to *than*. Zero deliberation needed.

⚡ He/him substitution for who/whom. Rephrase the clause with he/him: 'everyone trusts (him)' → whom; '(he) inspires everyone' → who. No ambiguity survives the substitution.

⚡ Despite/Although sorting. Full clause after the blank → Although/Though; noun or -ing → Despite/In spite of. The trap options swap these two families.

Confusable and homonym blanks

⚡ Slot first, meaning second. Decide noun/verb/adjective from the sentence frame; kill every option of the wrong class. Then choose by meaning among survivors — usually only two ever reach step two.

⚡ Context picture. Form a mental image of the sentence: a pile of paper in a godown → stationery; a parked van → stationary. The image picks the spelling instantly.

⚡ Triple-family cards. Make flash cards only for families of three or more: assure/ensure/insure, site/sight/cite, affect/effect/affective. Singles rarely appear in options.

Double blanks: two gaps, one sentence

⚡ Structure-pair recognition. too + adjective + to + base verb; so + adjective + that + clause; such + noun + that + clause; hardly...when; no sooner...than. Recognise the skeleton and both blanks fill at once.

⚡ Verb-agreement tiebreak. Plural verb with two subjects → Both...and; singular verb → Either...or / Neither...nor (matching the nearer noun). The verb alone often fixes the pair.

⚡ Fixed binomial bank. Revise binomials in pairs: safe and sound, part and parcel, null and void, first and foremost, ways and means. Options split genuine binomials with a wrong partner — the genuine one is the answer.

Synonyms & Antonyms

The root-word method: decode words you have never seen

⚡ Split-then-eliminate. Chop the unknown word at its joins (bene-vol-ent), translate the parts you know, and eliminate options contradicting them. One confident root kills two options.

⚡ Family clustering. Learn words in root families, not alphabetical order: dict family (dictate, verdict, indomitable? no — edict, indict), loqu family (eloquent, loquacious). One sitting per family.

⚡ Positive/negative split. Roots carry emotional charge: bene-, eu-, am- are warm; mal-, dys-, phob-, mis- are cold. When stuck, mark the word's charge and choose an option of matching charge.

Choosing the nearest meaning: elimination by class, degree, charge

⚡ Overlap test. The right synonym shares the core idea, not the neighbourhood. Ask 'can one replace the other in most sentences?' — if the sentence changes meaning, it is an associate, not a synonym.

⚡ Charge matching. Label the given word + or − (notorious −, celebrated +) and pick the option of the same sign. Half of synonym options fall to charge alone.

⚡ Degree ruler. Place the word and both surviving options on an intensity line (mild → extreme). The option at the same mark is the answer.

Antonyms: prefix flips and real opposites

⚡ Mini-sentence contradiction. Put the word and each candidate in the same short sentence ('The manager praised / criticised the report'). The option producing a genuine contradiction is the antonym.

⚡ Prefix flip check. Before scanning options, guess the flip yourself (conformity → nonconformity). If your guess is among the options, mark it — the options are built around the same flip.

⚡ Absence is not opposition. Kill options that merely lack the quality ('careful' for 'brave'). An antonym fights the word; it does not stand beside it.

Exam strategy for vocabulary questions

⚡ Two-pass vocabulary. Answer every certain vocabulary item in the first pass (15 s each); in the second pass, return to unknowns and root-split them with the remaining time.

⚡ Charge guess with options. Options reveal the axis: if options are (kind, cruel, wealthy, talkative), the axis is temperament — now guess the word's charge from its sound (mal-, dys- negative; bene-, eu- positive) and pick accordingly.

⚡ The 2-4 year echo. Words repeat across years. Solving the last 5 years' synonym/antonym questions is the single highest-yield vocabulary exercise — the bank above already encodes that history.

One Word Substitution

Suffix families: decode the word by its ending

⚡ Family filter. Classify the stem first: is it a killing word, a fear, a study, a ruler-system, a person-type, a place? Reject every option from a different family before reading further.

⚡ Root + suffix stack. Split rare answers: nycto-phobia, entomo-logy, carto-grapher. Two known parts beat one memorised word.

⚡ Counting-prefix drill. One evening per prefix set: mono-, bi-, tri-, poly-, pan-, omni-. Write each with three familiar words and the family stays for life.

One word for people: traits, trades and types

⚡ Object-anchor the trade. Never learn 'cobbler' alone — learn 'cobbler → shoes'. Stems name the object ('one who repairs shoes'), and your anchor matches it instantly.

⚡ Contrast pairs together. Store opposites and neighbours in one card: garrulous ↔ taciturn; miser ↔ spendthrift; novice ↔ veteran. The stem often points to one side of a pair.

⚡ Omni-triplet. One prefix (omni- = all), three endings: -potent (power), -scient (knowing), -present (place). One card covers all three SSC favourites.

One word for places and groups

⚡ Stored-object anchor. Places are defined by what is inside: bees→apiary, birds→aviary, grain→granary, coins→mint, corpses→mortuary. Match the object, not the sound.

⚡ Animal-first matching. For group stems, name the animal first, then recall its partner noun. Options scramble the five common pairs; the animal decides.

⚡ Near-pair cards. Make cards for one-letter pairs: apiary/aviary, granary/granite, monastery/convent (men/women). These are where marks leak.

Government systems, studies and word-craft

⚡ Ruler-first matching. Underline WHO rules in the stem (people, rich, officials, one, God) and match to democracy/plutocracy/bureaucracy/autocracy/theocracy. The stem hands you the answer's head word.

⚡ Verb-link the negatives. Chain each 'cannot-be' word to its verb: read→illegible, believe→incredible, see→invisible. Stems always quote the verb.

⚡ Dead-word precision. About the dead: praise → eulogy, mourning-poem → elegy, tomb-writing → epitaph. Three cards, three certain marks.

Exam strategy for one-word substitution

⚡ Specificity rule. When two options fit the family, the stem's detail decides: 'brother' → fratricide, not homicide; 'twice a year' → biannual, not annual.

⚡ Stem-object underline. Underline the noun the word must carry (bees, grain, king, tomb). One underline removes half the options.

⚡ Mock-word ledger. Keep one running sheet titled 'OWS from mocks'. Most repeats come from your own past mistakes, not from unknown words.

Idioms & Phrases

Why literal meaning is always wrong — and how to find the picture

⚡ Literal-first elimination. Scan the options and kill every one you could photograph. The remaining two or three are the true contest.

⚡ Picture-to-abstract translation. Name the picture's key action in the abstract: escape→reveal, cold water→trouble, midnight oil→late work, blue (sky)→unexpected.

⚡ Plug-back sentence. Fit each surviving option into your own sentence using the idiom. The one that sounds like ordinary English is the key.

Animal and body-part idioms

⚡ Animal-character match. Recall each animal's stock character (rat = cheat, wolf = hunger, bull = force). The idiom's meaning is the animal's character plus the verb.

⚡ Body-function match. Ear = listen, eye = watch, foot = stand firm, hand = help/work. Map the named part to its function before reading options.

⚡ Opaque-idiom cards. Ten idioms have lost their logic (bite the dust, cold shoulder, red tape). Card them separately — they are the only real memory load.

Colour and food idioms

⚡ Colour association. Blue = surprise, red = anger or bureaucracy, green = envy, white = useless grandness, black = disgrace. Half the colour idioms decode from this alone.

⚡ Food-judgement fit. Food idioms are judgements in disguise: cake→easy, nut→stubborn, pickle→difficulty, potato→controversy. Match the judgement, not the ingredient.

⚡ Pair-learning. Learn confusable idioms in pairs on one card: beans/milk, elephant/lion's share, red tape/red-letter day. The option list usually contains both members.

Idioms for situations and types of people

⚡ Family surfacing. Classify the stem's demand: sudden? trouble? effort? defeat? person-type? Recall the family's four members; the right one is usually among the options, and the wrong options are from *other* families.

⚡ Allusion anchors. A few idioms hide a story: Achilles' heel (the hero's one weak spot), bell the cat (the mouse that dared), pass the buck (shifting the deal-marker). One-line stories fix meanings permanently.

⚡ Outcome test. For action idioms, ask what changes at the end: throw in the towel → the fight stops (give up); turn the tables → positions reverse; face the music → consequences arrive.

Phrasal verbs (verb + particle phrases)

⚡ Particle meanings. *off* often means stop or cancel (call off, put off); *out* means finish or remove (run out, put out); *up* means complete or give up; *after* means follow or care for (look after, take after).

⚡ Same verb, four particles. When all four options share one verb (break down / out / up / in), say the sentence with each and keep the one whose meaning fits the whole sentence.

⚡ One-word swap. Replace the phrasal verb with a single formal word to check it: put out = extinguish, call off = cancel, put up with = tolerate, come across = meet by chance.

Exam strategy: banking 300 idioms in 30 days

⚡ Sentence production. For every idiom, speak a sentence about your own life ('I burned the midnight oil before my SSC mock'). Personal sentences are recalled under pressure; list-readings are not.

⚡ Family-first revision. Revise by bank (animals → body → colours → food → situations → people). Each bank is one mental room; in the exam you 'enter the room' and the members appear.

⚡ Bleeder list. Fail an idiom twice in self-tests? It goes to a 15-item list revised weekly. Most permanent marks come from this list, not from new words.

Spelling

i before e — and the exceptions that fill the paper

⚡ The c-test. Is there a c right before the pair? c → 'ei' (receive). No c → usually 'ie' (believe). Ten exceptions override: weird, seize, height, foreign, sovereign, leisure, either, neither, their, protein.

⚡ Rhyme-anchor pairs. height–weight rhyme and must match; believe–achieve match; receive–deceive match. Fix one of each pair forever and its partner comes free.

⚡ Say it in syllables. Pronounce the word slowly in your head — be-lieve, re-ceive. The mouth half-confirms what the eye doubts.

Doubling consonants and one-letter betrayals

⚡ Stress decides doubling. Say the word. If the last syllable is stressed, double before a vowel-suffix (begin→beginning, occur→occurred). If not, don't (benefit→benefited).

⚡ Double-double checklist. Five words need TWO doubled letters: accommodate (c,m), committee (m,t), embarrass (r,s), occurrence (c,r), millennium (l,n). Count them.

⚡ Letter census. In the options, count the repeated letters. accommodate must show 2 c's and 2 m's; embarrass 2 r's and 2 s's. Eliminate on census failure.

-able / -ible, -ance / -ence, -ant / -ent

⚡ Root trace. Strip the suffix and look at the root: exist→exist-ence, maintain→maintenance (ai shortens), resist→resist-ance, persist→persist-ence. The root's own spelling decides.

⚡ -fer always -ence. Words from 'fer' nouns take -ence: reference, preference, difference, inference. Verbs keep one f or double by stress: refer→referring, but reference.

⚡ Scient family. One family handles four words: sufficient, efficient, ancient, science (and conscience = con + science). All -cient/-ience.

The trap-word bank: silent letters and inherited mistakes

⚡ The n-test for -ment words. government = govern+ment, environment = environ+ment: the root's final n must survive. Ask 'what is the root?' — the letter reappears.

⚡ Three -ceed, one -sede. proceed, succeed, exceed = the only -ceed words; supersede = the only -sede. All other 'seed-sound' words are -cede.

⚡ No- doubling plurals. criteria, phenomena, stimuli, data already are plurals — any option adding -s to them is instantly wrong.

