class 12 Maths Practice paper free download - Set 1

 

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📝 CBSE Class 12 Mathematics Sample Question Paper 2026

(All Chapters | As Per Latest CBSE Pattern | With Answer Key | 80 Marks)

Time: 3 Hours
Maximum Marks: 80

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📌 General Instructions

1. All questions are compulsory.
2. The question paper consists of 5 Sections: A, B, C, D and E.
3. Use of calculator is not permitted.
4. Draw neat diagrams wherever required.

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🔹 SECTION A (1 × 20 = 20 Marks)

Very Short Answer Type Questions

1. If A is a 3×3 identity matrix, then |A| equals:
   (a) 0 (b) 1 (c) 3 (d) –1
2. If f(x) = |x|, then f is:
   (a) Differentiable everywhere
   (b) Not continuous at 0
   (c) Continuous but not differentiable at 0
   (d) Neither continuous nor differentiable
3. The derivative of sin⁻¹x is:
   (a) 1/√(1−x²)
   (b) −1/√(1−x²)
   (c) √(1−x²)
   (d) 1/(1+x²)
4. ∫ dx/(1+x²) equals:
   (a) tan⁻¹x + C
   (b) sin⁻¹x + C
   (c) ln|x| + C
   (d) cot⁻¹x + C
5. If vectors a and b are perpendicular, then a·b equals:
   (a) 1 (b) 0 (c) |a||b| (d) −1
6. The direction ratios of x/2 = y/3 = z/4 are:
   (a) (2,3,4)
   (b) (1/2,1/3,1/4)
   (c) (4,3,2)
   (d) (3,2,4)
7. The probability of getting a head in a fair coin toss is:
   (a) 1 (b) 0 (c) 1/2 (d) 1/4
8. If P(A) = 0.6, P(B) = 0.5 and P(A∩B) = 0.3, then P(A|B) equals:
   (a) 0.6 (b) 0.5 (c) 0.3 (d) 0.4
9. The order of differential equation dy/dx = x² + y² is:
   (a) 2 (b) 1 (c) 3 (d) 0
10. lim x→0 (sin x)/x equals:
    (a) 0 (b) 1 (c) ∞ (d) −1

11–20. (Continue similarly…)

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🔹 SECTION B (2 × 6 = 12 Marks)

Short Answer Type Questions

21. Find determinant of matrix
    | 1 2 |
    | 3 4 |
22. Differentiate y = x³ + 3x².
23. Evaluate ∫ x dx.
24. Find unit vector in direction of i + j + k.
25. Find equation of line passing through (1,2,3) and parallel to (2,1,1).
26. A die is thrown once. Find probability of getting a number greater than 4.

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🔹 SECTION C (3 × 8 = 24 Marks)

Short Answer Type Questions

27. Evaluate determinant:

| 1 1 1 |
| 1 2 3 |
| 1 3 6 |

28. Find dy/dx if y = sin x · cos x.
29. Evaluate ∫ x e^x dx.
30. Find area bounded by y = x and x-axis from x=0 to x=2.
31. Find angle between vectors
    a = i + 2j + 2k
    b = 2i + j + 2k
32. Find equation of plane passing through (1,1,1) and perpendicular to vector 2i + j + 3k.
33. Solve differential equation dy/dx = 2x.
34. A bag contains 3 red and 2 blue balls. Find probability of red ball.

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🔹 SECTION D (4 × 5 = 20 Marks)

Long Answer Type Questions

35. Find inverse of matrix

| 2 1 |
| 1 1 |

36. Find local maxima and minima of
    f(x) = x³ − 3x² + 2
37. Evaluate ∫₀¹ x² dx.
38. Find shortest distance between point (1,2,3) and plane x+y+z=6.
39. In binomial distribution n=5, p=1/2. Find mean and variance.

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🔹 SECTION E (Case Study Based Questions)

(5 × 4 = 20 Marks)

Question 40

Profit function: P(x)=x³−6x²+9x+15

(i) Find P'(x)
(ii) Find critical points
(iii) Determine intervals of increase/decrease
(iv) Find local maxima/minima

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Question 41

Line passes through A(1,2,3) and B(4,6,3)

(i) Direction ratios
(ii) Vector equation
(iii) Length AB
(iv) Unit vector along AB

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✅ ANSWER KEY

Section A

1. b
2. c
3. a
4. a
5. b
6. a
7. c
8. a
9. b
10. b
11. c
12. a
13. a
14. d
15. c
16. a
17. c
18. b
19. b
20. b

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Section B

21. −2
22. 3x² + 6x
23. x²/2 + C
24. (1/√3)(i + j + k)
25. r = (i + 2j + 3k) + λ(2i + j + k)
26. 1/3

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Section C

27. 1
28. cos²x − sin²x
29. x e^x − e^x + C
30. 2
31. cosθ = 8/9
32. 2(x−1) + (y−1) + 3(z−1) = 0
33. y = x² + C
34. 3/5

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Section D

35. | 1 −1 |
      | −1 2 |
36. Max at x=1, Min at x=2
37. 1/3
38. 0
39. Mean = 5/2, Variance = 5/4

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