Posts
CBSE – Class XII
Mathematics (041)
NCERT Based Question Paper
Time: 3 Hours | Maximum Marks: 80
General Instructions:
- All questions are compulsory.
- The question paper consists of Sections A, B, C, D and E.
- Use of calculator is not permitted.
- 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 —