Exam strategy: how to judge a spelling you have never seen

⚡ Difference-first reading. Don't read four full words — read the one letter-position where they differ. Options are built to differ in a single spot; that spot is the entire question.

⚡ Syllable count. Count syllables: mischievous (3), restaurant (2-3 as res-tau-rant), rhythm (2). A wrong spelling often forces a syllable that cannot be pronounced.

⚡ Write-from-memory drill. Recognition is not recall. Cover the bank, write the words, mark misses, repeat tomorrow — misses halve each cycle.

Active & Passive Voice

Voice basics: the three moves

⚡ Tense handover. The helping verb must carry the original tense: wrote becomes was, writes becomes is, will write becomes will be, has written becomes has been. A changed tense in an option is a planted error.

⚡ V3-only law. After be, been or being, the main verb is always V3 (written, given, built). Options like *is wrote* and *was build* are wrong on sight.

⚡ Pronoun mirror. The subject pronoun becomes the object pronoun behind by (he becomes him). The object pronoun becomes the subject pronoun in front (me becomes I).

Tense-by-tense conversions in action

⚡ Row recall. Name the active tense. Then say its passive row: present continuous is is/are being + V3, past perfect is had been + V3, future is will be + V3. Each wrong option breaks one cell of the row.

⚡ Agreement sweep. After converting, check is/are, was/were and has/have against the NEW subject before anything else.

⚡ Being or been in one look. Continuous in the active gives being. Perfect in the active gives been. A modal gives plain be.

Modals, imperatives, infinitives and questions

⚡ Let anchor. An order (no subject, base verb) points to Let + object + be + V3. An option with no Let and no modal cannot be right.

⚡ By whom first. Who-questions become *By whom + was/were/is + new subject + V3*. Check the word order first, because options scramble it.

⚡ Frame words. Please gives *You are requested to*. Advice gives *You are advised to*. A warning gives *You are warned not to*.

Reverse conversions: passive to active

⚡ First helping verb. Read only the first helping verb. Was means past. Is means present. Has or had means perfect. Will means future. Is being and was being mean continuous.

⚡ Doer hunt. The by-phrase gives the new subject. No by-phrase means *they*, *people* or *someone*.

⚡ Object-case check. The last pronoun must be in the object case: *praised him*, not *praised he*. Many options fail exactly here.

Exam strategy: the 20-second conversion protocol

⚡ Name the fault. Every wrong option breaks one rule: V-form, being/been, tense, agreement or pronoun. Name the fault and the option is out.

⚡ Five-row priority. Learn simple present, simple past, present perfect, present continuous and modals first. They cover most of the questions.

⚡ Write it, do not just say it. Voice errors hide on paper. Convert sentences in writing every day. The hand learns the pattern faster than the ear.

Direct & Indirect Speech

Reporting verbs and the mechanics of the shift

⚡ Read the tone first. Before you touch tense or pronouns, name the tone of the quote: order, advice, request, question, joy or sorrow. The matching verb alone often removes two options.

⚡ said to or told. Use *said to* + person, or *told* + person. Never mix them. Any option with *said me* or *told to me* is wrong.

⚡ Connector test. Statement: that. Yes/No question: if or whether. Wh-question: the wh-word alone. Order, request, advice: to + V1.

Backshift: the tense conversion table

⚡ Say the ladder aloud. Present goes to past. Present continuous goes to past continuous. Perfect goes to past perfect. Past goes to past perfect. Will goes to would. Can goes to could. May goes to might. Must (duty) goes to had to.

⚡ Frozen five. Could, would, should, might and ought to never change. If the quote has one of these, its tense is already right. Any shift is an error.

⚡ Truth test. Is the quote a law of nature, a habit or a proverb? Then do not backshift. Write: said that the earth revolves.

Pronouns, and the time/place word table

⚡ SONA sweep. Mark the three anchors before you change anything. First person goes to the reporter. Second person goes to the listener. Third person stays. Most wrong options break exactly one of these.

⚡ Yesterday and tomorrow pair. Yesterday becomes the previous day. Tomorrow becomes the next day. If the option shifts the verb but keeps the old time word, it is wrong.

⚡ Check the reporting verb first. If the reporting verb is 'says' (present), nothing shifts: not the tense, not the time words. Only the pronouns follow SONA.

Questions, commands and exclamations

⚡ Skeleton first. Name the type of the quote. Then write its skeleton: asked if ... / told ... to ... / exclaimed with joy that ... Pour the tense, pronoun and time-word changes into the skeleton.

⚡ Remove the helper do. In a direct question, do / does / did is only a helper. It vanishes in indirect speech. 'Do you know' becomes 'if I knew', never 'if I did know'.

⚡ Exclamation map. Hurrah: exclaimed with joy. Alas: exclaimed with sorrow. Bravo: applauded. Good morning: wished. Thank you: thanked.

Exam strategy: the four-check protocol

⚡ Name the fault. For each wrong option, name its fault: verb, tense, pronoun or time word. If you cannot name it, look again. If two options share no fault, you may have misread the tone verb.

⚡ Reverse-mode clues. In indirect to direct, 'the previous day' means the quote had 'yesterday'. 'Asked if' means the quote was a yes/no question with the helper verb first.

⚡ Write ten a day. Convert ten sentences a day by hand. Mix statements, questions and commands. In a week the four checks run together.

Sentence Rearrangement

The PQRS method: label, anchor, chain, eliminate

⚡ Pairs before strings. Do not read the four option strings first. Build 2–3 mandatory pairs from P/Q/R/S; then check which strings respect them. Strings die by broken links, not by vibes.

⚡ S1–S6 frame test. After chaining, read S1 + your chain + S6 as one paragraph. If any 'it/this/they' lacks its noun, the chain is wrong.

⚡ Adjacent-swap distractors. Wrong strings usually swap two adjacent parts. Find the ONE swap each wrong string commits and name its broken pair — that is your verification.

Opening and closing sentence clues

⚡ Pronoun-orphan test. Any part starting with he/she/it/they/this/these needs its noun earlier. If no earlier part in a string supplies the noun, that string dies.

⚡ Connector ban. But/However/Therefore/Thus/So cannot open. Two of four options often open with such parts — free elimination.

⚡ S6 pull-back. Read S6 and ask 'what sentence must come just before it?' — usually the one with Thus/That is why or the summarising judgement. Pin it, then build backwards.

Mandatory pairs: the links that never break

⚡ Noun-first scan. List the nouns/pronouns across P, Q, R, S. Circle pronouns and 'the'-nouns; draw arrows to the part that introduces each noun. The arrows ARE the answer skeleton.

⚡ This/That noun hunt. For every 'this + abstract noun', find which part contains that abstract noun. The order is forced.

⚡ a→the cascade. The part containing 'a + noun' must come before every part containing 'the + same noun'. Instant orderer for food-and-trick passages.

Time-sequence chains

⚡ Timeline first. Write the 4–6 events as a numbered list, then map numbers to letters. The string writes itself.

⚡ Tense radar. Past perfect = earlier; past continuous = background; simple past = the event line. Use tenses to order events when time words are missing.

⚡ By-time capping. 'By midnight/By dawn' closes the phase it describes — it follows, never opens, the event chain.

Exam strategy: elimination, budget and Tier 2 jumbles

⚡ Layered elimination. Layer 1: opener ban. Layer 2: mandatory pair. Layer 3: S6 check. Each layer is one question; two layers usually finish the item.

⚡ Local-difference test. Surviving strings differ in one part's position. Isolate that part and test only its two neighbours.

⚡ Odd-sentence radar. For odd-one-out items, mark each sentence's KEY noun. Four share it; the impostor only orbits it.

Grammar Essentials (Handbook)

Parts of speech: nouns, pronouns, adjectives, adverbs

⚡ Cover the other name. When a pronoun sits next to a noun (*Ravi and I / Ravi and me*), cover the noun. Read the sentence with the pronoun alone. The form that sounds right is the answer.

⚡ Scan every noun. In an error-spotting sentence, circle each noun. Ask two questions. Can I count it? Is it one of the special groups (no plural, always plural, cattle-type)? Words like informations, furnitures and advices are favourite e…

⚡ Few or a few: empty or half full. *Few* and *little* mean the glass is nearly empty (a negative feeling). *A few* and *a little* mean there is something (a positive feeling). Let the rest of the sentence show the feeling.

⚡ Adjective order and word families. Adjectives follow the order OSASCOMP: Opinion, Size, Age, Shape, Colour, Origin, Material, Purpose. Example: a beautiful small old round brown Indian wooden table. Some adjectives have no degrees: unique, perfect, ideal,…

⚡ Numbers with nouns. After a definite number, *dozen, hundred, thousand, million, score, lakh, crore* and *pair* take no -s: *three dozen eggs, five hundred rupees*. But *dozens of eggs* and *hundreds of people* keep the -s. A number + noun …

⚡ Where adverbs go. Frequency words go before the main verb but after *be*: *He always comes late. He is always late.* Use *very* with the plain adjective and with -ing adjectives (very tall, very interesting). Use *much* with comparatives …

Tenses and sequence of tenses

⚡ Scan for the signal word. Underline the time signal before reading the options. Since or for gives a perfect tense. Yesterday, ago or last gives the simple past. By + future time gives the future perfect. When, if or as soon as with future meanin…

⚡ No will after when or if. If a clause starts with when, if, unless, until or as soon as and talks about the future, remove will from that clause. Will stays in the main clause only.

Subject-verb agreement (all rules and exceptions)

⚡ Strike-through method. Draw a line through every phrase that starts with a preposition (of the boys, in the box, with his friends). Also strike every add-on phrase (as well as, along with, together with). What stays before the verb is the true…

⚡ Nearest subject for or / nor pairs. For either ... or, neither ... nor, not only ... but also, or and nor, look only at the subject next to the verb. Put the plural subject second so the plural verb sounds natural.

⚡ The number and a number. *The number* is one figure, so it is singular. *A number* means many, so it is plural. Memory hook: A = Abundant.

Articles (a, an, the, zero article) and determiners

⚡ Say it aloud. For a or an, whisper the next word. A vowel sound (on-est, em-el-ay, ex-ray) needs an. A y or w sound (you-niversity, won-day) needs a.

⚡ The geography rule of thumb. Plural names, water and chains take the: the Alps, the Ganga, the Indian Ocean, the Andamans. A single peak, a single lake, a city or most countries take no article: Mount Abu, Lake Dal, Delhi, Nepal.

Prepositions and conjunctions

⚡ Find the partner. When you see hardly, scarcely, no sooner, not only, neither, either, both, whether, lest or though, look ahead for its fixed partner. A wrong or missing partner is very often the error.

⚡ Group prepositions by meaning. Stopping: prevent, refrain, abstain, desist from. Dependence: depend, rely, count on. Skill: good, expert, adept at; proficient in. Latin comparatives: senior, junior, superior, inferior, prior, prefer …

Modals, conditionals and the subjunctive

⚡ Match the type by the if-clause. Read only the if-clause. Present simple: use will or can in the result. Past simple: use would in the result. Past perfect: use would have + V3.

⚡ Modal + base form. After any modal the next verb is bare. If you see a modal followed by to, -s, -ing or -ed, that is the error. The only exceptions are ought to and used to.

Degrees of comparison and question tags

⚡ Flip the polarity. Read the statement. Note whether it is positive or negative. Then use the helper verb with the opposite sign and a pronoun. Command? Use will you. Let's? Use shall we.

