CBSE Class 12 Maths Important Questions 2027 Chapter Wise with Solutions

संक्षेप में: CBSE Class 12 Maths का paper 80 marks का होता है। Calculus से 44% (35 marks), Algebra से 10% (8 marks), Vectors & 3D से 17% (14 marks), Linear Programming से 6% (5 marks), और Probability से 13% (10 marks) questions आते हैं। इस guide में सभी 13 chapters के most important questions with solutions दिए हैं।

CBSE Class 12 Maths Paper Pattern 2027

Unit-wise Marks Distribution

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Chapter 1: Relations and Functions

इस chapter से 3-5 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. Let A = {1, 2, 3}. The number of equivalence relations on A containing (1, 2) is:

   • (a) 1
   • (b) 2 ✓
   • (c) 3
   • (d) 4

2. If f: R → R is defined by f(x) = 3x + 2, then f⁻¹(x) is:

   • (a) (x - 2)/3 ✓
   • (b) (x + 2)/3
   • (c) 3x - 2
   • (d) 3x + 2

3. The function f: N → N defined by f(n) = n + 1 is:

   • (a) One-one and onto
   • (b) One-one but not onto ✓
   • (c) Onto but not one-one
   • (d) Neither one-one nor onto

4. If f(x) = x² and g(x) = √x, then (gof)(x) is:

   • (a) x²
   • (b) √x
   • (c) |x| ✓
   • (d) x

5. A relation R on set A is symmetric if:

   • (a) (a, b) ∈ R ⟹ (b, a) ∈ R ✓
   • (b) (a, a) ∈ R for all a
   • (c) (a, b), (b, c) ∈ R ⟹ (a, c) ∈ R
   • (d) None of these

Important Questions (2-5 marks)

Q1. (3 marks) Check whether the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive, symmetric, or transitive.

Solution:

• Reflexive: (1,1), (2,2), (3,3) ∈ R ✓ Yes
• Symmetric: (1,2) ∈ R but (2,1) ∉ R ✗ No
• Transitive: (1,2), (2,3) ∈ R but (1,3) ∉ R ✗ No

Q2. (3 marks) Show that the function f: R → R defined by f(x) = 2x + 3 is one-one and onto.

Solution:

• One-one: Let f(x₁) = f(x₂)
  • 2x₁ + 3 = 2x₂ + 3
  • x₁ = x₂ ✓
• Onto: For any y ∈ R, x = (y-3)/2 ∈ R
  • f(x) = 2((y-3)/2) + 3 = y ✓

Q3. (5 marks) Let A = R - {3} and B = R - {1}. Consider the function f: A → B defined by f(x) = (x - 2)/(x - 3). Show that f is one-one and onto. ⭐

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Chapter 2: Inverse Trigonometric Functions

इस chapter से 3-5 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. The principal value of sin⁻¹(1/2) is:

   • (a) π/6 ✓
   • (b) π/3
   • (c) 5π/6
   • (d) -π/6

2. tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) equals:

   • (a) 0
   • (b) π ✓
   • (c) π/2
   • (d) 2π

3. The value of cos⁻¹(cos(7π/6)) is:

   • (a) 7π/6
   • (b) 5π/6 ✓
   • (c) π/6
   • (d) π/3

4. sin⁻¹x + cos⁻¹x equals:

   • (a) 0
   • (b) π/2 ✓
   • (c) π
   • (d) 2π

5. The domain of sin⁻¹(2x) is:

   • (a) [-1, 1]
   • (b) [-1/2, 1/2] ✓
   • (c) [0, 1]
   • (d) R

Important Questions (2-5 marks)

Q1. (2 marks) Find the value of tan⁻¹(1) + cos⁻¹(-1/2) + sin⁻¹(-1/2).

Solution:

• tan⁻¹(1) = π/4
• cos⁻¹(-1/2) = 2π/3
• sin⁻¹(-1/2) = -π/6
• Sum = π/4 + 2π/3 - π/6 = 3π/12 + 8π/12 - 2π/12 = 9π/12 = 3π/4

Q2. (3 marks) 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) + tan⁻¹(1/3) = tan⁻¹((1/2 + 1/3)/(1 - 1/6)) = tan⁻¹((5/6)/(5/6)) = tan⁻¹(1) = π/4

Q3. (3 marks) Simplify: tan⁻¹(cosx/(1 + sinx))

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Chapter 3: Matrices

इस chapter से 4-6 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. If A is a square matrix such that A² = A, then (I + A)³ - 7A equals:

