CA Foundation PAPER 3: QUANTITATIVE APTITUDE Syllabus

What is the value of nC0, nCn, and what does nCr = nC(n−r) state?

nC0 = nCn = 1. The identity nCr = nC(n−r) states that choosing r items is equivalent to leaving out (n−r) items. Also nCr + nC(r−1) = (n+1)Cr (Pascal's rule).

How many ways can n distinct objects be arranged in a circle?

(n − 1)! ways for a circular arrangement. If clockwise and anticlockwise arrangements are considered the same (e.g., a garland/necklace), it is (n − 1)!/2.

How many permutations are there of n objects where there are repeated items (p alike, q alike, r alike)?

n! / (p! × q! × r!), where p, q, r are the counts of each kind of identical object.

State the formula for the nth term of an Arithmetic Progression (AP).

Tn = a + (n − 1)d, where a = first term, d = common difference, n = term number.

State the formula for the sum of the first n terms of an AP.

Sn = n/2 [2a + (n − 1)d] = n/2 (a + l), where a = first term, d = common difference, l = last term.

State the formula for the nth term and the sum of n terms of a Geometric Progression (GP).

nth term: Tn = a·r^(n−1). Sum: Sn = a(r^n − 1)/(r − 1) for r > 1, or a(1 − r^n)/(1 − r) for r < 1, where a = first term, r = common ratio.

What is the formula for the sum of an infinite GP, and when does it apply?

S∞ = a / (1 − r), valid only when |r| < 1 (the common ratio's absolute value is less than 1).

What is the relationship between the Arithmetic Mean (AM) and Geometric Mean (GM) of two positive numbers a and b?

AM = (a + b)/2, GM = √(ab). AM ≥ GM always (equal only when a = b). Also GM² = AM × HM.

Define a set, and distinguish between the subset (⊆) and proper subset (⊂) symbols.

A set is a well-defined collection of distinct objects. A ⊆ B means every element of A is in B (A may equal B). A ⊂ B means A is a subset of B and A ≠ B (proper subset).

If a set has n elements, how many subsets and how many proper subsets does it have?

Number of subsets = 2^n. Number of proper subsets = 2^n − 1 (excluding the set itself). The power set has 2^n elements.

State the formula for n(A ∪ B) and for n(A ∪ B ∪ C) (inclusion–exclusion).

n(A∪B) = n(A) + n(B) − n(A∩B). n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C).

What is the difference between a relation and a function?

A relation is any set of ordered pairs from A×B. A function is a special relation in which every element of the domain (A) is mapped to exactly one element of the codomain (B).

Define one-to-one (injective), onto (surjective), and bijective functions.

Injective (one-one): distinct inputs give distinct outputs. Surjective (onto): every element of the codomain is an image of some input. Bijective: both injective and onto (has an inverse).

What is the derivative of x^n, a constant, e^x, and ln x?

d/dx(x^n) = n·x^(n−1); d/dx(constant) = 0; d/dx(e^x) = e^x; d/dx(ln x) = 1/x.

State the product rule and quotient rule of differentiation.

Product rule: d/dx(uv) = u'v + uv'. Quotient rule: d/dx(u/v) = (u'v − uv') / v².

State the chain rule of differentiation.

If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). In Leibniz form: dy/dx = (dy/du)(du/dx).

How do you find the maxima or minima of a function using calculus?

Set the first derivative f'(x) = 0 to find critical points. Then use the second derivative: if f''(x) < 0 it is a maximum; if f''(x) > 0 it is a minimum; if f''(x) = 0 the test is inconclusive.

In economics, how are Marginal Cost and Marginal Revenue obtained from Total Cost and Total Revenue functions?

Marginal Cost = d(TC)/dx and Marginal Revenue = d(TR)/dx. Profit is maximized where MR = MC (and the slope of MC exceeds MR).

State the power rule for integration and the integral of 1/x and e^x.

∫x^n dx = x^(n+1)/(n+1) + C (n ≠ −1); ∫(1/x) dx = ln|x| + C; ∫e^x dx = e^x + C.

What does the definite integral ∫(from a to b) f(x) dx represent?

It represents the net area under the curve y = f(x) between x = a and x = b. By the Fundamental Theorem: ∫(a to b) f(x)dx = F(b) − F(a), where F is an antiderivative of f.

In a number/letter coding-decoding problem, what is the standard place value (rank) of letters A and Z, and the reverse rank of A?

Forward: A = 1, B = 2, ..., Z = 26. Reverse: A = 26, Z = 1. A useful check: forward rank + reverse rank of any letter = 27 (e.g., opposite of letter at position n is at position 27−n).

In odd-man-out problems, what is the basic approach?

Identify the common rule or property shared by most items (e.g., all prime, all perfect squares, same difference pattern) and pick the one item that does not follow that rule.

In direction tests, if you face North and turn right, then right again, which direction do you face? Give the clockwise order of directions.

Facing North, a right turn faces East, another right turn faces South. Clockwise order: North → East → South → West → North. A right turn is clockwise (90°), a left turn is anticlockwise.

In direction sense problems, how is the shortest (straight-line) distance between start and end points calculated?

Using the Pythagorean theorem: if net horizontal displacement is x and net vertical displacement is y, the shortest distance = √(x² + y²).