⚡ Watch than, that of and other. When a comparison feels odd, check two things. First: are two like things compared (use *that of* or *those of*)? Second: does the comparison come from the same group (add *other*)?

Non-finites (infinitive, gerund, participle), inversion and parallelism

⚡ Preposition then -ing. If the word before the blank is a preposition (of, in, at, without, after, before, by, about), the verb must end in -ing. Watch for to in look forward to and used to: it is a preposition there.

⚡ Negative start means flip. When a sentence opens with never, rarely, hardly, no sooner, not only or not until, flip the first helper before the subject. If there is no helper, add do, does or did.

General Awareness

Ancient Indian History

Indus Valley (Harappan) Civilisation

⚡ Four extremes — "Many Donkeys Are Slow". Manda (North, J&K) · Daimabad (South, Maharashtra) · Alamgirpur (East, UP) · Sutkagendor (West, Baluchistan).

⚡ Site-to-find pairing. Lothal = Launch (dockyard) · Kali-bangan = Kheti (ploughed field) · Mohenjo = Majestic Bath (Great Bath) · Chanhu = Chain of beads (bead factory) · Dholavira = Drop of water (reservoirs).

Vedic Age & Vedic Literature

⚡ Punjab rivers west→east: "Very Angry Parrots Visit Sutlej". Vitasta = Jhelum → Asikni = Chenab → Parushni = Ravi → Vipas = Beas → Sutudri = Sutlej. Same order as the modern J-C-R-B-S.

⚡ Veda → priest: "Rig-Hotri, Sama-Udgatri, Yajur-Adhvaryu, Atharva-Brahma". Think HUAB: Hotri recites, Udgatri sings, Adhvaryu performs, Brahma supervises.

⚡ Rigveda mandala anchors: 3-7-9-10. 3 = Gayatri, 7 = Ten Kings war, 9 = Soma, 10 = Purusha Sukta (varnas).

Mahajanapadas & Rise of Magadha

⚡ Capital pairs by first letter. Kosala–shravasti? No: use "Vatsa-Kaushambi, Vajji-Vaishali" (both V) and "Avanti-Ujjain, Anga-Champa" (both start with A). Kuru-Indraprastha, Gandhara-Taxila (G-T like 'GT road' ending in the north-we…

⚡ Magadha order: "Hari Shishu Nanda". Haryanka → Shishunaga → Nanda → Maurya. Councils: Ajatashatru (1st) is Haryanka, Kalashoka (2nd) is Shishunaga.

Buddhism & Jainism

⚡ Councils — places "Raja Vaishali Pata Kashmir", kings alternate "A-K-A-K". Places: Rajagriha → Vaishali → Pataliputra → Kashmir. Kings: Ajatashatru, Kalashoka, Ashoka, Kanishka. Presidents: "Maha-Saba-Moggali-Vasu".

⚡ Buddha's life symbols: "Lotus → Horse → Tree → Wheel → Stupa". Birth, renunciation, enlightenment, first sermon, death — in that order.

⚡ Tirthankara symbols: "Bull begins, Lion ends". 1st Rishabha = bull, 22nd = conch, 23rd Parshva = snake (Serpent), 24th Mahavira = lion.

Mauryan Empire

⚡ Edict numbers: "2-Hospital, 5-Officer, 12-Tolerance, 13-Kalinga". Think of a hospital ward 2, officer rank 5, the 12 faiths tolerated, and the unlucky 13 (war).

⚡ Mauryan dates chain: 322 → 185 → 73 BCE. Maurya starts 322, Shunga starts 185 (Pushyamitra), Kanva starts 73 (Vasudeva Kanva).

Post-Mauryan Age: Shungas to Kushanas

⚡ "57 minus, 78 plus". Vikram Samvat = 57 BCE (subtract), Shaka = 78 CE (add). To convert: CE = VS − 57; CE = Shaka + 78.

⚡ Satavahana = "S-P-G-H". Simuka founder · Pratishthana capital · Gautamiputra greatest · Hala poet.

Gupta Age & Harshavardhana

⚡ Gupta sequence: "Sri Ghat, Chandra-Samudra-Chandra, Kumar-Skand". Sri Gupta → Ghatotkacha → Chandragupta I → Samudragupta → Chandragupta II → Kumaragupta I → Skandagupta.

⚡ Prashasti authors: "H for Samudra, R for Pulakeshin". Harishena → Allahabad (Prayag) Prashasti of Samudragupta. Ravikirti → Aihole inscription of Pulakeshin II. Banabhatta → Harshacharita (biography, not an inscription).

⚡ Pilgrims in order: "Fa → Hiuen → I" (F-H-I). Fa-Hien (Chandragupta II) → Hiuen Tsang (Harsha) → I-Tsing (after Harsha). Alphabetical and chronological together.

Sangam Age & South Indian Dynasties

⚡ Sangam emblems: "Bow-Tiger-Fish = Chera-Chola-Pandya" (C-C-P). Alphabetical Bow, Tiger... simpler: Chera Bow (cheer with a bow), Chola Tiger (roar), Pandya Fish (Madurai's Meenakshi = 'fish-eyed').

⚡ Temple ↔ dynasty quick map. Kailasa Ellora = Rashtrakuta (Krishna I) · Kailasanatha Kanchi = Pallava (Rajasimha) · Brihadeeswara = Chola (Rajaraja I) · Virupaksha Pattadakal = Chalukya. Note 'Kailasa' vs 'Kailasanatha'.

⚡ Imperial Cholas: "VAPRR". Vijayalaya → Aditya I → Parantaka I → Rajaraja I → Rajendra I.

Ancient Literature & Authors

⚡ Drama trio: "Shudraka's Cart, Bhasa's Dream, Vishakha's Seal". *Mrichchhakatika* = little clay cart (Shudraka) · *Swapnavasavadatta* = dream of Vasavadatta (Bhasa) · *Mudrarakshasa* = Rakshasa's seal/ring (Vishakhadatta).

⚡ Medicine pair: "Charaka Cures, Sushruta Stitches". Charaka Samhita = general medicine (Kanishka's physician); Sushruta Samhita = surgery (plastic surgery, cataract).

Medieval Indian History

Early Medieval India (c. 750–1206)

⚡ Tripartite = "PPR fight for K". Pala, Pratihara, Rashtrakuta → Kannauj. East, West and South all wanted the centre.

⚡ Tarain: 1 to Prithviraj, 2 to Ghori. 1191 → 1st battle → Prithviraj wins. 1192 → 2nd battle → Ghori wins. The second number always goes to the invader.

Delhi Sultanate (1206–1526)

⚡ Dynasty order — "Sabka Khana Tum Sab Lo". Slave → Khalji → Tughlaq → Sayyid → Lodi. Starting years 1206 – 1290 – 1320 – 1414 – 1451 end at 1526 (Panipat).

⚡ Who did what — first letter hooks. Iltutmish = Iqta + Investiture; Balban = Bow down (sijda/paibos); Alauddin = All prices fixed; Muhammad bin Tughlaq = Move capital + Money tokens; Firoz = Fields & canals.

Vijayanagara & Bahmani Kingdoms

⚡ Vijayanagara dynasties — "Some Silly Tigers Attack". Sangama → Saluva → Tuluva → Aravidu. Krishnadevaraya sits in the third (Tuluva — "Top king").

⚡ Deccan five — "BiG AB B". Bijapur (Adil), Golconda (Qutb), Ahmadnagar (Nizam), Bidar (Barid), Berar (Imad). The first letters of the dynasty names: A-Q-N-B-I.

Mughal Empire & Sher Shah Suri

⚡ Mughal order — "Bahut Hi Accha Jalebi Shahi Aur". Babur → Humayun → Akbar → Jahangir → Shah Jahan → Aurangzeb. Sher Shah sits between Humayun's two reigns (1540–55).

⚡ Three Panipats — 26, 56, 61. 1526 Babur beats Ibrahim Lodi · 1556 Akbar (Bairam Khan) beats Hemu · 1761 Ahmad Shah Abdali beats the Marathas. Gap pattern: +30 years, then +205 years.

Marathas & Sikh Gurus

⚡ Guru order — "Nana Aur Amar Ram Arjun, Har-Har-Har Tegh Gobind". Nanak, Angad, Amar Das, Ram Das, Arjan → the three Hars (Hargobind, Har Rai, Har Krishan) → Tegh Bahadur → Gobind Singh.

⚡ Chauth vs Sardeshmukhi. Chauth = Chaar ana in the rupee (1/4); Sardeshmukhi = 10% — the Sardeshmukh claimed a 'tenth' as hereditary head.

Bhakti & Sufi Movements

⚡ Philosophy trio — "RaVi MaD". Ramanuja → Vishishtadvaita · Madhva → Dvaita · (Shankara → Advaita).

⚡ Language of the epic. Tulsidas = Awadhi (A for Ayodhya's Ram) · Surdas = Braj (Bal-Krishna).

Medieval Books & Foreign Travellers

⚡ Mughal chroniclers — "Abul = Ain, Badauni = Bitter". Abul Fazl wrote the official, praising Akbarnama/Ain; Badauni wrote the bitter, critical *Muntakhab*.

⚡ Travellers by century. 11th Al-Biruni → 13th Marco Polo → 14th Ibn Battuta → 15th Conti & Abdur Razzaq → 16th Paes & Nuniz → 17th Hawkins, Roe, Bernier, Tavernier.

Modern Indian History

Europeans in India & British Expansion

⚡ Plassey then Buxar — "57 buys, 64 seals". 1757 Plassey gave the British a foothold (Clive vs Siraj); 1764 Buxar sealed control (three Indian rulers beaten) and brought the Diwani (1765).

⚡ Governor-General reforms — "Cornwallis Settles, Wellesley Subsidises, Bentinck bans Sati, Ripon Rules locally". Permanent Settlement 1793 → Subsidiary Alliance 1798 → Sati abolition 1829 → Local self-government 1882.

Revolt of 1857 and Peasant & Tribal Uprisings

⚡ 1857 pairs — "Kanpur Nana, Lucknow Lady, Jhansi Rani, Bihar Kunwar". Kanpur–Nana Saheb · Lucknow–Begum Hazrat Mahal (the Lady) · Jhansi–Rani Lakshmibai · Bihar (Jagdishpur)–Kunwar Singh, the 80-year-old.

⚡ Tribal movements by letter. Santhal = Sidhu (1855) · Munda = Birsa (Millennium's end, 1899–1900).

Socio-Religious Reform Movements

⚡ 1875 — the year of three. Arya Samaj (Dayanand), Theosophical Society (Blavatsky–Olcott) and the Aligarh MAO school (Syed Ahmad Khan) all date to 1875.

⚡ Founder hooks. Ram Mohan = Reform of sati (Brahmo, 1828) · Dayanand = Discover the Vedas (Arya) · Phule = People of lower castes (Satyashodhak) · Vivekananda = Vedanta service (RK Mission, 1897).

Congress, Moderates & Extremists (1885–1916)

⚡ Five-six-seven-nine: "Split, League, Split, Separate". 1905 Bengal split → 1906 Muslim League → 1907 Congress split (Surat) → 1909 separate electorates (Morley–Minto).

⚡ Home Rule twins. Tilak first (April, Belgaum) → Besant next (Sept, Madras). "TAB then BSM".

Gandhian Era & Road to Independence (1915–1947)

⚡ Gandhi's first three — "C-A-K, 17-18-18". Champaran 1917 (indigo, Bihar) → Ahmedabad 1918 (mill workers, hunger strike) → Kheda 1918 (peasants, revenue).