   • (a) A
   • (b) I - A
   • (c) I ✓
   • (d) 3A

2. If A = [aᵢⱼ] is a 3×3 matrix and Aᵢⱼ is the cofactor of aᵢⱼ, then a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ equals:

   • (a) 0
   • (b) 1
   • (c) |A| ✓
   • (d) -|A|

3. If A is a 3×3 matrix such that |A| = 5, then |adj A| equals:

   • (a) 5
   • (b) 25 ✓
   • (c) 125
   • (d) 1/5

4. The number of all possible matrices of order 2×2 with each entry 0 or 1 is:

   • (a) 8
   • (b) 16 ✓
   • (c) 32
   • (d) 64

5. If A = [1 2; 3 4], then A⁻¹ exists if:

   • (a) |A| = 0
   • (b) |A| ≠ 0 ✓
   • (c) A is singular
   • (d) A is not square

Important Questions (2-5 marks)

Q1. (3 marks) If A = [2 3; 1 2], find A⁻¹ and verify that AA⁻¹ = I.

Solution:

• |A| = 4 - 3 = 1 ≠ 0, so A⁻¹ exists
• adj A = [2 -3; -1 2]
• A⁻¹ = (1/|A|) × adj A = [2 -3; -1 2]

Verification: AA⁻¹ = [2 3; 1 2][2 -3; -1 2] = [1 0; 0 1] = I ✓

Q2. (5 marks) Using matrices, solve the system of equations: ⭐

• x + 2y + z = 7
• x + 3z = 11
• 2x - 3y = 1

Solution: Matrix form: AX = B where A = [1 2 1; 1 0 3; 2 -3 0], X = [x; y; z], B = [7; 11; 1]

Find |A|, adj A, then X = A⁻¹B

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Chapter 4: Determinants

इस chapter से 4-6 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. If |A| = 2, then |3A| for a 3×3 matrix equals:

   • (a) 6
   • (b) 18
   • (c) 54 ✓
   • (d) 8

2. The value of |1 a a²; 1 b b²; 1 c c²| is:

   • (a) (a-b)(b-c)(c-a) ✓
   • (b) abc
   • (c) 0
   • (d) 1

3. If A is a 3×3 matrix and |A| = 8, then |A'| equals:

   • (a) 8 ✓
   • (b) -8
   • (c) 1/8
   • (d) 64

4. Area of triangle with vertices (0,0), (4,0), (0,3) using determinants is:

   • (a) 12
   • (b) 6 ✓
   • (c) 24
   • (d) 3

5. If |2x 5; 8 x| = |6 5; 8 3|, then x equals:

   • (a) 3
   • (b) ±3 ✓
   • (c) 6
   • (d) ±6

Important Questions (2-5 marks)

Q1. (3 marks) Using properties of determinants, prove that: |a-b-c 2a 2a; 2b b-c-a 2b; 2c 2c c-a-b| = (a+b+c)³ ⭐

Q2. (3 marks) Find the area of the triangle with vertices (1, 0), (6, 0), (4, 3) using determinants.

Solution: Area = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| = (1/2)|1(0-3) + 6(3-0) + 4(0-0)| = (1/2)|-3 + 18 + 0| = (1/2)(15) = 7.5 sq units

Q3. (5 marks) Solve using Cramer's rule: ⭐

• 2x + 3y + z = 9
• x + 2y + 3z = 6
• 3x + y + 2z = 8

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Chapter 5: Continuity and Differentiability

इस chapter से 6-8 marks के questions आते हैं। यह Calculus का important chapter है।

MCQ Questions (1 mark each)

1. The function f(x) = |x| is:

   • (a) Continuous everywhere ✓
   • (b) Differentiable everywhere
   • (c) Not continuous at x = 0
   • (d) None of these

2. If y = sin⁻¹x, then dy/dx equals:

   • (a) 1/√(1-x²) ✓
   • (b) -1/√(1-x²)
   • (c) 1/√(1+x²)
   • (d) -1/√(1+x²)

3. d/dx(eˣ) equals:

   • (a) eˣ ✓
   • (b) xeˣ⁻¹
   • (c) eˣ/x
   • (d) eˣ⁺¹

4. If y = log(sinx), then dy/dx equals:

   • (a) cotx ✓
   • (b) tanx
   • (c) secx
   • (d) cosecx

5. The derivative of xˣ is:

   • (a) xˣ(1 + logx) ✓
   • (b) xˣ⁻¹
   • (c) xˣlogx
   • (d) xxˣ⁻¹

Important Questions (2-5 marks)

Q1. (2 marks) Find the value of k if the function f(x) = {kx + 1, if x ≤ 5; 3x - 5, if x > 5} is continuous at x = 5.

