MHT CET Mathematics Syllabus

What are the product-to-sum formulas for 2 sinA cosB and 2 cosA cosB?

2 sinA cosB = sin(A+B) + sin(A-B); 2 cosA cosB = cos(A+B) + cos(A-B).

State the sine rule and cosine rule for a triangle with sides a, b, c.

Sine rule: a/sinA = b/sinB = c/sinC = 2R. Cosine rule: a² = b² + c² - 2bc cosA.

What is the general solution of sinθ = sinα?

θ = nπ + (-1)^n α, where n ∈ Z.

What is the general solution of cosθ = cosα and of tanθ = tanα?

cosθ = cosα ⇒ θ = 2nπ ± α; tanθ = tanα ⇒ θ = nπ + α, where n ∈ Z.

State the principal value ranges (domains of definition for output) of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

sin⁻¹x ∈ [-π/2, π/2]; cos⁻¹x ∈ [0, π]; tan⁻¹x ∈ (-π/2, π/2).

What is the value of sin⁻¹x + cos⁻¹x, and tan⁻¹x + cot⁻¹x?

sin⁻¹x + cos⁻¹x = π/2 (for x ∈ [-1,1]); tan⁻¹x + cot⁻¹x = π/2 (for all real x).

Give the addition formula for tan⁻¹x + tan⁻¹y (when xy < 1).

tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 - xy)], valid when xy < 1.

What is the slope of a line in terms of its inclination θ, and the slope through two points?

m = tanθ; through (x₁,y₁) and (x₂,y₂): m = (y₂ - y₁)/(x₂ - x₁).

State the conditions for two lines (slopes m₁, m₂) to be parallel and perpendicular.

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

What is the perpendicular distance from point (x₁, y₁) to the line ax + by + c = 0?

Distance = |a x₁ + b y₁ + c| / √(a² + b²).

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

Standard: (x - h)² + (y - k)² = r². General: x² + y² + 2gx + 2fy + c = 0, with centre (-g, -f) and radius √(g² + f² - c).

For the parabola y² = 4ax, state the focus, directrix, and length of latus rectum.

Focus: (a, 0); directrix: x = -a; length of latus rectum = 4a.

For the ellipse x²/a² + y²/b² = 1 (a > b), state the eccentricity and foci.

e = √(1 - b²/a²), with b² = a²(1 - e²); foci at (±ae, 0). Eccentricity e < 1.

For the hyperbola x²/a² - y²/b² = 1, state the eccentricity, foci, and asymptotes.

e = √(1 + b²/a²), with b² = a²(e² - 1); foci at (±ae, 0); asymptotes y = ±(b/a)x. Eccentricity e > 1.

What is the scalar (dot) product a·b in terms of the angle θ between vectors, and the condition for perpendicularity?

a·b = |a||b|cosθ. Vectors are perpendicular when a·b = 0.

What is the magnitude of the vector (cross) product a × b, and the condition for two vectors to be parallel?

|a × b| = |a||b|sinθ (a vector perpendicular to both). Vectors are parallel when a × b = 0.

What does the scalar triple product [a b c] = a·(b × c) represent geometrically and when is it zero?

Its absolute value is the volume of the parallelepiped with edges a, b, c. It is zero when the three vectors are coplanar.

Give the section formula for the point dividing the segment joining vectors a and b in ratio m:n internally.

Position vector = (m·b + n·a)/(m + n). Midpoint (1:1) = (a + b)/2.

What are the 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 angle between two lines in 3D with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂).

cosθ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)].

What is the perpendicular distance from point (x₁,y₁,z₁) to the plane ax + by + cz + d = 0?

Distance = |a x₁ + b y₁ + c z₁ + d| / √(a² + b² + c²).

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

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

State two standard limits: lim(x→0) sinx/x and lim(x→0) (e^x - 1)/x.

lim(x→0) sinx/x = 1; lim(x→0) (e^x - 1)/x = 1. Also lim(x→0) (1 + x)^(1/x) = e.

State the product rule and quotient rule for differentiation.

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

Give the derivatives of sin x, tan x, ln x, and e^x.

d/dx(sinx) = cosx; d/dx(tanx) = sec²x; d/dx(lnx) = 1/x; d/dx(e^x) = e^x.

What is the geometric meaning of the first derivative being zero, and the second derivative test for maxima/minima?

f'(x)=0 gives stationary (critical) points. If f''(x) < 0 it's a local maximum; if f''(x) > 0 it's a local minimum; if f''(x)=0 the test is inconclusive.

State the power rule for integration ∫xⁿ dx and ∫(1/x) dx.

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

State the formula for integration by parts.

∫u dv = uv - ∫v du, i.e. ∫u·v' dx = u·v - ∫u'·v dx (choose u by ILATE order).

What is the formula for the area bounded by the curve y = f(x), the x-axis, and lines x = a, x = b?

Area = ∫(from a to b) |f(x)| dx (take f(x) ≥ 0 over the interval, or split where it changes sign).

Define the order and degree of a differential equation.

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

What is the integrating factor method for a linear differential equation dy/dx + Py = Q?

Integrating Factor IF = e^(∫P dx); solution: y·(IF) = ∫Q·(IF) dx + C.

State the conditional probability formula and the multiplication rule.

P(A|B) = P(A∩B)/P(B), P(B) ≠ 0. Multiplication: P(A∩B) = P(B)·P(A|B) = P(A)·P(B|A).

State Bayes' Theorem.

P(A_i | B) = [P(A_i)·P(B|A_i)] / Σ[P(A_j)·P(B|A_j)], where the A_j form a partition of the sample space.

For a binomial distribution with n trials and success probability p, what is P(X = r), and the mean and variance?

P(X=r) = nCr · p^r · q^(n-r), where q = 1-p. Mean = np; variance = npq.

In mathematical logic, what is the truth value of a conditional p → q, and when is it false?

p → q is false only when p is true and q is false; it is true in all other cases (T→T, F→T, F→F are all true).

What is the contrapositive of p → q, and how does its truth value relate to the original?

The contrapositive is ~q → ~p; it is logically equivalent to p → q (same truth value).

State De Morgan's Laws in logic for the negation of a conjunction and a disjunction.

~(p ∧ q) ≡ ~p ∨ ~q; ~(p ∨ q) ≡ ~p ∧ ~q.

Distinguish a tautology, a contradiction, and a contingency in logic.

Tautology: a statement true for all truth values. Contradiction: false for all truth values. Contingency: true for some and false for others.

In Linear Programming, where does the optimal value of the objective function occur?

At a vertex (corner point) of the feasible region; for a bounded feasible region the optimum always occurs at one of its corner points.

Define the feasible region and the objective function in a Linear Programming Problem.

Feasible region: the set of all points satisfying every constraint (including non-negativity). Objective function: the linear expression (e.g. Z = ax + by) to be maximized or minimized over that region.