⚡ Session presidents that repeat in exams. Lahore 1929 = Jawaharlal (Purna Swaraj) · Karachi 1931 = Patel (Fundamental Rights) · Tripuri 1939 = Bose. "J-P-B" from 29 to 39.

⚡ Dandi numbers. 12 March start · 6 April salt law broken · 78 marchers · about 385 km (240 miles) · from Sabarmati Ashram.

Newspapers, Journals & Books of the Freedom Era

⚡ Gandhi's papers — "YNH". Young India · Navajivan · Harijan (plus *Indian Opinion* from South Africa). Anything else in the options is someone else's paper.

⚡ Jail-written classics. Tilak → *Gita Rahasya* in Mandalay; Nehru → *Discovery of India* in Ahmednagar Fort.

Art & Culture

Classical Dances & Their Exponents

⚡ Kerala has two, the rest one each. Kerala = Kathakali + Mohiniyattam. Every other classical dance has one home state: TN-Bharatanatyam, AP-Kuchipudi, Odisha-Odissi, Manipur-Manipuri, Assam-Sattriya, UP-Kathak.

⚡ Exponent hooks. Birju = Bol of Kathak · Kelucharan = Odissi (Konark poses) · Rukmini Devi = Revived Bharatanatyam (Kalakshetra) · Vempati = Village Kuchipudi · Vallathol = Kerala Kalamandalam (Kathaka…

Folk Dances by State

⚡ North-East quick five — "Cheraw Mizo, Hoja Tripura, Nong Khasi, Wang Garo, Bagu Bodo". Cheraw → Mizoram · Hojagiri → Tripura · Nongkrem → Meghalaya (Khasi) · Wangala → Meghalaya (Garo) · Bagurumba → Assam (Bodo).

⚡ Men vs women in Punjab. Bhangra = Bhaiya (men) · Giddha = Girls (women).

Music, Instruments & Maestros

⚡ Carnatic Trinity — "Tea, Milk, Sugar". Tyagaraja · Muthuswami Dikshitar · Syama Sastri. Purandara Dasa is the *father*, not part of the trinity.

⚡ Instrument families — "Tata Sings, Avan Drums, Ghana Clangs". Tata = strings, Sushira = air (wind), Avanaddha = skin drums, Ghana = solid (bells, cymbals, ghatam).

Paintings, Textiles & Handicrafts

⚡ "Kalam in Andhra, Patta in Odisha, Phad in Rajasthan". Kalamkari → Andhra (the pen = *kalam*) · Pattachitra → Odisha (cloth = *patta*) · Phad → Rajasthan (Pabuji scroll). Madhubani → Bihar's Mithila.

⚡ Bidri = Bidar. The craft is named after its town — Bidriware from Bidar (Karnataka). Same trick: Patola from Patan, Chanderi from Chanderi.

Temple & Rock-cut Architecture

⚡ "Shikhara North, Gopuram South, Vesara in Between". See a gopuram → Dravida. See a curved shikhara → Nagara. A star-shaped Hoysala plan → Vesara (Deccan).

⚡ Odisha pagodas. Black Pagoda = Konark (dark stone, Sun temple) · White Pagoda = Puri (whitewashed Jagannath).

Festivals & Fairs

⚡ Nagaland December — "Hornbill closes the year". Hornbill Festival runs 1–10 December (Nagaland Statehood Day is 1 December).

⚡ Two cattle fairs. Pushkar = Pink city state (Rajasthan) camels; Sonepur = Saran/Bihar cattle & elephants.

UNESCO Heritage & Cultural Institutions

⚡ WHS count ladder. Remember the last few by year: 2021 → 40 (Dholavira), 2023 → 42 (Hoysala), 2024 → 43 (Moidams), 2025 → 44 (Maratha forts), 2026 → 45 (Sarnath).

⚡ Three Akademis — "Sangeet first, then Sahitya & Lalit together". SNA 1952, then Sahitya and Lalit Kala both in 1954.

Indian Polity & Constitution

Making of the Constitution & borrowed features

⚡ Committee-chairman pairing. Nehru took the *Union/States/Constitution* committees (the 'national' subjects), Patel took *Provinces and Rights* (the 'practical' subjects), Prasad took *procedure and flag*, Ambedkar took the *Drafting…

⚡ Borrowed features speed-pair. UK-Parliament • USA-Rights • Ireland-Duties-of-state (DPSP) • Canada-Centre-strong • Australia-List-3 • USSR-Duties-of-citizens • France-Republic • Germany-Emergency-rights • Japan-law-procedure.

Preamble, Parts and the 12 Schedules

⚡ Schedule story-line. 1-Names, 2-Salaries, 3-Oaths, 4-RS seats, 5-Areas(SC/ST), 6-Tribal NE, 7-Three Lists, 8-Languages, 9-Land laws, 10-Defection, 11-Panchayat(29), 12-Municipality(18).

⚡ Preamble keyword count. Sovereign-Socialist-Secular-Democratic-Republic = 5 descriptors; J-L-E-F = Justice-Liberty-Equality-Fraternity, with only Justice split three ways (social-economic-political).

Fundamental Rights, DPSP and Fundamental Duties

⚡ FR article blocks. 14-18 Equality • 19-22 Freedom • 23-24 Exploitation • 25-28 Religion • 29-30 Culture-Education • 32 Remedies. Count blocks as 5+4+2+4+2+1.

⚡ Duty amendment pair. 42nd gave 10 duties; 86th added the 11th (education of 6-14 year-olds). Same 86th also inserted Art. 21A — one amendment, twin gifts.

Union and State Executive

⚡ Electoral college contrast. President = elected MPs + elected MLAs (no nominated). Vice-President = ALL MPs of both Houses, including the 12 nominated. 'Vice' reaches where 'President' does not.

⚡ Pardon ladder. President's Art. 72 pardons court-martial and death sentences; the Governor's Art. 161 covers neither. 'Governor = softer ladder'.

Parliament and the Judiciary

⚡ Committee trio numbers. PAC 22 (opposition chair), CPU 22 (15+7), Estimates 30 (all LS). 'Estimates = 30, exclusively LS.'

⚡ Writ first letters. H-M-P-C-Q: Habeas(body), Mandamus(command), Prohibition(stop-before), Certiorari(quash-after), Quo warranto(by what authority). Prohibition stops a case early; certiorari quashes after.

Important Constitutional Amendments

⚡ Amendment decade clusters. 1950s structural (1st, 7th) • 1970s power (24, 42, 44) • 1980s politics (52 anti-defection, 61 voting age) • 1990s grass-roots (73, 74) + language (71, 92) • 2000s rights (86, 91, 97) • 2010s-20s economy-equality (101 GS…

⚡ 42 vs 44 tug-of-war. Whatever the 42nd tightened, the 44th loosened: term 6→5 years, 'internal disturbance'→'armed rebellion', property FR→300A legal right.

Panchayati Raj, Municipalities and Emergency

⚡ Committee tier-count rhyme. Balwant-3 (1957), Ashok-2 (1977), Rao-district (1985), Singhvi-Constitution (1986). Decade rhyme: 57-77-85-86.

⚡ Emergency digits. 352-National (3 proclamations: 62, 71, 75), 356-President's Rule (Kerala 1959 first), 360-Financial (zero uses). Approvals: 1 month + every 6 months.

Constitutional & Statutory Bodies (with current heads)

⚡ Article blocks for bodies. 76-AG • 280-FC • 148-CAG • 315-UPSC • 324-EC. Descending ladder: 324 > 315 > 280 > 148 > 76 reads 'EC-UPSC-FC-CAG-AG'.

⚡ Statutory vs constitutional filter. If the source is a dated Act (1993 NHRC, 2003 CVC, 2005 CIC, 2013 Lokpal) it is statutory; if the source is an Article number it is constitutional; NITI Aayog (Cabinet resolution, 2015) is neither.

Indian Geography

Location, extent, neighbours and islands

⚡ Tropic of Cancer states — 'GRM CJ WTM'. West to east: Gujarat, Rajasthan, Madhya Pradesh, Chhattisgarh, Jharkhand, West Bengal, Tripura, Mizoram. Read it as 'Great Royal Mango, Cool Juice, With Tasty Mangoes'. Odisha and Bih…

⚡ Border order: 'Big Cats Prowl Near My Big Area'. Longest to shortest border — Bangladesh > China > Pakistan > Nepal > Myanmar > Bhutan > Afghanistan.

⚡ Channel numbers go down as you go south-west. Ten Degree (Andaman | Nicobar) → Nine Degree (Minicoy | Lakshadweep) → Eight Degree (Minicoy | Maldives). The Andaman group has the bigger number.

Himalayas, peaks, passes, plateau, coasts and lakes

⚡ Pass → state by first letter group. Z-K-K (Zoji, Khardung, Karakoram) = Ladakh; R-S-B (Rohtang, Shipki, Baralacha) = Himachal; L-M (Lipulekh, Mana) = Uttarakhand; N-J (Nathu, Jelep) = Sikkim; Bomdi = Arunachal; Pal…

⚡ Highest-peak ladder. India claimed: K2 → India administered: Kangchenjunga → South India: Anamudi → Nilgiris: Doddabetta → Aravallis: Guru Shikhar → Satpura: Dhupgarh.

⚡ Lake superlatives: 'Fresh Wular, Brackish Chilika, Salty Sambhar, Floating Loktak'. Say it as one chain. Examiners swap the adjectives between the four lakes.

Rivers, dams and waterfalls

⚡ Panch Prayag order — 'Very Nice Kids Read Daily'. Vishnuprayag → Nandprayag → Karnaprayag → Rudraprayag → Devprayag (upstream to downstream). At Devprayag the river becomes the Ganga.

⚡ West-flowers: 'Nobody Takes Mahi Seriously'. Narmada, Tapi, Mahi, Sabarmati flow into the Arabian Sea and form estuaries. Narmada and Tapi flow through rift valleys.

⚡ Waterfall ↔ river pairs. Jog–Sharavati, Chitrakote–Indravati, Dudhsagar–Mandovi, Athirappilly–Chalakudy, Shivanasamudra/Hogenakkal–Kaveri, Dhuandhar–Narmada.

Monsoon, local winds and soils

⚡ Black soil rich/poor list. Rich in 'LIMP' — Lime, Iron, Magnesium, Potash. Poor in 'NPH' — Nitrogen, Phosphorus, Humus.

⚡ Shower names by crop. Mango showers → mango (Kerala); Blossom showers → coffee (Karnataka); Kalbaisakhi → tea/jute/rice (Bengal, Assam). Western disturbances → wheat (rabi) in the north-west.

Agriculture, revolutions, minerals and ports

⚡ Revolution colours. Colour of the product: White = milk, Blue = water/fish, Yellow = oilseeds (mustard flower), Silver = eggs (shell shine), Round = potato, Green = grains/fields.

⚡ Kharif = 'Monsoon crops need water'. Water-loving rice, jute, cotton → kharif. Cool-weather wheat, gram, mustard → rabi. Summer melons → zaid.

National parks, tiger reserves and wildlife

⚡ Animal-first recall. Rhino → Kaziranga; Lion → Gir; Cheetah → Kuno; Snow leopard → Hemis; Hangul → Dachigam; Sangai → Keibul Lamjao; Nilgiri tahr → Eravikulam; Lion-tailed macaque → Silent Valley; Saltwater cr…

⚡ Year chain 36-73-86-92. 1936 first NP (Hailey/Corbett) → 1973 Project Tiger → 1986 first biosphere reserve (Nilgiri) → 1992 Project Elephant.

