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CBSE Class 12 Mathematics
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.
- Diagrams must be neat and properly labeled.
Section A (1 × 20 = 20 Marks)
1. Find the value of
\[
\begin{vmatrix}
1 & 2 \\
3 & 6
\end{vmatrix}
\]
2. Find the degree of the differential equation
\[
\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0
\]
3. Write the domain of the function
\[
f(x) = \sqrt{5 - x}
\]
4. Evaluate
\[
\sin^{-1}\left(\sin\frac{\pi}{6}\right)
\]
5. If \( |A| = 5 \), find \( |3A| \).
6. Evaluate
\[
\int_0^1 x \, dx
\]
7. Write the vector form of the line parallel to
\[
\hat{i} + 2\hat{j} + 3\hat{k}
\]
8. Find the distance between the points
\[
(0,0,0) \text{ and } (1,2,2)
\]
9. If A and B are mutually exclusive events, find
\[
P(A \cap B)
\]
10. Find the order of the matrix
\[
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
\]
Section B (2 × 5 = 10 Marks)
11. Find the inverse of
\[
A =
\begin{bmatrix}
2 & 1 \\
5 & 3
\end{bmatrix}
\]
12. Evaluate
\[
\int (2x^2 + 3x)\,dx
\]
13. Find the equation of the line passing through
\[
(1,2,3)
\]
with direction ratios
\[
(2,-1,1)
\]
14. If
\[
P(A)=\frac{1}{2}, \quad P(B)=\frac{1}{3}
\]
find
\[
P(A \cap B)
\]
15. Find the derivative of
\[
y = \sin x \cos x
\]
Section C (3 × 6 = 18 Marks)
16. Solve the system of equations using matrix method:
\[
\begin{aligned}
x + y + z &= 6 \\
2x + 3y + z &= 10 \\
x + 2y + 3z &= 14
\end{aligned}
\]
17. Evaluate
\[
\int \frac{1}{x^2 + 4}\,dx
\]
18. Find the area bounded by
\[
y = x^2 \quad \text{and} \quad y = 4
\]
19. A die is thrown twice. Find the probability that the sum is
\[
8
\]
20. Solve the differential equation
\[
\frac{dy}{dx} + y = e^x
\]
Section D (5 × 4 = 20 Marks)
21. Evaluate
\[
\int_0^{\pi/2} \sin^2 x \, dx
\]
22. Find the maximum value of
\[
y = x^3 - 3x^2 + 4
\]
23. Find the equation of the plane passing through
\[
(1,0,0), (0,1,0), (0,0,1)
\]
Section E (Case Study Based Questions)
24. The profit function is
\[
P(x) = -x^2 + 12x - 20
\]
(i) Find the value of \(x\) for maximum profit.
(ii) Find the maximum profit.
25. A box contains 3 red, 4 blue and 5 green balls.
(i) Find probability of drawing a blue ball.
(ii) Find probability of drawing neither red nor blue.
— End of Question Paper —
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