CBSE CLASS 12 MATHEMATICS PAPER

CBSE – Class XII

Mathematics (041)

NCERT Based Question Paper

Time: 3 Hours    |    Maximum Marks: 80

General Instructions:

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

Section A (1 × 20 = 20 Marks)

1. Evaluate \( \sin^{-1}\left(\sin \frac{\pi}{6}\right) \).

2. Find the domain of \( f(x) = \sqrt{4-x} \).

3. Find the order of a matrix having 3 rows and 4 columns.

4. If \( |A| = 3 \), find \( |2A| \).

5. Find \( \dfrac{d}{dx}(e^{2x}) \).

6. Evaluate \( \int_0^1 x\,dx \).

7. Write the principal value of \( \tan^{-1}(\sqrt{3}) \).

8. Find the distance between points \( (0,0,0) \) and \( (1,2,2) \).

9. If events A and B are mutually exclusive, find \( P(A \cap B) \).

10. Write one application of derivatives.

11. Find \( \cos 60^\circ \).

12. Find the degree of \( \left(\dfrac{d^2y}{dx^2}\right)^2 + y = 0 \).

13. Write the vector form of a line parallel to \( \hat{i}+2\hat{j}+3\hat{k} \).

14. Find \( P(\bar{A}) \) if \( P(A)=0.35 \).

15. Write the general solution of \( \dfrac{dy}{dx}=0 \).

16. Find the mean of first 10 natural numbers.

17. Evaluate \( \cos^{-1}(-1) \).

18. Write one example of a continuous function.

19. Find \( \dfrac{d}{dx}(\ln x) \).

20. Write the order of matrix \( [2\;\;3\;\;4] \).

Section B (2 × 5 = 10 Marks)

21. Find the inverse of \[ \begin{bmatrix} 2 & 1\\ 5 & 3 \end{bmatrix} \]

22. Evaluate \[ \int (3x^2 + 4x)\,dx \]

23. Find the equation of a line passing through \( (1,2,3) \) with direction ratios \( (2,-1,1) \).

24. If \( P(A)=\frac{1}{2} \) and \( P(B)=\frac{1}{3} \), find \( P(A\cap B) \).

25. Find \( \dfrac{dy}{dx} \) if \( y=\sin x \cos x \).

Section C (3 × 6 = 18 Marks)

26. Solve the system of equations using matrix method: \[ \begin{aligned} x+y+z &= 6\\ 2x+3y+z &= 10\\ x+2y+3z &= 14 \end{aligned} \]

27. Evaluate \[ \int \frac{1}{x^2+9}\,dx \]

28. Find the area bounded by \( y=x^2 \) and \( y=9 \).

29. Find the shortest distance between the skew lines.

30. A die is thrown twice. Find the probability that the sum is 7.

31. Solve \[ \frac{dy}{dx}+y=e^x \]

Section D (5 × 4 = 20 Marks)

32. Find the inverse of the matrix using elementary row operations: \[ \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 4\\ 5 & 6 & 0 \end{bmatrix} \]

33. Evaluate \[ \int_0^{\pi/2} \sin^2 x\,dx \]

34. Find the maximum value of \[ y=x^3-6x^2+9x \]

35. Find the equation of the plane passing through \( (1,0,0), (0,1,0), (0,0,1) \).

Section E (Case Study Based Questions) (5 × 2 = 10 Marks)

36. The profit function of a company is \[ P(x)=-x^2+10x-16 \] (i) Find the value of \(x\) for which profit is maximum. (ii) Find the maximum profit.

37. A box contains 4 red, 5 blue and 3 green balls. (i) Find the probability of drawing a blue ball. (ii) Find the probability of drawing neither red nor blue.

— End of Question Paper —