KCET Mathematics Syllabus

State the formula for the number of combinations of n distinct objects taken r at a time.

C(n, r) = nCr = n!/(r!(n − r)!).

State the relation between nPr and nCr.

nPr = nCr × r!, i.e., permutations equal combinations times the arrangements of the chosen r objects.

How many distinct permutations are there of n objects where some are alike (p alike of one kind, q of another)?

n!/(p! q! ...), dividing by the factorials of the counts of identical objects.

State the general (r+1)th term in the binomial expansion of (a + b)^n.

T(r+1) = nCr · a^(n−r) · b^r, for r = 0, 1, ..., n.

How is the middle term of the binomial expansion of (a + b)^n determined?

If n is even, there is one middle term, the (n/2 + 1)th term. If n is odd, there are two middle terms, the ((n+1)/2)th and ((n+3)/2)th terms.

State the formula for the nth term and sum of the first n terms of an arithmetic progression (AP).

nth term: a_n = a + (n−1)d. Sum: S_n = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

State the formula for the nth term and sum of the first n terms of a geometric progression (GP).

nth term: a_n = ar^(n−1). Sum: S_n = a(r^n − 1)/(r − 1) for r ≠ 1. Infinite sum (|r| < 1): S∞ = a/(1 − r).

What is the relationship (inequality) between the arithmetic mean (AM) and geometric mean (GM) of two positive numbers?

AM ≥ GM, i.e., (a + b)/2 ≥ √(ab), with equality when a = b.

State the principle of mathematical induction.

To prove P(n) for all n ≥ 1: (1) Base step — prove P(1) is true. (2) Inductive step — assume P(k) is true and prove P(k+1) is true. Then P(n) holds for all natural numbers.

Give the sum formulas for the first n natural numbers, their squares, and their cubes.

Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

State the slope-intercept and point-slope forms of the equation of a straight line.

Slope-intercept: y = mx + c. Point-slope: y − y₁ = m(x − x₁).

State the formula for the distance of a point (x₁, y₁) from the line Ax + By + C = 0.

Distance = |Ax₁ + By₁ + C| / √(A² + B²).

What is the condition for two lines with slopes m₁ and m₂ to be parallel or perpendicular?

Parallel: m₁ = m₂. Perpendicular: m₁·m₂ = −1.

State the standard equation of a circle with centre (h, k) and radius r.

(x − h)² + (y − k)² = r².

State the standard equation of a parabola opening rightward and its focus and directrix.

y² = 4ax; focus at (a, 0); directrix x = −a; vertex at origin.

State the standard equation of an ellipse (a > b) and the formula for its eccentricity.

x²/a² + y²/b² = 1; eccentricity e = √(1 − b²/a²), with 0 < e < 1.

State the standard equation of a hyperbola and its eccentricity.

x²/a² − y²/b² = 1; eccentricity e = √(1 + b²/a²), with e > 1.

State the distance formula between two points in three-dimensional space.

Distance = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).

What are direction cosines of a line, and what relation do they satisfy?

Direction cosines l, m, n are the cosines of the angles the line makes with the x, y, z axes. They satisfy l² + m² + n² = 1.

State the formula for the section that divides the join of two points internally in ratio m:n in 3D.

Point = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)).

State the conditions for a function f to be continuous at a point x = a.

f is continuous at a if: (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) lim(x→a) f(x) = f(a).

State the standard limit lim(x→0) sin(x)/x and lim(x→0) (1 − cos x)/x.

lim(x→0) sin(x)/x = 1; lim(x→0) (1 − cos x)/x = 0.

State the derivatives of sin x, cos x, tan x, and ln x.

d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x; d/dx(ln x) = 1/x.

State the product rule and quotient rule for differentiation.

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

State the chain rule for differentiating a composite function y = f(g(x)).

dy/dx = f'(g(x)) · g'(x), i.e., dy/dx = (dy/du)(du/dx).

What is the first derivative test for determining local maxima and minima?

At a critical point where f'(x) = 0: if f' changes from + to −, it is a local maximum; if from − to +, a local minimum; if no sign change, neither.

State the second derivative test for local extrema.

At a critical point c where f'(c) = 0: if f''(c) < 0, local maximum; if f''(c) > 0, local minimum; if f''(c) = 0, the test is inconclusive.

State the integrals of 1/x, e^x, and sec²x.

∫(1/x)dx = ln|x| + C; ∫e^x dx = e^x + C; ∫sec²x dx = tan x + C.

State the power rule for integration of x^n.

∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ −1.

State the integration by parts formula.

∫u dv = uv − ∫v du.

State the Fundamental Theorem of Calculus (evaluation part).

If F is an antiderivative of f on [a, b], then ∫(a to b) f(x) dx = F(b) − F(a).

What is the order and degree of a differential equation?

Order = highest derivative present. Degree = power of the highest-order derivative, after the equation is made polynomial (free of radicals/fractions) in derivatives.

How do you solve a linear first-order differential equation dy/dx + Py = Q?

Find the integrating factor IF = e^(∫P dx). Then solution: y·(IF) = ∫Q·(IF) dx + C.

How do you solve a variable-separable differential equation?

Write it as f(y)dy = g(x)dx, then integrate both sides: ∫f(y)dy = ∫g(x)dx + C.

Define the dot (scalar) product of two vectors and state its formula.

a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃, where θ is the angle between them. The result is a scalar.

Define the cross (vector) product of two vectors and its magnitude.

a × b is a vector perpendicular to both a and b, with magnitude |a||b|sin θ. It equals the determinant with rows i,j,k; a₁,a₂,a₃; b₁,b₂,b₃.

State the conditional probability formula P(A|B).

P(A|B) = P(A ∩ B)/P(B), provided P(B) ≠ 0.

State Bayes' theorem for two events.

P(A|B) = [P(B|A)·P(A)] / P(B), where P(B) = P(B|A)P(A) + P(B|A')P(A').

State the multiplication theorem of probability for independent events A and B.

If A and B are independent, P(A ∩ B) = P(A)·P(B).

State the formulas for mean and variance of a binomial distribution B(n, p).

Mean = np; Variance = npq, where q = 1 − p.

State the formula for the variance of a set of n observations using the mean.

Variance σ² = (1/n)Σ(xᵢ − x̄)², where x̄ is the mean. Standard deviation σ = √(variance).

What is the relationship between mean, median, and mode in a moderately skewed distribution?

Empirical relation: Mode = 3·Median − 2·Mean.

State the condition for two matrices to be multipliable and the order of the product.

Matrix A (m×n) can multiply B (p×q) only if n = p; the product AB has order m×q.

What is the formula for the inverse of a square matrix A in terms of its adjoint and determinant?

A⁻¹ = (1/|A|)·adj(A), provided |A| ≠ 0 (A is non-singular).

State the property of determinants when two rows (or columns) are interchanged.

Interchanging two rows or two columns changes the sign of the determinant.

Define a symmetric and a skew-symmetric matrix.

Symmetric: A' = A (aᵢⱼ = aⱼᵢ). Skew-symmetric: A' = −A (aᵢⱼ = −aⱼᵢ, with all diagonal elements zero).

State Cramer's rule conditions for a system of linear equations using determinants.

For a system AX = B, if |A| = D ≠ 0, a unique solution exists: x = D₁/D, y = D₂/D, etc. If D = 0, the system has no unique solution (inconsistent or infinitely many).