Solution: For continuity at x = 5:

• LHL = k(5) + 1 = 5k + 1
• RHL = 3(5) - 5 = 10
• f(5) = 5k + 1

5k + 1 = 10 k = 9/5

Q2. (3 marks) If y = (sinx)ˣ, find dy/dx. ⭐

Solution: Taking log: log y = x log(sinx)

Differentiating: (1/y)(dy/dx) = log(sinx) + x(cosx/sinx) dy/dx = y[log(sinx) + x cotx] dy/dx = (sinx)ˣ[log(sinx) + x cotx]

Q3. (5 marks) If y = (cos⁻¹x)², prove that (1-x²)y₂ - xy₁ - 2 = 0 ⭐

Q4. (3 marks) Differentiate: tan⁻¹((√(1+x²) - 1)/x) with respect to x.

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Chapter 6: Application of Derivatives

इस chapter से 6-8 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. The rate of change of area of a circle with respect to its radius r when r = 3 cm is:

   • (a) 6π cm²/cm ✓
   • (b) 3π cm²/cm
   • (c) 9π cm²/cm
   • (d) 12π cm²/cm

2. The function f(x) = x³ - 3x is:

   • (a) Increasing in (-1, 1)
   • (b) Decreasing in (-1, 1) ✓
   • (c) Increasing everywhere
   • (d) Decreasing everywhere

3. The maximum value of sinx + cosx is:

   • (a) 1
   • (b) √2 ✓
   • (c) 2
   • (d) 1/√2

4. The slope of tangent to y = x² at x = 2 is:

   • (a) 2
   • (b) 4 ✓
   • (c) 8
   • (d) 1

5. If f(x) = x² - 4x + 5, then the minimum value of f(x) is:

   • (a) 0
   • (b) 1 ✓
   • (c) 4
   • (d) 5

Important Questions (2-5 marks)

Q1. (3 marks) Find the intervals in which f(x) = 2x³ - 9x² + 12x - 5 is increasing or decreasing.

Solution: f'(x) = 6x² - 18x + 12 = 6(x² - 3x + 2) = 6(x-1)(x-2)

Critical points: x = 1, 2

Q2. (5 marks) Find the equation of tangent and normal to the curve y = x³ at the point (1, 1). ⭐

Solution:

• dy/dx = 3x²
• At (1, 1): slope = 3(1)² = 3

Tangent: y - 1 = 3(x - 1) → 3x - y - 2 = 0

Normal: y - 1 = (-1/3)(x - 1) → x + 3y - 4 = 0

Q3. (5 marks) A rectangular sheet of tin 45 cm × 24 cm is to be made into a box without top by cutting off squares from corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum? ⭐

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Chapter 7: Integrals

इस chapter से 8-10 marks के questions आते हैं। यह सबसे important chapter है।

MCQ Questions (1 mark each)

1. ∫eˣ(sinx + cosx)dx equals:

   • (a) eˣsinx + C ✓
   • (b) eˣcosx + C
   • (c) eˣ(sinx - cosx) + C
   • (d) eˣ + C

2. ∫dx/(x² + a²) equals:

   • (a) (1/a)tan⁻¹(x/a) + C ✓
   • (b) tan⁻¹(x/a) + C
   • (c) (1/a)sin⁻¹(x/a) + C
   • (d) log|x + √(x²+a²)| + C

3. ∫₀^π sinx dx equals:

   • (a) 0
   • (b) 1
   • (c) 2 ✓
   • (d) -2

4. ∫x²eˣdx equals:

   • (a) eˣ(x² - 2x + 2) + C ✓
   • (b) eˣ(x² + 2x + 2) + C
   • (c) x²eˣ + C
   • (d) eˣx² - 2xeˣ + C

5. ∫₀^(π/2) log(tanx)dx equals:

   • (a) 0 ✓
   • (b) 1
   • (c) π/2
   • (d) π/4

Important Questions (2-5 marks)

Q1. (3 marks) Evaluate: ∫dx/(x² - 6x + 13)

Solution: x² - 6x + 13 = (x-3)² + 4

∫dx/((x-3)² + 2²) = (1/2)tan⁻¹((x-3)/2) + C

Q2. (3 marks) Evaluate: ∫(x + 1)/(x² + x + 1)dx ⭐

Solution: Let x + 1 = A(2x + 1) + B Comparing: A = 1/2, B = 1/2

∫(x+1)/(x²+x+1)dx = (1/2)∫(2x+1)/(x²+x+1)dx + (1/2)∫dx/(x²+x+1)

