ACET (Institute of Actuaries of India Exams) cheat sheet

Quadratic formula: x = [-b ± √(b²−4ac)] / 2a; the discriminant b²−4ac decides two real, one real, or no real roots.

Laws of indices: a^m·a^n = a^(m+n); a^m/a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1; a^(−n) = 1/a^n.

Logarithm rules: log(ab) = log a + log b; log(a/b) = log a − log b; log(a^n) = n·log a; log_b b = 1; log_b 1 = 0.

Arithmetic progression: nth term a + (n−1)d; sum = (n/2)[2a + (n−1)d]. Geometric progression: nth term ar^(n−1); sum = a(r^n−1)/(r−1).

Sum identities: 1+2+…+n = n(n+1)/2; 1²+2²+…+n² = n(n+1)(2n+1)/6.

Circle: area = πr², circumference = 2πr; sector area = ½r²θ and arc length = rθ when θ is in radians.

Trigonometry: sin²θ + cos²θ = 1; sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc·cos A.

Coordinate geometry: distance = √[(x₂−x₁)² + (y₂−y₁)²]; midpoint = ((x₁+x₂)/2, (y₁+y₂)/2); slope = (y₂−y₁)/(x₂−x₁).

Permutations and combinations: nPr = n!/(n−r)! (order matters); nCr = n!/[r!(n−r)!] (order does not matter); nCr = nC(n−r).

Mean = sum of values / count; it is sensitive to outliers, while the median is not.

Median: for n values in order, take the middle value when n is odd and the average of the two middle values when n is even.

Mode is the most frequent value; a data set can be bimodal or have no mode at all.

Range = maximum − minimum; the interquartile range IQR = Q3 − Q1 covers the middle 50% of the data.

Variance of a population = Σ(x−μ)²/N; standard deviation is the positive square root of variance.

For a normal distribution, roughly 68% of values lie within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.

Probability of an event = favourable outcomes / total equally likely outcomes; P(A∪B) = P(A) + P(B) − P(A∩B).

Independent events satisfy P(A∩B) = P(A)·P(B); mutually exclusive events satisfy P(A∩B) = 0.

Correlation r lies between −1 and +1; r near 0 means weak linear association, and correlation does not imply causation.

Percentage change = ((new − old)/old) × 100; a rise of x% followed by a fall of x% leaves a net fall of x²/100 percent.

If A is x% more than B, then B is (100x/(100+x))% less than A; the base of the percentage always matters.

Ratio scaling: if a:b = 3:5 and the total is T, the parts are 3T/8 and 5T/8; keep units consistent before comparing.

Average speed for equal distances = 2xy/(x+y) (harmonic mean), not the arithmetic mean of the two speeds.

In a pie chart, a sector angle = (component/total) × 360°; in a bar chart, compare bar heights only from a zero baseline.

Syllogism rule: a conclusion follows only if it holds in every arrangement of the premises; a statement like 'some A are B' does not imply 'some B are not A'.

For seating or ordering puzzles, build a table of positions and fix the most constrained clue first, then eliminate contradictions.

Number-series rule: check successive differences, then ratios, then alternating or prime-based patterns before assuming a single rule.

Subject–verb agreement: a singular subject takes a singular verb; phrases between subject and verb (e.g. 'along with') do not change the number.

Collective nouns (team, committee) take a singular verb when acting as one unit and a plural verb when members act individually.

'Fewer' modifies countable nouns; 'less' modifies uncountable nouns; 'amount' pairs with uncountable, 'number' with countable.

Active voice: subject performs the action; passive voice: object becomes subject and the verb uses be + past participle.

Direct to indirect speech: shift tense back one step and change pronouns and time words (now → then, today → that day).

Conditionals: first (if + present, will), second (if + past, would), third (if + had + past participle, would have + past participle).

Punctuation: a semicolon joins two independent clauses; a colon introduces a list or explanation; commas set off non-defining clauses.

Formal communication uses full forms, impersonal tone, and precise verbs; contractions and slang belong to informal registers.