CBSE Class 12 Maths 30 Most Expected Questions 2027 — Board Exam

In short: These 30 questions cover the most frequently asked topics in CBSE Class 12 Maths board exams. Based on previous year analysis and CBSE sample papers — practice these for maximum marks.

Paper Pattern Overview

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Relations and Functions (8 Marks)

Q1. Equivalence Relation ⭐⭐⭐

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a - b| is even} is an equivalence relation.

Approach:

• Prove Reflexive: (a, a) ∈ R for all a
• Prove Symmetric: (a, b) ∈ R ⟹ (b, a) ∈ R
• Prove Transitive: (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R

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Q2. Inverse of Function ⭐⭐

If f: R → R is defined by f(x) = 3x + 2, show that f is bijective and find f⁻¹.

Solution:

• One-one: f(x₁) = f(x₂) ⟹ x₁ = x₂
• Onto: For every y ∈ R, ∃ x = (y-2)/3
• f⁻¹(x) = (x - 2)/3

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Q3. Composition of Functions ⭐⭐

If f(x) = x² and g(x) = 2x + 1, find (fog)(x) and (gof)(x).

Solution:

• (fog)(x) = f(g(x)) = f(2x+1) = (2x+1)²
• (gof)(x) = g(f(x)) = g(x²) = 2x² + 1

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Inverse Trigonometric Functions (4 Marks)

Q4. Simplification ⭐⭐⭐

Prove that: tan⁻¹(1/2) + tan⁻¹(1/3) = π/4

Solution: Using tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] = tan⁻¹[(1/2 + 1/3)/(1 - 1/6)] = tan⁻¹[(5/6)/(5/6)] = tan⁻¹(1) = π/4

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Q5. Principal Value ⭐⭐

Find the principal value of: sin⁻¹(sin 3π/5)

Solution: 3π/5 is not in [-π/2, π/2] sin(3π/5) = sin(π - 3π/5) = sin(2π/5) Principal value = 2π/5

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Matrices and Determinants (10 Marks)

Q6. Matrix Equation ⭐⭐⭐

If A = [2 3; 1 2], find A⁻¹ and verify that AA⁻¹ = I.

Solution: |A| = 4 - 3 = 1 adj(A) = [2 -3; -1 2] A⁻¹ = adj(A)/|A| = [2 -3; -1 2]

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Q7. System of Equations ⭐⭐⭐

Solve using matrices: 2x + 3y + z = 9, x + 2y + 3z = 6, 3x + y + 2z = 8

Approach:

• Write as AX = B
• Find A⁻¹
• X = A⁻¹B

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Q8. Properties of Determinants ⭐⭐

Without expanding, prove that:

|1  a  a²|
|1  b  b²| = (a-b)(b-c)(c-a)
|1  c  c²|

Approach: Use row operations R₂ → R₂ - R₁, R₃ → R₃ - R₁

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Continuity and Differentiability (8 Marks)

Q9. Continuity ⭐⭐⭐

Find the value of k for which f(x) is continuous at x = 0:

f(x) = (sin 2x)/(5x), x ≠ 0
     = k,            x = 0

Solution: lim(x→0) sin2x/5x = (2/5) × lim(x→0) sin2x/2x = 2/5 For continuity: k = 2/5

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Q10. Logarithmic Differentiation ⭐⭐⭐

Differentiate: y = xˣ + (sin x)^(cos x)

Approach:

• Let u = xˣ, v = (sin x)^(cos x)
• Take log, differentiate
• dy/dx = du/dx + dv/dx

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Q11. Second Derivative ⭐⭐

If y = sin⁻¹x, prove that (1-x²)y₂ - xy₁ = 0

Solution: y₁ = 1/√(1-x²) Differentiate again and simplify

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Applications of Derivatives (8 Marks)

Q12. Rate of Change ⭐⭐

A ladder 5m long is leaning against a wall. The bottom slides away at 2 cm/s. How fast is the top sliding down when the bottom is 4m from the wall?

Solution: x² + y² = 25 2x(dx/dt) + 2y(dy/dt) = 0 When x = 4, y = 3 dy/dt = -8/3 cm/s

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Q13. Maxima and Minima ⭐⭐⭐

Find the maximum and minimum values of f(x) = sin x + cos x on [0, π].

Solution: f'(x) = cos x - sin x = 0 ⟹ x = π/4 f(0) = 1, f(π/4) = √2, f(π) = -1 Maximum = √2 at x = π/4 Minimum = -1 at x = π

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Q14. Increasing/Decreasing ⭐⭐

Find intervals where f(x) = x³ - 3x² + 4x is strictly increasing.

