MP Board 12th Maths Model Paper 2027 — Full Paper With Solutions

This is a complete model paper for MP Board Class 12th Mathematics 2027 based on the official MPBSE blueprint. Solve this under timed conditions — 3 hours for 100 marks.

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Paper Structure

Section	Question Type	Marks Each	Total
Section A	Objective (MCQ)	1	5
Section B	Very Short Answer	2	10
Section C	Short Answer	3	15
Section D	Long Answer	4	16
Section E	Numericals/Proofs	Variable	54

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SECTION A — Objective Questions (1 Mark Each)

Q1. The value of sin⁻¹(sin 3π/5) is:

(a) 3π/5 (b) 2π/5 (c) π/5 (d) −2π/5

Answer: (b) 2π/5

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Q2. If A is a square matrix of order 3 and |A| = 4, then |2A| is:

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

Answer: (c) 32
(Reason: |kA| = k³|A| for a 3×3 matrix, so |2A| = 8 × 4 = 32)

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Q3. The derivative of sin(x²) with respect to x is:

(a) cos(x²) (b) 2x cos(x²) (c) 2cos(x²) (d) x cos(x²)

Answer: (b) 2x cos(x²)

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Q4. The value of ∫₀¹ x² dx is:

(a) 1 (b) 1/2 (c) 1/3 (d) 2/3

Answer: (c) 1/3

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Q5. If P(A) = 0.4 and P(B) = 0.5, and A and B are independent events, then P(A∩B) is:

(a) 0.9 (b) 0.2 (c) 0.1 (d) 0.45

Answer: (b) 0.2
(Reason: P(A∩B) = P(A) × P(B) = 0.4 × 0.5 = 0.2 for independent events)

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SECTION B — Very Short Answer (2 Marks Each)

Q6. Find the principal value of tan⁻¹(−√3).

Solution: tan⁻¹(−√3) = −tan⁻¹(√3) = −π/3

Since the principal value range of tan⁻¹ is (−π/2, π/2), the answer is −π/3.

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Q7. If the matrix A = [[2, 3], [1, 4]], find the transpose of A.

Solution: Transpose A^T = [[2, 1], [3, 4]]
(Rows become columns and columns become rows.)

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Q8. Differentiate log(sin x) with respect to x.

Solution: d/dx [log(sin x)] = (1/sin x) × cos x = cos x / sin x = cot x

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Q9. Evaluate ∫ (1 + cos 2x) dx.

Solution: 1 + cos 2x = 2cos²x
∫ 2cos²x dx = 2 × (x/2 + sin2x/4) + C = x + sin2x/2 + C

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Q10. Find the direction cosines of the line joining (1, 2, 3) and (4, 6, 3).

Solution: Direction ratios = (4−1, 6−2, 3−3) = (3, 4, 0)
Magnitude = √(9 + 16 + 0) = √25 = 5
Direction cosines = (3/5, 4/5, 0)

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SECTION C — Short Answer (3 Marks Each)

Q11. Prove that 2 sin⁻¹(3/5) = tan⁻¹(24/7).

Solution:
Let sin⁻¹(3/5) = θ, so sin θ = 3/5, cos θ = 4/5, tan θ = 3/4
Then 2θ has: sin 2θ = 2(3/5)(4/5) = 24/25, cos 2θ = 1 − 2(9/25) = 7/25
Therefore tan 2θ = 24/7
Thus 2θ = tan⁻¹(24/7), i.e., 2 sin⁻¹(3/5) = tan⁻¹(24/7) ∎

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Q12. Find the intervals where f(x) = x³ − 12x is increasing and decreasing.

Solution:
f'(x) = 3x² − 12 = 3(x² − 4) = 3(x−2)(x+2)
f'(x) > 0 when x < −2 or x > 2 → increasing on (−∞, −2) ∪ (2, ∞)
f'(x) < 0 when −2 < x < 2 → decreasing on (−2, 2)

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Q13. Evaluate ∫ x/(x² + 1) dx.

Solution:
Let t = x² + 1, then dt = 2x dx, so x dx = dt/2
∫ x/(x²+1) dx = ∫ (1/t)(dt/2) = (1/2) log|t| + C = (1/2) log(x² + 1) + C

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Q14. Find the area of the region bounded by the parabola y = x² and the line y = x.

Solution:
Intersection: x² = x → x = 0 or x = 1
Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 sq. units

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Q15. A bag contains 3 red and 4 blue balls. Two balls are drawn at random without replacement. Find the probability that both are red.

Solution:
P(both red) = C(3,2)/C(7,2) = 3/21 = 1/7

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SECTION D — Long Answer (4 Marks Each)

Q16. If A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find A⁻¹ using elementary row transformations.

Solution:
|A| = 1(0−24) − 2(0−20) + 3(0−5) = −24 + 40 − 15 = 1 ≠ 0, so inverse exists.

Using cofactor method:
A₁₁ = −24, A₁₂ = 20, A₁₃ = −5
A₂₁ = 18, A₂₂ = −15, A₂₃ = 4
A₃₁ = 5, A₃₂ = −4, A₃₃ = 1

A⁻¹ = (1/|A|) × adjA =
[[-24, 18, 5], [20, -15, -4], [-5, 4, 1]] (since |A| = 1)

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Q17. Find the equation of the plane passing through points (1, 1, 0), (1, 2, 1) and (−2, 2, −1).

Solution:
Let plane equation be ax + by + cz = d
From the three points:

• a + b = d
• a + 2b + c = d
• −2a + 2b − c = d

Solving: From eq1 and eq2: b + c = 0, so b = −c
From eq2 and eq3: −3a + 0 − 2c = 0 → a = −2c/3
Let c = −3: a = 2, b = 3, d = 2+3 = 5
Equation: 2x + 3y − 3z = 5

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SECTION E — Practice Problems (Self Test — No Answers)

Solve these independently to test your preparation:

1. Using matrices, solve the system: 2x + y = 5, 3x + 2y = 8
2. Find the maximum and minimum values of f(x) = 2x³ − 15x² + 36x + 1
3. Evaluate ∫₀^(π/2) sin²x dx
4. Find the general solution of the differential equation dy/dx + y = e^x
5. Two cards are drawn from a deck of 52. Find P(both are kings)

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Important Chapters by Marks

Chapter	Marks in Exam
Calculus (Integrals + Derivatives)	44
Algebra (Matrices + Determinants)	13
Vectors and 3D Geometry	17
Linear Programming	6
Probability	10
Relations and Functions	10

Strategy: Calculus alone is 44% of the paper. If you master integration and differentiation completely, you secure nearly half the paper.