World Geography

Solar system, Earth's motions, latitudes and time

Time difference from longitude

\Delta t = \Delta\lambda \times 4\ \text{minutes}

1° of longitude = 4 min; 15° = 1 hour

Local time

T_{\text{local}} = T_{\text{GMT}} \pm \frac{\lambda}{15}\ \text{h}

+ for east longitudes, − for west

⚡ Planet superlatives in one line. Mercury smallest–nearest, Venus hottest–brightest–backward spin, Mars red, Jupiter largest (Ganymede biggest moon), Saturn rings–lightest, Uranus on its side, Neptune farthest–windiest.

⚡ IST from the meridian. India's meridian is 82½°E → 82.5 × 4 = 330 minutes = 5 h 30 min ahead of GMT.

Earth's interior, earthquakes, rocks and volcanoes

⚡ Discontinuity order: 'Come Make Good Lunch'. Conrad (inside crust) → Moho (crust|mantle) → Gutenberg (mantle|core) → Lehmann (outer|inner core).

⚡ Metamorphic pairs: 'Lime-Marble, Sand-Quartz, Shale-Slate, Granite-Gneiss'. The first letters of the new rock are easy: Marble from Limestone is the most asked (Taj Mahal marble).

Atmosphere, pressure belts, planetary and local winds

⚡ Layer order: 'The Strong Man Throws Eggs'. Troposphere → Stratosphere (ozone, jets) → Mesosphere (coldest, meteors) → Thermosphere (ionosphere, radio) → Exosphere.

⚡ Local wind by country. Chinook–USA/Canada, Foehn–Alps, Sirocco–Sahara→Italy, Harmattan–West Africa, Mistral–France, Khamsin–Egypt, Santa Ana–California, Loo–India.

Oceans, currents, straits and canals

⚡ Cold current → desert pairs. Humboldt–Atacama, Benguela–Namib, California–Sonoran. West coasts + cold current = dry air = desert.

Continents, superlatives, grasslands and tribes

⚡ Grasslands: 'Pretty Pam Visits Steppe Down'. Prairies (N America), Pampas (Argentina), Veld (S Africa), Steppes (Eurasia), Downs (Australia).

⚡ Lake superlatives. Caspian largest (salt), Superior largest freshwater, Baikal deepest, Titicaca highest navigable, Dead Sea lowest.

Nicknames and boundary lines

⚡ Sun pair. Rising Sun = Japan (east of Asia, first sunrise); Midnight Sun = Norway (Arctic summer).

⚡ Line → neighbours: 'Radcliffe-Rift, McMahon-Mountains, Durand-Afghan'. Radcliffe partitioned India–Pakistan, McMahon runs along the Himalayas with China, Durand divides Pakistan–Afghanistan.

Indian Economy

National income, growth and base years

NNP

NNP = GNP - \text{Depreciation}

GNP already includes net factor income from abroad

GDP deflator

\text{Deflator} = \frac{\text{Nominal GDP}}{\text{Real GDP}} \times 100

divide nominal by (deflator/100) to get real

Market price bridge

MP = FC + \text{Indirect taxes} - \text{Subsidies}

moving between factor cost and market price

⚡ G-D-N ladder. GDP (territory) + NFIA = GNP; GNP − depreciation = NNP. 'Territory → Nation → Net'.

⚡ New base-year pairs. Production takes financial years: GDP and IIP = 2022-23; prices of consumers take a calendar year: CPI = 2024; wholesale moved to 2022-23 with PPI twins in June 2026.

Five Year Plans and NITI Aayog

⚡ Plan-era story line. 1-farm (1951) → 2-machines (Mahalanobis) → 3-wars → holiday (66-69) → 4-stability + bank nationalisation → 5-Garibi Hatao → rolling (Janata) → 6-7 growth → 8-reforms (1992) → 9-12 inclusive → NITI (2015).

⚡ Plan-break tags. Two gaps: Plan Holiday 1966-69 (wars/drought) and Annual Plans 1990-92 (transition after the 8th was delayed).

Money, banking and the RBI

⚡ Corridor arithmetic. Repo is the middle: SDF = repo − 0.25 (floor), MSF = Bank Rate = repo + 0.25 (ceiling). With repo 5.25 → 5.00 / 5.25 / 5.50.

⚡ Nationalisation anchors. 1935 born, 1949 nationalised (RBI), 1955 SBI, 1969 fourteen, 1975 RRBs, 1980 six.

⚡ MPC fixed points. 6 members; 4 meetings a year minimum; target CPI 4% ± 2%; Governor's casting vote.

Inflation and price indices

⚡ Index-compiler pairs. CPI-IIP-GDP = MoSPI/NSO; WPI-PPI = DPIIT's Office of Economic Adviser. 'M for MoSPI, W for DPIIT(OEA)'.

⚡ Core = strip the volatile. Core inflation = headline − food − fuel. Stagflation = stagnation + inflation together.

Budget and fiscal policy

Fiscal deficit

FD = \text{Total Expenditure} - \text{Total Receipts (excl. borrowings)}

equals government borrowing requirement

Revenue deficit

RD = \text{Revenue Expenditure} - \text{Revenue Receipts}

revenue items only

Primary deficit

PD = FD - \text{Interest Payments}

strips out past borrowing costs

⚡ Budget 2026-27 number sheet. Spend 53.5 • Receipts 36.5 • Borrow 16.9 (₹ lakh crore); FD 4.3% • RD 1.5% • PD 0.7%; Capex 12.2; Debt 55.6%.

⚡ Article trio for money matters. 112-Budget • 266-Consolidated Fund • 267-Contingency Fund; 265-no tax without law.

Taxation and GST

⚡ GST article trio. 246A power • 269A inter-state • 279A Council. 101st Amendment, 1 July 2017.

⚡ GST 2.0 slab story. Two working slabs — 5 (merit) and 18 (standard) — with 40 kept aside for sin/luxury; 12% and 28% abolished on 22 Sep 2025.

⚡ Tax-year switch. Income-tax Act, 2025 → effective 1 April 2026; one 'tax year' replaces previous/assessment year pair; 536 sections, 23 chapters.

Government schemes and missions

⚡ Launch-year clusters. 2014: Jan Dhan, Make in India • 2015: BBBP, APY, MUDRA, PMAY • 2016: Ujjwala • 2018: Ayushman • 2019: KISAN • 2020: SVANidhi, PLI • 2023: Vishwakarma, Drone Didi • 2024: Surya Ghar • 2026: VB-G RAM G live (125 days).

⚡ MGNREGA → RAM G swap. 100 → 125 days; 15-day → weekly wages; Act of 2025, in force 1 July 2026.

Regulators, exchanges and international organisations

⚡ Regulator HQ pairs. RBI-SEBI-NABARD = Mumbai; IRDAI = Hyderabad; SIDBI = Lucknow; PFRDA = New Delhi.

⚡ Bretton-Woods twins. IMF + World Bank both born at Bretton Woods (1944), both in Washington DC; WTO (1995, Geneva) replaced GATT (1947).

⚡ Asian banks map. ADB 1966 → Manila; AIIB 2016 → Beijing; NDB (BRICS) 2015 → Shanghai. India founding member in all three.

Physics

SI units, conversions and measuring instruments

Kilowatt-hour

1\ \text{kWh} = 3.6 \times 10^{6}\ \text{J}

1 'unit' of electricity

Horsepower

1\ \text{hp} \approx 746\ \text{W}

Light year

1\ \text{ly} \approx 9.46 \times 10^{15}\ \text{m}

a unit of distance

⚡ Base-unit roll call: 'Mighty Kings Seldom Answer Kind Monks Correctly'. Metre, Kilogram, Second, Ampere, Kelvin, Mole, Candela — the 7 SI base units. Anything else (newton, joule, volt, watt…) is derived.

⚡ 'Meter' by what it measures. Hygro = humidity (hygiene/water), Hydro = liquid density, Lacto = milk, Anemo = wind (Greek *anemos*), Sphygmo = pulse/BP, Pyro = fire/high temperature, Seismo = earthquake, Alti = altitud…

Motion, gravitation, work-energy, fluids and levers

First equation of motion

v = u + at

Second equation of motion

s = ut + \tfrac{1}{2}at^{2}

Third equation of motion

v^{2} = u^{2} + 2as

Newton's second law

F = ma = \frac{\Delta p}{\Delta t}

p = mv

Law of gravitation

F = \frac{G m_1 m_2}{r^{2}}

G = 6.67 × 10⁻¹¹ N m² kg⁻²

Kinetic and potential energy

KE = \tfrac{1}{2}mv^{2},\quad PE = mgh

Work and power

W = Fs\cos\theta,\quad P = \frac{W}{t}

1 W = 1 J/s

Pressure

P = \frac{F}{A},\quad P_{\text{liquid}} = h\rho g

unit pascal

⚡ Lever class by the middle item: 'FLE = 1-2-3'. Whatever sits in the middle decides the class: Fulcrum → 1, Load → 2, Effort → 3. Nutcracker has the load (nut) in the middle → class 2; tongs are squeezed in the middle → class 3.

⚡ Squares in energy. KE ∝ v²: double the speed → 4× KE; triple → 9×. Momentum ∝ v: double speed → 2× momentum.

Heat, temperature and sound

Temperature scales

\frac{C}{100} = \frac{F-32}{180} = \frac{K-273}{100}

−40 °C = −40 °F

Heat absorbed

Q = mc\,\Delta T

c = specific heat

Latent heat

Q = mL

no temperature change during phase change

Wave speed

v = f\lambda

speed = frequency × wavelength

Echo distance

d = \frac{v\,t}{2}

sound travels to the wall and back

⚡ Sound speed order: 'Steel Shouts, Water Whispers, Air Awaits'. Solids fastest, liquids next, gases slowest; vacuum — no sound at all (astronauts use radio).

⚡ Pitch–frequency, Loudness–amplitude. 'Pitch = Frequency' (PF like 'Provident Fund'), 'Loudness = Amplitude' (LA like 'Los Angeles').

Light — mirrors, lenses, eye, dispersion and scattering

Power of a lens

P = \frac{1}{f\,(\text{m})} = \frac{100}{f\,(\text{cm})}

unit dioptre

Lens formula

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Cartesian sign convention

Mirror formula

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

f = R/2

Refractive index

n = \frac{c}{v}

c = 3 × 10⁸ m/s

⚡ Eye defect lens: 'My-Con, Hyper-Vex'. Myopia → Concave; Hypermetropia → convex. Short-sighted people cannot see far — a diverging lens pushes the image back onto the retina.

⚡ Mirror by job. Need a wide view → convex (vehicle rear view). Need an enlarged or focused image/beam → concave (shaving, dentist, headlight).

Electricity, magnetism and the EM spectrum

Ohm's law

V = IR

Series resistance

R_s = R_1 + R_2 + \cdots

Parallel resistance

\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

two resistors: R₁R₂/(R₁+R₂)

Electric power

P = VI = I^{2}R = \frac{V^{2}}{R}

Joule heating

H = I^{2}Rt

Resistance of a wire

R = \rho\frac{L}{A}

ρ = resistivity

⚡ Fleming hands: 'Left for Motor, Right for Generator' (L-M, R-G). Remember as the alphabet — L comes before M; G is 'generated' by the Right hand.

