ACET (Institute of Actuaries of India Exams) flashcards

31 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)] / 2a. The sign before the square root gives two solutions.

Discriminant

The value b² − 4ac from a quadratic. Positive gives two distinct real roots, zero gives one repeated real root, negative gives no real roots.

Arithmetic progression sum

For a first term a, common difference d and n terms, the sum is (n/2)[2a + (n−1)d]. The nth term is a + (n−1)d.

Geometric progression sum

For first term a, common ratio r (r ≠ 1) and n terms, the sum is a(rⁿ − 1)/(r − 1). The nth term is arⁿ⁻¹.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its matching angle. Use it when you know a side and its opposite angle.

Cosine rule

a² = b² + c² − 2bc·cos A. It finds a third side from two sides and the included angle, or an angle from all three sides.

Combination formula

nCr = n!/[r!(n−r)!] counts selections where order does not matter. Use nPr = n!/(n−r)! when order does matter.

Logarithm product rule

log(ab) = log a + log b for positive a and b. Related rules: log(a/b) = log a − log b and log(aⁿ) = n·log a.

Sector arc length

With radius r and central angle θ in radians, arc length = rθ and sector area = ½r²θ. Convert degrees first by multiplying by π/180.

Median

Arrange the values in order. With an odd count, take the middle value; with an even count, average the two middle values. It resists outliers.

Standard deviation

The positive square root of the variance. It expresses spread in the same units as the data, so it is easier to interpret than variance.

Interquartile range

IQR = Q3 − Q1, the spread of the middle 50% of ordered data. It is less affected by extreme values than the range.

Mean sensitivity to outliers

The mean uses every value, so one extreme observation can shift it strongly. The median changes little, which makes it robust for skewed data.

Normal distribution empirical rule

About 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Addition rule of probability

P(A∪B) = P(A) + P(B) − P(A∩B). Subtract the overlap so outcomes counted in both events are not double-counted.

Independent events

Two events are independent when P(A∩B) = P(A)·P(B), meaning the occurrence of one does not change the probability of the other.

Mutually exclusive events

Events that cannot occur together, so P(A∩B) = 0 and P(A∪B) = P(A) + P(B). They are not the same as independent events.

Correlation coefficient

A number r between −1 and +1 measuring the strength and direction of linear association. Values near zero indicate weak linear relationship.

Percentage change

Percentage change = ((new − old)/old) × 100. Always divide by the original value, which is the base of the comparison.

Successive percentage change

A rise of x% then a fall of x% gives a net fall of x²/100 percent, not zero, because the second change uses a different base.

Average speed, equal distances

For two equal distances covered at speeds x and y, average speed = 2xy/(x+y). This harmonic mean is lower than the arithmetic mean.

Pie chart sector angle

Each sector angle equals (component ÷ total) × 360°. Multiply the angle by total/360 to recover an unknown component value.

Syllogism validity rule

A conclusion is valid only if it must be true whenever the premises are true. A single counterexample arrangement makes it invalid.

Ratio part value

If a quantity splits in ratio a:b, the parts are aT/(a+b) and bT/(a+b), where T is the total. Convert units before applying the ratio.

Subject–verb agreement

A verb agrees with its subject in number. Intervening phrases such as 'along with' or 'as well as' do not make a singular subject plural.

Fewer versus less

Use 'fewer' with countable nouns (fewer chairs) and 'less' with uncountable nouns (less water). 'Number' pairs with countable, 'amount' with uncountable.

Active versus passive voice

Active voice has the subject performing the action. Passive voice makes the object the subject and uses be plus a past participle.

Third conditional

Form: if + past perfect, would have + past participle. It describes an unreal past situation and its imagined result, which did not occur.

Semicolon versus colon

A semicolon joins two independent clauses. A colon introduces a list, quotation or explanation and must follow a complete clause.

Collective noun agreement

A collective noun such as team or committee takes a singular verb when acting as one unit, and a plural verb when members act individually.

Direct to indirect speech

Shift the tense back one step, adjust pronouns to the reporter, and change time words: now becomes then, today becomes that day.