What are the main types of seating arrangement problems?

Linear arrangement (in a row, all facing one direction or facing each other), circular arrangement (facing centre or facing outward), and rectangular/square table arrangements. Direction of facing determines left/right interpretation.

In blood relation problems, who is your father's father's only son, and your mother's brother?

Father's father's only son = your father. Mother's brother = your maternal uncle. (Key skill: break the chain step by step and track gender.)

What are the main forms of statistical (graphical) representation of data?

Bar diagram, pie chart (circular diagram), histogram, frequency polygon, ogive (cumulative frequency curve), and line/time-series graphs. Tabulation and textual presentation are non-graphical forms.

What is the difference between an inclusive and an exclusive class interval?

Exclusive: upper limit of one class equals lower limit of next (e.g., 10–20, 20–30); upper limit not included. Inclusive: limits do not overlap (e.g., 10–19, 20–29); must be converted to exclusive (class boundaries) before drawing a histogram.

Distinguish between a census and a sample survey.

Census: data collected from every unit of the entire population (complete enumeration). Sample survey: data collected from a representative subset (sample) of the population, then generalized. Sampling saves time and cost but has sampling error.

Differentiate between a parameter and a statistic.

A parameter is a numerical characteristic of the population (e.g., population mean μ, population SD σ). A statistic is a numerical characteristic computed from a sample (e.g., sample mean x̄, sample SD s) used to estimate the parameter.

Name common probability/random sampling methods.

Simple random sampling (with/without replacement), stratified sampling, systematic sampling, cluster sampling, and multistage sampling. These are probability sampling methods; quota and convenience sampling are non-probability methods.

State the formulas for Mean, Median (for grouped data), and Mode.

Mean = Σfx/Σf. Median (grouped) = L + [(N/2 − cf)/f] × h. Mode = most frequent value; grouped Mode = L + [(f1−f0)/(2f1−f0−f2)] × h.

State the empirical relationship between Mean, Median, and Mode.

Mode = 3 Median − 2 Mean (for a moderately asymmetrical/skewed distribution). For a perfectly symmetrical distribution, Mean = Median = Mode.

What are the main measures of dispersion?

Range, Quartile Deviation (semi-interquartile range), Mean Deviation, Standard Deviation, and Variance. Range and QD are absolute measures based on position; SD and MD are based on all values.

State the formula for Standard Deviation and Variance for a frequency distribution.

Variance σ² = Σf(x − x̄)²/N = Σfx²/N − (Σfx/N)². Standard Deviation σ = √(Variance). SD is the positive square root of variance.

What is the Coefficient of Variation and what does it indicate?

CV = (Standard Deviation / Mean) × 100. It is a relative measure of dispersion used to compare variability of two or more series; a higher CV means less consistency/more variability.

State the classical (mathematical) definition of probability.

P(A) = (Number of favourable outcomes) / (Total number of equally likely outcomes) = m/n. Probability lies between 0 and 1 inclusive.

State the addition theorem of probability for two events.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

State the multiplication theorem and the condition for independent events.

P(A ∩ B) = P(A) × P(B|A). For independent events, P(B|A) = P(B), so P(A ∩ B) = P(A) × P(B).

State Bayes' Theorem and conditional probability formula.

Conditional: P(A|B) = P(A∩B)/P(B). Bayes: P(Ai|B) = [P(Ai)·P(B|Ai)] / Σ[P(Aj)·P(B|Aj)]. It revises prior probabilities using observed evidence.

State the mean and variance of the Binomial distribution.

For Binomial B(n, p): Mean = np, Variance = npq (where q = 1−p), Standard Deviation = √(npq). Note: Mean > Variance always (since q < 1).

State the conditions/properties of a Binomial distribution.

Fixed number of trials n; each trial has only two outcomes (success/failure); constant probability p of success; trials are independent. P(X=r) = nCr · p^r · q^(n−r).

State the mean and variance of the Poisson distribution.

For Poisson with parameter m: Mean = Variance = m. P(X = r) = (e^(−m) · m^r)/r!. It is the limiting case of binomial when n→∞, p→0, np = m (rare events).

State the key properties of the Normal distribution.

Bell-shaped and symmetrical about the mean; Mean = Median = Mode; total area = 1; defined by μ and σ. Empirical rule: about 68% of data lie within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ.

What is the standard normal variate (z-score) and its distribution parameters?

z = (x − μ)/σ. The standard normal distribution has mean = 0 and standard deviation = 1. The z-score measures how many standard deviations a value is from the mean.

What does the correlation coefficient (r) measure and what is its range?

It measures the degree and direction of linear relationship between two variables. r ranges from −1 to +1: +1 = perfect positive, −1 = perfect negative, 0 = no linear correlation.

State the formula for Karl Pearson's coefficient of correlation.

r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)²·Σ(y−ȳ)²], or r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

State the formula for Spearman's rank correlation coefficient.

R = 1 − [6Σd²] / [n(n² − 1)], where d = difference between ranks of paired items and n = number of pairs.

What are the two regression coefficients, and how do they relate to r?

byx = r·(σy/σx) (regression of y on x) and bxy = r·(σx/σy) (regression of x on y). Their product byx × bxy = r², so r = ±√(byx × bxy) (sign same as the regression coefficients).