⚡ EM spectrum order: 'Good X-rays Use Visible Infra-Micro Radios'. Gamma → X-ray → UV → Visible → IR → Microwave → Radio: wavelength increases, frequency and energy decrease.

Nuclear physics, inventions and scientists

Mass–energy equivalence

E = mc^{2}

Einstein

⚡ Fission vs fusion. Fission = split (heavy → lighter, reactors, atom bomb). Fusion = fuse (light → heavier, Sun, hydrogen bomb).

⚡ Particle discoverers: 'Tom Ran Chasing'. Electron — Thomson (1897); nucleus/proton — Rutherford; neutron — Chadwick (1932).

Chemistry

Matter, separation methods and atomic structure

Moles from mass

n = \frac{m}{M}

M = molar mass in g/mol

Number of particles

N = n \times N_A,\quad N_A = 6.022 \times 10^{23}

Avogadro number

Boyle's law

P_1V_1 = P_2V_2

temperature constant

Charles's law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

pressure constant, T in kelvin

Ideal gas equation

PV = nRT

R = 8.314 J mol⁻¹ K⁻¹

Maximum electrons in a shell

2n^{2}

K = 2, L = 8, M = 18

⚡ Iso-words by the letter. Isotopes — same atomic number (same protons, 'P for isoto-P-es'). Isobars — same A (mass number). Isotones — same neutrons.

⚡ Sublimation set: 'Camphor, Naphthalene, Iodine, Ammonium chloride, Dry ice' — 'CNIAD'. If an option is one of these, it is the sublimating substance.

Periodic table, elements and record-holders

⚡ Crust order: 'OSAIC' — Oxygen, Silicon, Aluminium, Iron, Calcium. So the most abundant element is O and the most abundant metal is Al.

⚡ Two liquids at room temperature. Mercury — metal; Bromine — non-metal. Gallium and caesium melt just above room temperature.

Acids, bases, pH, common compounds and reactions

\text{pH} = -\log_{10}[\text{H}^{+}]

pH + pOH = 14 at 25 °C

Neutralisation

\text{Acid} + \text{Base} \rightarrow \text{Salt} + \text{H}_2\text{O}

Setting of Plaster of Paris

\text{CaSO}_4\cdot\tfrac{1}{2}\text{H}_2\text{O} + 1\tfrac{1}{2}\,\text{H}_2\text{O} \rightarrow \text{CaSO}_4\cdot 2\text{H}_2\text{O}

PoP → gypsum

⚡ Kitchen acid map: 'Vinegar-Acetic, Lemon-Citric, Tamarind-Tartaric, Tomato-Oxalic, Curd-Lactic, Apple-Malic, Ant-Formic'. Pairs that start alike help: Tamarind–Tartaric, Apple–mAlic (think 'mAlus', the apple genus), Formica (Latin for ant)–Formic.

⚡ Phenolphthalein goes 'Pink in Base'. 'Phenol → Pink' only in a base; in acid it stays colourless.

⚡ Vitriol colours by metal. Blue — copper (CuSO₄ crystals are blue), Green — iron (ferrous), White — zinc; Oil of vitriol — H₂SO₄.

Metals, ores, metallurgy and alloys

Gold purity

\text{Purity}\,(\%) = \frac{\text{carat}}{24} \times 100

22 ct ≈ 91.67%

⚡ Reactivity series: 'Please Stop Calling Me A Zebra, I Like Her Calling Me Smart Goat'. Potassium, Sodium, Calcium, Magnesium, Aluminium, Zinc, Iron, Lead, Hydrogen, Copper, Mercury, Silver, Gold.

⚡ Brass vs Bronze. Br-a-ss has Zinc (think 'brass band with Zing'); Bronze has Tin (bronze medal — 'third'/tin). Both are copper-based.

Carbon compounds, fuels, polymers, industry and pollution

Alkane / alkene / alkyne

C_nH_{2n+2},\quad C_nH_{2n},\quad C_nH_{2n-2}

Haber process

N_2 + 3H_2 \rightleftharpoons 2NH_3

iron catalyst

Complete combustion of methane

CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O

⚡ Process → product: 'Haber-Ammonia, Contact-Sulphuric, Ostwald-Nitric, Solvay-Soda'. Pair the catalysts too: Haber — Fe, Contact — V₂O₅, Ostwald — Pt.

⚡ Japanese diseases: 'Mina-Mercury, Itai-Cadmium'. Minamata — Mercury; Itai-itai ('ouch-ouch', bone pain) — Cadmium.

Biology

Cell, organelles and genetics

⚡ Organelle nicknames: 'Power-Mito, Suicide-Lyso, Protein-Ribo, Packing-Golgi'. These four nicknames answer most organelle questions.

⚡ Cell discoverers in order. Hooke saw it (1665, dead cork) → Leeuwenhoek saw it alive → Brown saw the nucleus → Schleiden–Schwann made the theory → Virchow said cells come from cells.

Human body — organs, blood, digestion, glands

Body mass index

\text{BMI} = \frac{\text{mass (kg)}}{\text{height (m)}^{2}}

18.5–24.9 is the normal range (WHO)

⚡ Blood group giving: 'O gives, AB takes'. O has no A/B antigens → can give to all. AB has no anti-A/anti-B antibodies → can take from all.

⚡ Brain parts: 'Cerebrum thinks, Cerebellum balances, Medulla keeps you alive'. Thinking/memory — cerebrum; balance/posture — cerebellum; heartbeat/breathing — medulla oblongata.

Vitamins, minerals and nutrition

⚡ Fat-soluble vitamins: 'KADE' (or 'ADEK'). A, D, E, K dissolve in fat and are stored in the body; B and C dissolve in water.

⚡ Deficiency pairs: 'A-Andha (night blind), B1-Beriberi, C-sCurvy, D-Deformed bones (rickets), K-Klotting'. Add B3 — Pellagra ('3 D's) and B12 — Pernicious anaemia (cobalt).

Diseases, pathogens, vectors and vaccines

⚡ Bacterial diseases: 'TB, Cholera, Typhoid, Diphtheria, Tetanus, Leprosy, Plague, Pertussis'. If the disease is in this list it is bacterial; the rest of the common ones (polio, measles, AIDS, dengue, rabies, chickenpox, hepatitis) are viral. Malaria and kala-azar are protozoan.

⚡ Mosquito map: 'Anopheles-Malaria, Aedes-Dengue, Culex-Filaria'. Aedes also spreads chikungunya, yellow fever and Zika (day-biting). Only the female mosquito bites for blood.

Classification, plants, hormones and animal facts

Photosynthesis

6CO_2 + 6H_2O \xrightarrow[\text{chlorophyll}]{\text{sunlight}} C_6H_{12}O_6 + 6O_2

O₂ comes from water

Aerobic respiration

C_6H_{12}O_6 + 6O_2 \rightarrow 6CO_2 + 6H_2O + \text{energy (ATP)}

⚡ Underground but not a root. Potato, ginger, turmeric, onion grow underground but are stems (they have nodes/buds or 'eyes'). Carrot, radish, sweet potato, beetroot are true roots.

⚡ Five kingdoms: 'My Pet Fish Plays Alone'. Monera, Protista, Fungi, Plantae, Animalia — Whittaker, 1969.

Branches of biology and ecology

10% law of energy transfer

E_{n} = E_{1} \times (0.1)^{\,n-1}

E₁ = energy at the producer level, n = trophic level

⚡ Culture words by root. Api = bee (apiary), Seri = silk, Pisci = fish (Pisces), Viti = vine/grapes, Pomo = fruit (pomegranate), Olera = vegetables.

⚡ Study-of roots. Ornitho = bird, Ento = insect, Ichthyo = fish, Herpeto = creeping animals, Myco = fungus, Phyco = seaweed/algae, Onco = tumour, Nephro = kidney.

Static GK

Books & Authors

⚡ Booker ladder — "Rushdie 81, Roy 97, Desai 06, Adiga 08". Four India-connected Booker winners in order: Midnight's Children → God of Small Things → Inheritance of Loss → White Tiger. Geetanjali Shree (2022) won the International Booker (for translated fiction).

⚡ Sports book hooks. Sunny Days = Sunil Gavaskar · Race of My Life = Milkha (the 'Flying Sikh' raced) · Unbreakable = Mary Kom (boxer) · Ace = Sania (tennis ace) · Test of My Life = Yuvraj (cancer battle).

Sports: Trophies, Terms, Players & Venues

⚡ Badminton cups — "Thomas the man, Uber the woman". Thomas Cup = men's team, Uber Cup = women's team, Sudirman = mixed. Tennis: Davis = Dudes (men), Billie Jean King = women.

⚡ Team size — "Basket 5, Volley 6, Kabaddi 7". Count up: basketball 5, volleyball 6, kabaddi 7; then cricket/football/hockey 11.

Awards & Honours

⚡ First Bharat Ratnas — "RRR 1954". Rajagopalachari, Radhakrishnan, Raman — three Rs in 1954.

⚡ Award decade ladder. 50s Bharat Ratna (54) → 60s Arjuna (61), Jnanpith (65), Phalke (69) → 80s Dronacharya (85) → 90s Khel Ratna (91–92).

Important Days

⚡ Birthday days — pair the person. 12 Jan Vivekananda → youth · 29 Aug Dhyan Chand → sports · 5 Sep Radhakrishnan → teachers · 15 Sep Visvesvaraya → engineers · 14 Nov Nehru → children · 22 Dec Ramanujan → maths.

⚡ Two 'Feb–Nov' science/constitution anchors. 28 Feb = Raman effect (Science Day). 26 Nov = Constitution adopted (26 Jan = enforced → Republic Day).

National Symbols & Firsts

⚡ Women firsts — "Sarojini Governs, Sucheta Chiefs". Governor → Sarojini Naidu (1947) · Chief Minister → Sucheta Kriplani (1963). Both from UP.

⚡ Flag numbers — "3:2 and 24". Length : width = 3 : 2; chakra spokes = 24; designer = Pingali Venkayya.

Organisations, Capitals, Currencies & Parliaments

⚡ Geneva vs Rome vs Nairobi. Geneva = health, trade, labour (WHO, WTO, ILO). Rome = food (FAO, WFP, IFAD). Nairobi = environment (UNEP) — the only major UN HQ in Africa/Global South.

⚡ Neighbour currencies — "Taka, Ngul, Kyat". East side of India: Bangladesh Taka, Bhutan Ngultrum, Myanmar Kyat. Nepal, Sri Lanka and Pakistan use the rupee.

Current Affairs — How to Prepare

How current-affairs questions are framed

⚡ Pairing drill. Read any news item and write its two halves — e.g. 'X award → field', 'Y summit → host'. Exams ask the halves, not the story.

Award categories to track — structure, not winners

⚡ Ladder memory. Civilian: Ratna > Vibhushan > Bhushan > Shri. Sports: Khel Ratna > Arjuna (players) > Dronacharya (coaches) — teachers below students is the joke that fixes the order.

Sports events and how tournament news is asked

⚡ Cycle split. Four-yearly: Olympics, Winter, CWG, Asian Games, FIFA, ODI WC. Two-yearly: T20 WC, Thomas/Uber. Annual: IPL, Ranji, Khelo India.

Summits, groupings and organisations to track

⚡ HQ trio. SCO — Beijing; SAARC — Kathmandu; BIMSTEC — Dhaka — the three most-swapped headquarters. ASEAN sits at Jakarta as the fourth.