= (1/2)log|x²+x+1| + (1/2) × (2/√3)tan⁻¹((2x+1)/√3) + C

Q3. (5 marks) Evaluate: ∫₀^(π/2) (sinx - cosx)/(1 + sinx cosx)dx ⭐

Q4. (5 marks) Evaluate using integration by parts: ∫x²sin⁻¹x dx ⭐

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Chapter 8: Application of Integrals

इस chapter से 5-6 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. Area bounded by y = x², x-axis and x = 2 is:

   • (a) 4/3
   • (b) 8/3 ✓
   • (c) 2/3
   • (d) 16/3

2. Area of region bounded by y = |x| and y = 2 is:

   • (a) 2
   • (b) 4 ✓
   • (c) 8
   • (d) 1

3. Area enclosed by y = sinx, x-axis from 0 to π is:

   • (a) 1
   • (b) 2 ✓
   • (c) π
   • (d) 0

Important Questions (5 marks)

Q1. (5 marks) Find the area of the region bounded by the curve y² = 4x and the line x = 3. ⭐

Solution: y² = 4x → y = ±2√x

Area = 2∫₀³ 2√x dx = 4[x^(3/2)/(3/2)]₀³ = 4 × (2/3) × 3^(3/2) = (8/3) × 3√3 = 8√3 sq units

Q2. (5 marks) Find the area of the region bounded by the ellipse x²/16 + y²/9 = 1. ⭐

Solution: Area = 4 × ∫₀⁴ (3/4)√(16-x²) dx = 3∫₀⁴ √(16-x²) dx = 3 × (1/2)[x√(16-x²) + 16sin⁻¹(x/4)]₀⁴ = 3 × (1/2) × 16 × (π/2) = 12π sq units

Q3. (5 marks) Find the area of the region bounded by y = x² and y = x.

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Chapter 9: Differential Equations

इस chapter से 5-6 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. The order of the differential equation (d²y/dx²)³ + (dy/dx)² + y = 0 is:

   • (a) 1
   • (b) 2 ✓
   • (c) 3
   • (d) 6

2. The degree of dy/dx + sin(dy/dx) = 0 is:

   • (a) 1
   • (b) 2
   • (c) Not defined ✓
   • (d) 0

3. The integrating factor of dy/dx + y/x = x² is:

   • (a) x ✓
   • (b) 1/x
   • (c) eˣ
   • (d) logx

4. Solution of dy/dx = y/x is:

   • (a) y = cx ✓
   • (b) y = c/x
   • (c) xy = c
   • (d) y = x + c

5. The general solution of dy/dx = eˣ⁺ʸ is:

   • (a) eˣ + e⁻ʸ = c ✓
   • (b) eˣ + eʸ = c
   • (c) e⁻ˣ + eʸ = c
   • (d) eˣ - eʸ = c

Important Questions (2-5 marks)

Q1. (3 marks) Solve: dy/dx + 2y tanx = sinx

Solution: This is linear DE: dy/dx + Py = Q P = 2tanx, Q = sinx

IF = e^∫2tanx dx = e^(2log|secx|) = sec²x

Solution: y × sec²x = ∫sinx × sec²x dx y sec²x = ∫secx tanx dx = secx + C y = cosx + C cos²x

Q2. (5 marks) Solve: (x² - y²)dx + 2xy dy = 0 ⭐

Q3. (5 marks) Solve: x dy/dx + y = x logx ⭐

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Chapter 10: Vector Algebra

इस chapter से 5-6 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. If |a⃗| = 3, |b⃗| = 4 and a⃗·b⃗ = 6, then |a⃗ × b⃗| equals:

   • (a) 6
   • (b) 6√3 ✓
   • (c) 12
   • (d) 6√2

2. If a⃗ × b⃗ = c⃗ × d⃗ and a⃗ × c⃗ = b⃗ × d⃗, then:

   • (a) a⃗ - d⃗ is parallel to b⃗ - c⃗ ✓
   • (b) a⃗ + d⃗ is parallel to b⃗ + c⃗
   • (c) a⃗ - b⃗ is parallel to c⃗ - d⃗
   • (d) None of these

3. The projection of a⃗ = 2î - ĵ + k̂ on b⃗ = î - 2ĵ + k̂ is:

   • (a) 5/√6 ✓
   • (b) √6/5
   • (c) 5/6
   • (d) 6/5

4. If a⃗ and b⃗ are perpendicular, then a⃗·b⃗ equals:

   • (a) 1
   • (b) -1
   • (c) 0 ✓
   • (d) |a⃗||b⃗|

5. The area of parallelogram with adjacent sides a⃗ and b⃗ is:

   • (a) a⃗·b⃗
   • (b) |a⃗ × b⃗| ✓
   • (c) a⃗ × b⃗
   • (d) |a⃗·b⃗|

Important Questions (2-5 marks)

Q1. (3 marks) Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), C(1, 5, 5).