Solution: f'(x) = 3x² - 6x + 4 = 3(x-1)² + 1 > 0 for all x f(x) is strictly increasing on R

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Integrals (12 Marks)

Q15. Integration by Parts ⭐⭐⭐

Evaluate: ∫ x² eˣ dx

Solution: Using ILATE: ∫ x²eˣ dx = x²eˣ - 2∫xeˣ dx = x²eˣ - 2(xeˣ - eˣ) + C = eˣ(x² - 2x + 2) + C

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Q16. Partial Fractions ⭐⭐⭐

Evaluate: ∫ (x+1)/[(x-1)(x-2)] dx

Solution: (x+1)/[(x-1)(x-2)] = A/(x-1) + B/(x-2) A = -2, B = 3 = -2ln|x-1| + 3ln|x-2| + C

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Q17. Definite Integral Property ⭐⭐

Evaluate: ∫₀^(π/2) log(tan x) dx

Solution: Using property: ∫₀^a f(x)dx = ∫₀^a f(a-x)dx I = ∫₀^(π/2) log(cot x) dx = -I 2I = 0 ⟹ I = 0

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Q18. Area Under Curve ⭐⭐⭐

Find the area bounded by y = x² and y = x.

Solution: Intersection: x² = x ⟹ x = 0, 1 Area = ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/6 sq units

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Differential Equations (8 Marks)

Q19. Variable Separable ⭐⭐

Solve: dy/dx = (1 + y²)/(1 + x²)

Solution: dy/(1+y²) = dx/(1+x²) tan⁻¹y = tan⁻¹x + C

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Q20. Linear Differential Equation ⭐⭐⭐

Solve: dy/dx + y/x = x²

Solution: IF = e^∫(1/x)dx = x y × x = ∫x³ dx = x⁴/4 + C y = x³/4 + C/x

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Q21. Homogeneous Equation ⭐⭐

Solve: (x² - y²)dx + 2xy dy = 0

Approach: Put y = vx, dy = v dx + x dv Separate and integrate

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Vectors (6 Marks)

Q22. Scalar Triple Product ⭐⭐⭐

Find the volume of parallelepiped with edges a = 2i - 3j, b = i + j - k, c = 3i - k

Solution: Volume = |[a b c]| = |determinant|

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Q23. Vector Product ⭐⭐

Find a unit vector perpendicular to both a = 2i + j + k and b = i - j + 2k

Solution: a × b = |i j k | |2 1 1 | |1 -1 2 | = 3i - 3j - 3k Unit vector = (i - j - k)/√3

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Three Dimensional Geometry (8 Marks)

Q24. Shortest Distance ⭐⭐⭐

Find the shortest distance between lines:

r = (1-t)i + (t-2)j + (3-2t)k
r = (s+1)i + (2s-1)j - (2s+1)k

Formula: d = |[(a₂-a₁)·(b₁×b₂)]|/|b₁×b₂|

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Q25. Equation of Plane ⭐⭐

Find the equation of plane passing through (1, 2, 3) and perpendicular to planes x + y + z = 1 and 2x - y + z = 2.

Approach: Normal = n₁ × n₂ Use point-normal form

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Linear Programming (5 Marks)

Q26. LPP ⭐⭐⭐

Maximize Z = 3x + 4y subject to:

x + y ≤ 4
x ≥ 0, y ≥ 0

Solution: Corner points: (0,0), (4,0), (0,4) Z values: 0, 12, 16 Maximum Z = 16 at (0, 4)

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Probability (8 Marks)

Q27. Bayes' Theorem ⭐⭐⭐

Bag I has 3 red, 4 black balls. Bag II has 5 red, 6 black balls. One bag is chosen at random and a ball is drawn. If the ball is red, find the probability it came from Bag I.

Solution: P(I|R) = P(R|I)P(I) / [P(R|I)P(I) + P(R|II)P(II)] = (3/7 × 1/2) / (3/7 × 1/2 + 5/11 × 1/2) = 33/68

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Q28. Probability Distribution ⭐⭐⭐

A random variable X has the following distribution:

X:    0    1    2    3
P(X): 1/8  3/8  3/8  1/8

Find E(X) and Var(X).

Solution: E(X) = Σx·P(x) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 12/8 = 1.5 E(X²) = 0 + 3/8 + 12/8 + 9/8 = 24/8 = 3 Var(X) = E(X²) - [E(X)]² = 3 - 2.25 = 0.75

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Q29. Binomial Distribution ⭐⭐

If X follows binomial distribution with n = 5 and P(X = 0) = 1/32, find P(X = 2).

Solution: P(X = 0) = C(5,0)p⁰q⁵ = q⁵ = 1/32 q = 1/2, p = 1/2 P(X = 2) = C(5,2)(1/2)²(1/2)³ = 10/32 = 5/16

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Q30. Conditional Probability ⭐⭐

Two cards are drawn without replacement from a deck. Find the probability that both are kings.

Solution: P(K₁ ∩ K₂) = P(K₁) × P(K₂|K₁) = (4/52) × (3/51) = 12/2652 = 1/221

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Chapter-wise Question Distribution

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Frequently Asked Questions

Which chapters are most important for Class 12 Maths?

1. Calculus (Integration, Applications) — 25+ marks
2. Matrices & Determinants — 10 marks
3. Probability — 8 marks
4. Vectors & 3D Geometry — 10 marks

How to score 90+ in Maths board exam?

1. Master NCERT examples and exercises
2. Practice previous year questions
3. Focus on presentation and steps
4. Attempt all questions, don't leave any blank

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