Important days — national and UN framework

⚡ Person-day strings. Chain them by date: 12 Jan Vivekananda → 1 Jul B.C. Roy → 29 Aug Dhyan Chand → 5 Sep Radhakrishnan → 31 Oct Patel → 14 Nov Nehru → 22 Dec Ramanujan.

Indexes and reports — index ↔ publisher

⚡ UN vs NGO split. UN bodies: UNDP (HDI), SDSN (Happiness), WIPO (Innovation), IMF (WEO). NGOs: Hunger = Concern + Welthungerhilfe, Peace = IEP, Press = RSF, ASER = Pratham, Corruption = TI.

Computer Knowledge

Computer Fundamentals

Five generations of computers

⚡ Generation ladder mnemonic. Vacuum tube -> Transistor -> IC -> Microprocessor -> AI = "Very Tiny ICs Made AI". Five words = five generations, in order.

⚡ Firsts trio. ENIAC 1946 = first electronic general-purpose computer; UNIVAC-I 1951 = first *commercial* one; Intel 4004 1971 = first *microprocessor*. Remember 1946 -> 1951 -> 1971 as "built, sold, shrunk".

Types of computers

⚡ Size ladder. "Some Ministers Must Promise Speed" - Supercomputer > Mainframe > Minicomputer > PC (Micro) > Smartphone/embedded, largest to smallest.

⚡ PARAM pin. PARAM = India's supercomputer series, built by C-DAC, Pune; PARAM 8000 came in 1991. Pair it with 'Vijay Bhatkar = father of Indian supercomputing'.

Basic organisation: CPU, memory, I/O

⚡ FDES chant. "Fat Dogs Eat Snacks" = Fetch, Decode, Execute, Store - the machine cycle order. Every instruction, every time.

⚡ ALU vs CU. ALU = Actual Labour Unit (does the sums); CU = traffic Constable, Untouched by maths (only directs). If a question says 'performs calculations' -> ALU; 'coordinates/controls' -> CU.

Pioneers and programming-language levels

⚡ Compiler vs Interpreter. Compiler = Complete (whole program first, fast execution, shows all errors together); Interpreter = Instant (line by line, stops at first error). C/C++ -> compiler; Python/JS -> interpreter.

⚡ Who-does-what chain. Babbage Built (the idea), Ada Added (first program), Turing Thought (theory), von Neumann Noted (stored program), Berners-Lee Browsed (WWW).

Core abbreviations and full forms

⚡ ROM family ladder. P -> EP -> EEPROM: each generation gets *easier* to erase - PROM (once, at manufacture/programming), EPROM (erase with UV light), EEPROM (erase electrically, in circuit). "UV for EPROM, Electricity for EEPROM."

⚡ OCR-OMR-MICR trio. OCR reads Characters (printed text -> editable text), OMR reads Marks (exam bubbles), MICR reads Ink on cheques (banks). Characters, Marks, Ink = CRM.

Hardware, Memory & Number Systems

Memory hierarchy, RAM/ROM and cache

⚡ Volatile = Vanishes. Volatile memories Vanish when power goes: registers, cache, RAM. Everything you *keep files on* (ROM, SSD, HDD, pen drive) is non-volatile.

⚡ ROM family eraser. PROM (once) -> EPROM (Erase with Photons = UV) -> EEPROM (Electrically Erasable). The more E's, the more Electrical.

⚡ Cache position. Cache is the CPU's personal notepad between CPU and RAM: L1 inside the core (fastest), then L2, then L3 shared. Any question 'memory between CPU and main memory' = cache.

Memory units and conversions

Step multiplier

1\ \text{unit}_{next} = 2^{10} = 1024\ \text{units}_{prev}

KB->MB->GB->TB->PB all multiply by 1024

Byte identity

1\ \text{Byte} = 8\ \text{bits},\quad 1\ \text{Nibble} = 4\ \text{bits}

half a byte = nibble

Common ladders

1\ \text{GB} = 1024\ \text{MB} = 2^{20}\ \text{KB} = 2^{30}\ \text{B}

4 GB = 4096 MB = 2^22 KB

bits vs bytes

1\ \text{MB} = 8\ \text{Mb}

capital B = byte, small b = bit (speeds are usually bits)

⚡ x1024 ladder. Every step up multiplies by 1024 (2^10), never 1000 in SSC convention. 4 GB = 4 x 1024 = 4096 MB. Half a KB = 512 B.

⚡ Bit-vs-byte speed check. Line speeds are in bits (Mbps), file sizes in bytes (MB). To convert a download speed to MB/s, divide by 8: 8 Mbps = 1 MB/s.

Number systems: binary, octal, decimal, hexadecimal

Positional value

N = \sum d_i \cdot b^{i}

digit d_i at position i (from 0, rightmost), base b

Decimal to base b

N = (\ldots r_2 r_1 r_0)_b\ \text{from repeated division by } b

read remainders bottom to top

Grouping shortcuts

1\ \text{octal digit} \leftrightarrow 3\ \text{bits},\quad 1\ \text{hex digit} \leftrightarrow 4\ \text{bits}

group binary from the RIGHT; pad with leading zeros

⚡ 3-4 grouping. Octal = 3-bit groups, Hex = 4-bit groups, always made from the right (pad the left with zeros). Binary -> octal/hex needs no division at all.

⚡ Hex letter wheel. A=10, B=11, C=12, D=13, E=14, F=15. So 2F = 2x16+15 = 47; FF = 15x16+15 = 255 = 11111111. 'F fills: F=15, FF=255'.

⚡ Power-of-2 positions. Memorise 1,2,4,8,16,32,64,128 - then any binary->decimal is just picking positions: 11010111 = 128+64+16+8+4+2+1 = 215. No working needed beyond addition.

Input and output devices

⚡ Mark-Character-Ink. OMR = Marks (bubbles), OCR = Characters (text), MICR = Ink on cheques (banks). Marks, Characters, Ink - match the noun in the question.

⚡ Impact = hit. If the printer physically *hits* paper (dot matrix, daisy wheel) it is impact - noisy but makes carbon copies. Inkjet/Laser/Thermal never touch with force: non-impact.

Ports and connectors

⚡ Serial = Single lane. Serial sends a Single stream (1 bit at a time, slow, long distance); Parallel sends a Packet (whole byte across 8 wires, fast, short distance - old printers).

⚡ VGA vs HDMI. VGA = Video only (15 pins, analog); HDMI = Home-theatre cable: audio + video, digital. If the question says 'sound and picture together' -> HDMI.

Storage and backup media

⚡ Optical capacity ladder. CD 700 MB -> DVD 4.7 GB -> Blu-ray 25 GB (single layer). Roughly: CD 0.7, DVD 4.7, BR 25 - '700, 4.7, 25'.

⚡ SSD vs HDD one-liner. SSD = no moving parts (flash), silent, shock-proof, faster, costlier per GB. HDD = spinning platters, cheap bulk storage. 'No moving parts' in a question = SSD.

Software & Operating Systems

Types of software: system, application, utility

⚡ SUApp test. Ask: does it Serve the machine (system), Unclog/maintain it (utility), or do the user's App-job (application)? Windows = system, antivirus = utility, Excel = application.

⚡ Licence 4-word key. Freeware = Free forever (closed source); Shareware = Sample first (trial); Open source = Open code; Proprietary = Pay.

Operating system functions, types and booting

⚡ PMF-SUIE services. OS gives six services: Process, Memory, File, Security, UI, Error handling. 'Manage + platform' answers are OS; anything computing payroll is application.

⚡ Boot words. Cold boot = Current was off (switched on from cold); Warm boot = Without power cut (restart). POST always comes first: 'Power On, Self-Test, then load'.

Windows OS features

⚡ PnP = Plug 'n' Pray-free. Plug and Play means the OS auto-detects and auto-configures new hardware - no driver hunt. The phrase 'automatically detected' in a question = PnP.

⚡ Task Manager 3-2-Esc. Fastest route to Task Manager: Ctrl+Shift+Esc (direct). Ctrl+Alt+Del only opens the security menu first. 'Direct' is what SSC tests.

File systems and file management

⚡ FAT32's four-wall. FAT32 cannot hold a single file bigger than 4 GB - the classic exam line. NTFS breaks that wall and adds security permissions + journaling.

⚡ Wildcards. In Windows search, ? masks exactly one character and * masks any number: *.docx finds all Word files, s?t finds sat/set/sit.…

MS Word

Word interface, views and basics

⚡ Views ladder. Default view = Print Layout (what you print is what you see). Reading = book mode; Web = browser look; Outline = headings hierarchy; Draft = bare text. If a question says 'default', Print Layout is the answer.

⚡ Backstage tab. New, Open, Save, Print, Options live under the File tab (Backstage view), not the Home ribbon. Any 'where do you find Print/Options' question -> File tab.

Formatting text, paragraphs and pages

⚡ Alignment letters. The shortcut letter sits inside the word: ctrE? No - simply: Left=Ctrl+L, CEnter=Ctrl+E, Right=Ctrl+R, Justify=Ctrl+J. Only Centre breaks the pattern with E.

⚡ Spacing numbers. Line-spacing shortcuts ARE the spacing: Ctrl+1 single, Ctrl+2 double, Ctrl+5 1.5. (Five looks like S for single-and-a-half.)

⚡ Case cycle. Shift+F3 cycles case: start at 'hello world' -> HELLO WORLD -> Hello World. Press it repeatedly to see all five styles; no menu needed.

Mail merge, references, review and macros

⚡ No style, no TOC. An automatic Table of Contents is built from Heading styles. If the headings were bolded by hand (not styled), the TOC comes out empty - the classic trap.

⚡ Footnote = F below, Endnote = D for 'enD'. Alt+Ctrl+F = Footnote (bottom of page); Alt+Ctrl+D = enDnote (end of document/section).

⚡ Merge field marks. Mail-merge placeholders appear inside guillemets - «Name». Data comes from the data source; the letter layout lives in the main document. Two parts, always.

File types and defaults

⚡ X = XML = 2007+. Across Office: docx/xlsx/pptx = 2007 and later (Office Open XML, zip-compressed); the 3-letter .doc/.xls/.ppt = 97-2003. If a question mentions the X, it means XML.

⚡ Font eras. Word 2003 default = Times New Roman 12. Word 2007-2021 = Calibri 11. Office 365 (mid-2023) & Office 2024 = Aptos 12. Exams still ask Calibri 11 most often.

MS Excel

Workbook, cells and references

⚡ Dollar locks. A $ before the letter locks the column; before the number locks the row. $A$1 = both locked ('dollar = padlock'). F4 taps through the four states.…

⚡ Rows x Cols. 2007+: 2^20 rows (10,48,576) x 2^14 columns (16,384); last column XFD. Old Excel: 65,536 x 256 ('256 = IV roman-ish, XFD = 16384').

Functions you must know

⚡ COUNT counts Numbers, COUNTA counts Anything. COUNT = digits only; COUNTA = everything non-empty (text too); COUNTBLANK = the gaps. Test any MCQ by tagging each cell in the range N (number), T (text), E (empty).

⚡ V of VLOOKUP = Vertical. VLOOKUP hunts down the first column; HLOOKUP along the first row. The 4th argument FALSE = exact match ('F for Full match').

⚡ TODAY vs NOW. TODAY() = date only; NOW() = date + time. Both refresh on recalculation (unlike Ctrl+; which stamps a fixed date).