Solution: AB⃗ = î + 2ĵ + 3k̂ AC⃗ = 0î + 4ĵ + 3k̂

AB⃗ × AC⃗ = |î ĵ k̂; 1 2 3; 0 4 3| = î(6-12) - ĵ(3-0) + k̂(4-0) = -6î - 3ĵ + 4k̂

|AB⃗ × AC⃗| = √(36 + 9 + 16) = √61

Area = (1/2)|AB⃗ × AC⃗| = √61/2 sq units

Q2. (3 marks) Find the unit vector perpendicular to both a⃗ = 2î + ĵ + k̂ and b⃗ = î - ĵ + 2k̂. ⭐

Q3. (5 marks) Prove that [a⃗ + b⃗, b⃗ + c⃗, c⃗ + a⃗] = 2[a⃗, b⃗, c⃗] ⭐

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Chapter 11: Three Dimensional Geometry

इस chapter से 6-8 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. The direction cosines of a line equally inclined to the axes are:

   • (a) (1, 1, 1)
   • (b) (1/√3, 1/√3, 1/√3) ✓
   • (c) (1/3, 1/3, 1/3)
   • (d) (√3, √3, √3)

2. The angle between the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-4)/5 = (y-1)/2 = z/1 is:

   • (a) cos⁻¹(20/√(29×30))
   • (b) cos⁻¹(20/(√29 × √30)) ✓
   • (c) 90°
   • (d) 0°

3. The distance of point (1, 2, 3) from the plane x + y + z = 11 is:

   • (a) 5/√3 ✓
   • (b) √3/5
   • (c) 5√3
   • (d) 3/√5

4. The equation of plane passing through (1, 2, 3) and parallel to xy-plane is:

   • (a) x = 1
   • (b) y = 2
   • (c) z = 3 ✓
   • (d) x + y + z = 6

5. The shortest distance between the lines r⃗ = î + ĵ + λ(2î - ĵ + k̂) and r⃗ = 2î + ĵ - k̂ + μ(3î - 5ĵ + 2k̂) is:

   • (a) 10/√59 ✓
   • (b) √59/10
   • (c) 59/10
   • (d) 10/59

Important Questions (2-5 marks)

Q1. (3 marks) Find the equation of the plane passing through (1, 1, 1), (1, -1, 1) and (-7, -3, -5).

Q2. (5 marks) Find the shortest distance between the lines: ⭐

• r⃗ = (1-t)î + (t-2)ĵ + (3-2t)k̂
• r⃗ = (s+1)î + (2s-1)ĵ - (2s+1)k̂

Q3. (5 marks) Find the equation of the plane passing through the intersection of planes x + y + z = 1 and 2x + 3y + 4z = 5 and perpendicular to the plane x - y + z = 0. ⭐

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Chapter 12: Linear Programming

इस chapter से 5 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. The corner points of the feasible region determined by x + y ≤ 4, x ≥ 0, y ≥ 0 are:

   • (a) (0, 0), (4, 0), (0, 4) ✓
   • (b) (0, 0), (4, 4)
   • (c) (4, 0), (0, 4)
   • (d) (0, 0), (4, 0), (4, 4), (0, 4)

2. The maximum value of Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0 is:

   • (a) 12
   • (b) 16 ✓
   • (c) 7
   • (d) 0

3. In a LPP, if the objective function Z = ax + by has the same maximum value on two corner points, then:

   • (a) The maximum occurs at every point on the line joining them ✓
   • (b) The maximum occurs only at one point
   • (c) There is no maximum
   • (d) None of these

Important Questions (5 marks)

Q1. (5 marks) Maximize Z = 5x + 3y subject to: ⭐

• 3x + 5y ≤ 15
• 5x + 2y ≤ 10
• x ≥ 0, y ≥ 0

Solution: Corner points: (0, 0), (2, 0), (0, 3), (20/19, 45/19)

