Mathematics · Sample Paper 1

General instructions

Section A · Multiple choice

1. Question 14 marks

   If $n(A) = 3$ and $n(B) = 4$, the number of elements in $A\times B$ is

   1. A.7
   2. B.12
   3. C.64
   4. D.81
2. Question 24 marks

   The number of bijective functions from a set with 4 elements to itself is

   1. A.16
   2. B.24
   3. C.256
   4. D.4
3. Question 34 marks

   The value of $\cos^2 15^\circ - \sin^2 15^\circ$ is

   1. A.$\tfrac12$
   2. B.$\tfrac{\sqrt3}{2}$
   3. C.$\tfrac{1}{\sqrt2}$
   4. D.1
4. Question 44 marks

   If $\alpha, \beta$ are roots of $x^2 - 5x + 6 = 0$, then $\alpha^2 + \beta^2$ is

   1. A.13
   2. B.25
   3. C.11
   4. D.37
5. Question 54 marks

   The modulus of $\dfrac{1 + i}{1 - i}$ is

   1. A.0
   2. B.1
   3. C.2
   4. D.$\sqrt2$
6. Question 64 marks

   The number of ways 5 people can sit around a round table is

   1. A.120
   2. B.24
   3. C.60
   4. D.5
7. Question 74 marks

   The coefficient of $x^5$ in $(1 + x)^{10}$ is

   1. A.210
   2. B.252
   3. C.120
   4. D.45
8. Question 84 marks

   The sum $1 + 3 + 5 + \cdots + 49$ is

   1. A.625
   2. B.600
   3. C.650
   4. D.576
9. Question 94 marks

   The angle between the lines y = x and y = √3 x is

   1. A.$15^\circ$
   2. B.$30^\circ$
   3. C.$45^\circ$
   4. D.$60^\circ$
10. Question 104 marks

    The eccentricity of the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ is

    1. A.$\tfrac54$
    2. B.$\tfrac45$
    3. C.$\tfrac53$
    4. D.$\tfrac34$
11. Question 114 marks

    $\lim_{x\to0}\dfrac{1 - \cos 2x}{x^2}$ equals

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

    If $A = \begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}$, then $A + A'$ is

    1. A.$\begin{pmatrix}2 & 5\\ 5 & 8\end{pmatrix}$
    2. B.$\begin{pmatrix}2 & 4\\ 6 & 8\end{pmatrix}$
    3. C.$\begin{pmatrix}1 & 5\\ 5 & 4\end{pmatrix}$
    4. D.$\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$
13. Question 134 marks

    If $A$ is a $3\times3$ matrix with $|A| = 2$, then $|3A|$ is

    1. A.6
    2. B.18
    3. C.54
    4. D.27
14. Question 144 marks

    If $y = \log(\sin x)$, then $\dfrac{dy}{dx}$ is

    1. A.$\tan x$
    2. B.$\cot x$
    3. C.$\sec x$
    4. D.$-\cot x$
15. Question 154 marks

    The maximum value of $f(x) = x(12 - x)$ is

    1. A.36
    2. B.72
    3. C.144
    4. D.24
16. Question 164 marks

    $\int_0^{\pi/2}\sin^2x\,dx$ equals

    1. A.$\tfrac\pi2$
    2. B.$\tfrac\pi4$
    3. C.1
    4. D.$\pi$
17. Question 174 marks

    The integrating factor of $\dfrac{dy}{dx} + 2y = e^x$ is

    1. A.$e^x$
    2. B.$e^{2x}$
    3. C.$e^{-2x}$
    4. D.$2x$
18. Question 184 marks

    If $|\vec a| = 2$, $|\vec b| = 3$ and $\vec a\cdot\vec b = 3$, the angle between them is

    1. A.$30^\circ$
    2. B.$45^\circ$
    3. C.$60^\circ$
    4. D.$90^\circ$
19. Question 194 marks

    The lines $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$ and $\dfrac{x}{4} = \dfrac{y}{6} = \dfrac{z}{8}$ are

    1. A.perpendicular
    2. B.parallel
    3. C.intersecting at the origin
    4. D.skew
20. Question 204 marks

    A and B are independent with $P(A) = 0.3$ and $P(B) = 0.4$. Then $P(A\cup B)$ is

    1. A.0.70
    2. B.0.58
    3. C.0.12
    4. D.0.82

Section B · Numerical value

1. Question 214 marks

   Find the number of distinct arrangements of the letters of the word ALLAHABAD.

2. Question 224 marks

   Evaluate $\int_0^3 (2x + 1)\,dx$.

3. Question 234 marks

   Find the value of $\begin{vmatrix}2 & 1 & 3\\ 0 & 4 & 1\\ 1 & 0 & 2\end{vmatrix}$.

4. Question 244 marks

   Find the variance of the data 2, 4, 6, 8, 10.

5. Question 254 marks

   Find the area (in square units) of the region bounded by $y = 3x^2$, the x-axis and the lines $x = 0$ and $x = 2$.