Formula errors, charts and data tools

⚡ ###### is not an error. Widen the column - done. Real errors start with #: DIV/0 (zero), NAME (spelling), VALUE (type), REF (deleted), N/A (not found). Match the cause table above.

⚡ Chart chooser. Pie = Parts, Line = Line of time (trend), Column = Compare, Scatter = relationship. 'Trend over months' questions always end at Line.

⚡ Goal Seek direction. Goal Seek: you give the answer, it finds the input. Pivot Table: you give the data, it gives summaries.

Excel shortcuts and file facts

⚡ Date stamp duo. Ctrl+; = date, Ctrl+Shift+; = time - the Shift makes it work-o'clock (time). 'Current date in a cell' questions answer Ctrl+;.

⚡ F-key row for Excel. F2 edit, F4 lock, F7 spell, F9 recalc, F11 chart sheet, F12 Save As - the F-row reads like Excel's toolbar.

⚡ Alt+= = AutoSum. Alt with the plus key writes =SUM() over the adjacent numbers instantly - the fastest marks in the section.

MS PowerPoint

Slides, placeholders and file types

⚡ X-family again. pptx/docx/xlsx = 2007+; ppt/doc/xls = 97-2003; the extra M (pptm/xlsm/docm) = macros; ppsx = the self-running show ('s for show).

⚡ M for More slides. Ctrl+M = More slide (new slide). Ctrl+D = Duplicate the current slide. Remember M-then-D: More, then Duplicate.

Views, slide master, transitions vs animations

⚡ Transition = TRansfer. Transition transfers you between slides; animation acts on atoms (objects) inside one slide. If the question says 'between slides', the answer is transition.

⚡ Sorter = order desk. Slide Sorter shows every slide as a thumbnail - the view for rearranging, duplicating and deleting slides quickly. Normal edits content, Sorter edits order.

Slideshow and editing shortcuts

⚡ F5 = Full show. F5 from the First slide; Shift+F5 from where you Stand. Esc = Exit. The F-key is PowerPoint's signature - no other app uses F5 to present.

⚡ B/W = Blackout/Washout. Mid-show, B blanks the screen to black and W washes it to white - for pauses/questions. Pressing the same key again brings the slide back.

Internet, Browsers & E-mail

Internet, WWW, URL and DNS

⚡ Internet vs WWW. The Internet = hardware network (cables, routers, servers); the WWW = content service (hyperlinked pages) riding on it. Invented later (1989) by ONE man - Berners-Lee; the Internet is older (1969 ARPANET).

⚡ DNS = phonebook. DNS translates names to numbers - never 'assigns IPs' (that is DHCP) and never 'sends mail' (that is SMTP). Any question phrased 'converts domain names into IP addresses' = DNS.

Browsers, search engines, downloading and uploading

⚡ Browser vs engine. Browser = the car; search engine = the GPS you ask inside it. Chrome/Edge/Firefox/Safari/Opera = cars. Google/Bing/Yahoo/DuckDuckGo = GPS. If it is a program you install, browser; if it is a website you visit, engine…

⚡ Ctrl-letter browser map. Tab new, Wipe (close) tab, Drop a bookmark, History, Junk... no - Jetisoned files (Downloads), Normal window, N+Shift No-trace (incognito). F5 = freshen.

⚡ Up vs Down. Upload = YOU send up (to the server); Download = server sends down to you. The direction is always relative to YOUR machine.

E-mail: structure, To/CC/BCC and protocols

⚡ CC vs BCC. CC = Carbon Copy - Everyone sees. BCC = Blind - eyes covered, so To/CC recipients cannot see the blind list. 'Who was secretly copied?' - nobody but the sender knows.

⚡ SMTP Sends, POP Picks, IMAP In-sync. SMTP = Send (outgoing). POP3 = Pick up on one device (downloads, server emptied). IMAP = In All devices (server copy stays, everything syncs).

⚡ @ = at. In ray@gmail.com the @ separates 'who' from 'where' - user at domain. Options without @, or with a space/comma, are dead giveaways.

e-Banking and digital payments

⚡ NEFT vs RTGS. RTGS = Really To Go, in batches of one, 2 lakh+ (real-time, gross = one-by-one, big amounts). NEFT = Nice Easy Batches (settlement in half-hourly batches, any amount).

⚡ IFSC 11. IFSC = 11 characters (4 bank code + 0 + 6 branch code), used on NEFT/RTGS/UPI branch routing. MICR is the 9-digit cheque strip.

⚡ 2FA two keys. Two-factor = password (know) + OTP/biometric (have/are). One stolen password is not enough - that is the whole point.

Networking: Devices, Topologies & Protocols

Network types and topologies

⚡ Mesh half-row sum. Full-mesh links for n nodes = n(n-1)/2: 6 nodes -> 6x5/2 = 15. Half of n rows of (n-1) - compute, don't memorise per-n.

⚡ PLMW span ladder. PAN < LAN < MAN < WAN - Personal, Local, Metropolitan, Wide. The Internet is the WAN of WANs.

⚡ Star = spine. Modern offices run star topology around a switch. If a question says 'central device fails, whole network down', it is describing star's weakness.

Network devices

⚡ Hub-Switch-Router ladder. Hub hears and shouts to all (layer 1); Switch = Selective, by MAC (layer 2, same network); Router = Routes between networks by IP (layer 3). Exam verbs: 'broadcasts to all' -> hub; 'MAC address table'…

⚡ Modem = modulator + demodulator. The full form itself is the answer: digital computer data is modulated onto an analog carrier and demodulated back. 'Digital to analog conversion' in a question = modem.

OSI and TCP/IP reference models

⚡ OSI mnemonics. Bottom-up: "Please Do Not Throw Sausage Pizza Away" (Physical, Data link, Network, Transport, Session, Presentation, Application). Top-down: "All People Seem To Need Da…

⚡ Layer-device pairs. Router = layer 3 (both R words rhyme with 'routing/IP'); Switch/Bridge = layer 2 (frames/MAC); Hub/Repeater = layer 1 (bits). Any 'works at which layer' question resolves by the device's address type: IP …

IP addressing: IPv4 classes and IPv6

⚡ Class ruler. Remember three cut-points: 1 - 126 - 128 - 191 - 192 - 223. A starts at 1, B at 128, C at 192, D (multicast) at 224. 127 is the loopback no-man's land between A and B.

⚡ Loopback = 127. 127.0.0.1 = localhost - pinging it tests your own TCP/IP stack, never the cable. Any question with 127.x wants 'loopback'.

⚡ IPv4 vs IPv6 size. IPv4 = 32-bit dotted decimal; IPv6 = 128-bit colon-hex. 'Runs out of addresses' questions -> IPv6 as the fix.

Protocols and port numbers; TCP vs UDP

⚡ Port anchors. Memorise six: 80 HTTP, 443 HTTPS, 25 SMTP, 53 DNS, 110 POP3, 21 FTP - these cover most CKT options. Then '22 SSH, 23 Telnet' (Telnet = the insecure 23).

⚡ TCP = Tracked, UDP = Untracked. TCP Tracks every packet (reliable, slower - web/mail/files); UDP is Unreliable but quick (streaming, gaming, DNS). 'Guaranteed delivery' in the question = TCP.

Cyber Security

Malware types

⚡ V-W-T triangle. Virus needs a Vehicle (host file); Worm Wanders alone; Trojan Tricks you into installing. Replication + host = virus; replication without host = worm; disguise without replication = trojan.

⚡ Ransom = Ransomware. Encrypts your files, demands money -> ransomware (WannaCry). 'Keystrokes recorded' -> keylogger. 'Hides with admin rights' -> rootkit. 'Zombie army' -> botnet.

Cyber attacks and social engineering

⚡ The -ishing family. Phishing = fake mail/site; Vishing = Voice; SMishing = SMS; Pharming = fake site with NO click (DNS poisoned). If the victim 'entered a real-looking site without clicking any link', think pharming.

⚡ MITM = postman reading letters. Man-in-the-Middle: both parties think they talk to each other; the attacker relays (and reads/edits) everything - classic on open Wi-Fi. Pure listening without relaying = sniffing.

Preventive measures and security tools

⚡ AV detects, FW decides. Antivirus = detective (scans files for malware already arriving); Firewall = gatekeeper (allows/blocks traffic by rules). 'Filters network traffic' -> firewall; 'removes virus from a file' -> antivirus.

⚡ 2FA two doors. Two-factor = knowledge (password) + possession/inherence (OTP, token, fingerprint). One stolen password opens only the first door.

IT Act 2000, offences and authorities

⚡ Section story-line. 43 = damage fine, 65 = source code, 66 = computer crime, 66C = identity (C for Credential theft), 66D = cheating by personation (D for Donning a false identity), 66F = terrorism (F for Fearsome), …

⚡ 2000-17-10. IT Act passed and effective in 2000 (in force 17 October 2000), UNCITRAL-based, amended 2008. CERT-In follows in 2004.

⚡ Helpline 1930. Money lost to online fraud? Dial 1930 fast and report at cybercrime.gov.in - the hotline aims to freeze the siphoned money in the banking chain.

Keyboard Shortcuts

Windows 10/11 and File Explorer shortcuts

⚡ Win + first letter. Most Win shortcuts use the first letter of the action: Explorer, Lock, Run, I (settIngs), Search, Desktop, Minimise, Project, V (paste history, like Ctrl+V). Learn the letter and you k…

⚡ Alt = Active window. Alt shortcuts act on the window in front of you: Alt+Tab switches it, Alt+F4 closes it, Alt+PrtScn captures only it, Alt+Enter shows the selected item's properties.

Universal Ctrl keys and MS Word shortcuts

⚡ L-E-R-J alignment. Left, cEntre, Right, Justify. C is already taken by Copy, so centre borrows the E from cEntre.

⚡ Spacing = the number itself. Ctrl+1 gives single spacing, Ctrl+2 gives double, and Ctrl+5 gives 1.5. Five is the odd one out: read it as 1.5.

MS Excel and PowerPoint shortcuts

⚡ Same key, different app. Shift+F3: change case in Word, Insert Function in Excel. Ctrl+D: Font dialog in Word, fill down in Excel, duplicate in PowerPoint, bookmark in a browser. F5: Go To in Word/Excel, slide show in PowerPoint, ref…

⚡ Page keys move pages (sheets). In Excel, Ctrl+PageDown moves to the next sheet and Ctrl+PageUp to the previous one, as if turning pages of the workbook.

Web browser shortcuts (Chrome, Edge, Firefox)

⚡ Shift = undo the tab. Ctrl+T opens a tab and Ctrl+Shift+T brings back the one you closed. Shift adds 'the opposite' or 'the stronger version': Ctrl+Shift+N is a stronger (private) new window, and Ctrl+Shift+Delete is a stronger delete (al…

⚡ Alt+arrows = time travel. Alt+Left goes back in history and Alt+Right goes forward. Backspace no longer goes back in Chrome.

Function keys F1-F12

⚡ 1-2-5-7-12 ladder. F1 = help (first thing you need), F2 = rename (give it a 2nd name), F5 = refresh/show (5 = S for Show), F7 = spell check (7 letters in 'SPELLER'), F12 = Save As (end of the row, you save at the end).

⚡ Shift+F10 = right-click. Shift+F10 opens the context menu, the same one you get with a right mouse click. The Menu key next to right Ctrl does the same.