Maximum Z = 235/19 at (20/19, 45/19)

Q2. (5 marks) A manufacturer produces two types of steel trunks. He has two machines A and B. For completing the first type, it requires 3 hours on machine A and 3 hours on machine B. For the second type, it requires 3 hours on machine A and 2 hours on machine B. Machine A can work for 18 hours and B for 15 hours per day. He earns a profit of ₹30 and ₹25 per trunk of first and second type respectively. How many trunks of each type should he make to maximize profit? ⭐

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Chapter 13: Probability

इस chapter से 8-10 marks के questions आते हैं।

MCQ Questions (1 mark each)

1. If P(A) = 0.4, P(B) = 0.3 and P(A∩B) = 0.2, then P(A|B) equals:

   • (a) 2/3 ✓
   • (b) 1/2
   • (c) 1/3
   • (d) 1/4

2. If A and B are independent events with P(A) = 0.3 and P(B) = 0.4, then P(A∩B) equals:

   • (a) 0.12 ✓
   • (b) 0.7
   • (c) 0.1
   • (d) 0.5

3. The mean of a binomial distribution is 5 and variance is 4. Then n equals:

   • (a) 20
   • (b) 25 ✓
   • (c) 10
   • (d) 15

4. If X follows binomial distribution with n = 5 and p = 1/2, then P(X = 3) equals:

   • (a) 5/16
   • (b) 10/32 ✓
   • (c) 1/32
   • (d) 5/32

5. For two events A and B, P(A∪B) = P(A) + P(B) if:

   • (a) A and B are independent
   • (b) A and B are mutually exclusive ✓
   • (c) P(A|B) = P(A)
   • (d) P(B|A) = P(B)

Important Questions (2-5 marks)

Q1. (3 marks) A bag contains 4 red and 4 black balls. Another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag. Find the probability that the ball drawn is red.

Solution: P(Bag 1) = 1/2, P(Bag 2) = 1/2 P(Red|Bag 1) = 4/8 = 1/2 P(Red|Bag 2) = 2/8 = 1/4

P(Red) = P(Bag 1) × P(Red|Bag 1) + P(Bag 2) × P(Red|Bag 2) = (1/2)(1/2) + (1/2)(1/4) = 1/4 + 1/8 = 3/8

Q2. (5 marks) There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it was the two-headed coin? ⭐

Solution: Let A₁ = two-headed, A₂ = biased, A₃ = unbiased P(A₁) = P(A₂) = P(A₃) = 1/3

P(H|A₁) = 1, P(H|A₂) = 3/4, P(H|A₃) = 1/2

P(H) = (1/3)(1) + (1/3)(3/4) + (1/3)(1/2) = 1/3 + 1/4 + 1/6 = 9/12 = 3/4

By Bayes' theorem: P(A₁|H) = P(A₁) × P(H|A₁) / P(H) = (1/3 × 1) / (3/4) = (1/3) × (4/3) = 4/9

Q3. (5 marks) A random variable X has the following probability distribution: ⭐

Find: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) Mean of X

Q4. (5 marks) Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that: (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade?

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Case Study Questions — Sample

Case Study 1 (4 marks): A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is ₹50 each for type A and ₹60 each for type B.

(i) Formulate the LPP. (ii) Find the maximum profit. (iii) How many souvenirs of each type should be made?

Case Study 2 (4 marks): In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random:

(i) Find P(student opted for NCC or NSS) (ii) Find P(student opted for NCC but not NSS) (iii) Find P(student opted for NSS given that student opted for NCC)

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Frequently Asked Questions

CBSE Class 12 Maths में 80 में से 70+ marks कैसे लाएं?

1. Calculus (35 marks) — Integration और Differentiation पर focus करो
2. 3D Geometry (14 marks) — Formulas याद करो, numericals practice करो
3. Probability (8 marks) — Bayes' theorem और Binomial distribution master करो
4. Case Studies (12 marks) — Previous year case studies solve करो

कौन से chapters सबसे ज्यादा marks carry करते हैं?

1. Integrals (Ch 7) — 8-10 marks
2. Three Dimensional Geometry (Ch 11) — 6-8 marks
3. Probability (Ch 13) — 8-10 marks
4. Application of Derivatives (Ch 6) — 5-6 marks

Integration की तैयारी कैसे करें?

• सभी standard formulas याद करो
• Integration by parts के लिए ILATE rule याद करो
• Partial fractions के सभी cases practice करो
• Daily 10-15 integration problems solve करो

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