Mathematics · Sample Paper 2

General instructions

Section A · Multiple choice

1. Question 14 marks

   If $n(U)=50$, $n(A)=20$, $n(B)=25$ and $n(A\cap B)=10$, then $n(A^{\prime }\cap B^{\prime })$ is

   1. A.5
   2. B.15
   3. C.35
   4. D.10
2. Question 24 marks

   If $f(x)=\frac{x}{x-1}$, $x\ne 1$, then $f(f(x))$ equals

   1. A.x
   2. B.1/x
   3. C.−x
   4. D.$\frac{x}{1-x}$
3. Question 34 marks

   The value of $\sin 15^{\circ }\cos 15^{\circ }$ is

   1. A.$\frac{1}{4}$
   2. B.$\frac{1}{2}$
   3. C.$\frac{\sqrt{3}}{4}$
   4. D.$\frac{1}{8}$
4. Question 44 marks

   If one root of $x^2-kx+12=0$ is three times the other, then k is

   1. A.$\pm 6$
   2. B.$\pm 8$
   3. C.$\pm 4$
   4. D.$\pm 12$
5. Question 54 marks

   The value of $i^{2025}$ is

   1. A.1
   2. B.−1
   3. C.i
   4. D.−i
6. Question 64 marks

   The number of ways to choose a committee of 3 from 5 men and 4 women with at least one woman is

   1. A.74
   2. B.84
   3. C.64
   4. D.80
7. Question 74 marks

   The term independent of x in ${\left(x^2+\frac{1}{x}\right)}^6$ is

   1. A.15
   2. B.20
   3. C.6
   4. D.1
8. Question 84 marks

   If a, b, c are in GP and $a=2$, $c=18$, then b is

   1. A.$\pm 6$
   2. B.10
   3. C.9
   4. D.$\pm 3$
9. Question 94 marks

   The distance between the parallel lines $3x+4y=5$ and $3x+4y=15$ is

   1. A.1
   2. B.2
   3. C.3
   4. D.10
10. Question 104 marks

    The length of the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is

    1. A.$\frac{32}{5}$
    2. B.$\frac{16}{5}$
    3. C.8
    4. D.$\frac{25}{4}$
11. Question 114 marks

    ${\lim }_{x\to 0}\frac{e^{3x}-1}{x}$ equals

    1. A.1
    2. B.3
    3. C.e
    4. D.0
12. Question 124 marks

    If $A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, then $A^2$ is

    1. A.A
    2. B.the zero matrix
    3. C.the identity matrix
    4. D.−A
13. Question 134 marks

    If $\left|\begin{array}{cc}x& 2\\ 18& x\end{array}\right|=\left|\begin{array}{cc}6& 2\\ 18& 6\end{array}\right|$, then x is

    1. A.6
    2. B.$\pm 6$
    3. C.−6
    4. D.0
14. Question 144 marks

    If $y={\sin }^{-1}x$, then $\frac{dy}{dx}$ is

    1. A.$\frac{1}{\sqrt{1-x^2}}$
    2. B.$\frac{-1}{\sqrt{1-x^2}}$
    3. C.$\frac{1}{1+x^2}$
    4. D.$\sqrt{1-x^2}$
15. Question 154 marks

    The equation of the tangent to $y=x^2$ at (1, 1) is

    1. A.y = 2x − 1
    2. B.y = x
    3. C.y = 2x + 1
    4. D.y = x + 1
16. Question 164 marks

    $\int \frac{dx}{x\log x}$ equals

    1. A.$\log (\log x)+C$
    2. B.$\frac{1}{\log x}+C$
    3. C.$(\log x{)}^2+C$
    4. D.$x\log x+C$
17. Question 174 marks

    The solution of $\frac{dy}{dx}=\frac{y}{x}$ is

    1. A.y = cx
    2. B.xy = c
    3. C.$y=c{x}^2$
    4. D.$y=c{e}^x$
18. Question 184 marks

    The area of the parallelogram with adjacent sides $\hat{i}+2\hat{j}$ and $3\hat{i}+\hat{k}$ is

    1. A.$\sqrt{41}$
    2. B.7
    3. C.$\sqrt{29}$
    4. D.5
19. Question 194 marks

    The angle between the lines with direction ratios (1, 1, 0) and (0, 1, 1) is

    1. A.$30^{\circ }$
    2. B.$45^{\circ }$
    3. C.$60^{\circ }$
    4. D.$90^{\circ }$
20. Question 204 marks

    Two cards are drawn without replacement from a pack of 52. The probability that both are kings is

    1. A.$\frac{1}{221}$
    2. B.$\frac{1}{169}$
    3. C.$\frac{4}{221}$
    4. D.$\frac{1}{13}$

Section B · Numerical value

1. Question 214 marks

   Find the number of diagonals of a decagon (10-sided polygon).

2. Question 224 marks

   Evaluate ${\int }_0^2{x}^{3}dx$.

3. Question 234 marks

   Find the sum of the first 20 terms of the AP 3, 7, 11, ….

4. Question 244 marks

   Find the minimum value of $f(x)=x^2-6x+13$.

5. Question 254 marks

   If $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+4\hat{j}+\lambda \hat{k}$ are perpendicular, find λ.