Mathematics · Sample Paper 3

General instructions

Section A · Multiple choice

1. Question 14 marks

   If A and B are sets with $n(A)=10$, $n(B)=15$ and $n(A\cup B)=20$, then $n(A\cap B)$ is

   1. A.5
   2. B.10
   3. C.15
   4. D.25
2. Question 24 marks

   The domain of $f(x)=\frac{1}{\sqrt{x^2-9}}$ is

   1. A.$(-3,3)$
   2. B.$(-\infty ,-3)\cup (3,\infty )$
   3. C.$[-3,3]$
   4. D.$\mathbb{R}$
3. Question 34 marks

   The maximum value of $3\sin x+4\cos x$ is

   1. A.3
   2. B.4
   3. C.5
   4. D.7
4. Question 44 marks

   If $\alpha ,\beta$ are roots of $x^2-3x+1=0$, then $\frac{1}{\alpha }+\frac{1}{\beta }$ equals

   1. A.3
   2. B.1
   3. C.1/3
   4. D.−3
5. Question 54 marks

   The argument of $-1+i\sqrt{3}$ is

   1. A.$\frac{\pi }{3}$
   2. B.$\frac{2\pi }{3}$
   3. C.$-\frac{\pi }{3}$
   4. D.$\frac{5\pi }{6}$
6. Question 64 marks

   The number of ways in which 4 boys and 3 girls can sit in a row so that no two girls are together is

   1. A.1440
   2. B.720
   3. C.144
   4. D.5040
7. Question 74 marks

   The sum of the coefficients in the expansion of $(1+2x{)}^5$ is

   1. A.32
   2. B.243
   3. C.64
   4. D.125
8. Question 84 marks

   The sum to infinity of the GP $6+2+\frac{2}{3}+\cdots$ is

   1. A.9
   2. B.12
   3. C.8
   4. D.18
9. Question 94 marks

   The equation of the line through (2, 3) perpendicular to $3x+4y=5$ is

   1. A.4x − 3y + 1 = 0
   2. B.3x + 4y − 18 = 0
   3. C.4x + 3y − 17 = 0
   4. D.4x − 3y − 1 = 0
10. Question 104 marks

    The length of the latus rectum of the parabola $x^2=12y$ is

    1. A.3
    2. B.6
    3. C.12
    4. D.24
11. Question 114 marks

    ${\lim }_{x\to 0}\frac{\tan 3x}{\sin 2x}$ equals

    1. A.$\frac{3}{2}$
    2. B.$\frac{2}{3}$
    3. C.1
    4. D.0
12. Question 124 marks

    If A is a square matrix with $A^2=A$, then $(I+A{)}^2-3A$ equals

    1. A.I
    2. B.A
    3. C.0
    4. D.2A
13. Question 134 marks

    If A is a $3\times 3$ invertible matrix with $|A|=3$, then $|A^{-1}|$ is

    1. A.3
    2. B.$\frac{1}{3}$
    3. C.9
    4. D.$\frac{1}{9}$
14. Question 144 marks

    If $y=\log (\sec x+\tan x)$, then $\frac{dy}{dx}$ is

    1. A.$\sec x$
    2. B.$\tan x$
    3. C.${\sec }^{2}x$
    4. D.$\sec x\tan x$
15. Question 154 marks

    The function $f(x)=x^3-3x^2+3x$ is

    1. A.increasing on ℝ
    2. B.decreasing on ℝ
    3. C.neither
    4. D.constant
16. Question 164 marks

    ${\int }_0^{\pi }{\sin }^{2}xdx$ equals

    1. A.$\frac{\pi }{2}$
    2. B.$\pi$
    3. C.1
    4. D.0
17. Question 174 marks

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

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

    If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, the angle between $\vec{a}$ and $\vec{b}$ is

    1. A.$0^{\circ }$
    2. B.$45^{\circ }$
    3. C.$90^{\circ }$
    4. D.$180^{\circ }$
19. Question 194 marks

    The distance of the point (1, 2, 3) from the origin is

    1. A.$\sqrt{14}$
    2. B.6
    3. C.$\sqrt{12}$
    4. D.14
20. Question 204 marks

    A bag has 4 red and 6 black balls. Two balls are drawn with replacement. The probability that both are red is

    1. A.$\frac{4}{25}$
    2. B.$\frac{2}{15}$
    3. C.$\frac{16}{25}$
    4. D.$\frac{2}{5}$

Section B · Numerical value

1. Question 214 marks

   Find the number of ways to select 3 people from a group of 8.

2. Question 224 marks

   Find the value of $\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 10\end{array}\right|$.

3. Question 234 marks

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

4. Question 244 marks

   The sum of the first n natural numbers is 210. Find n.

5. Question 254 marks

   Find ${\lim }_{x\to 2}\frac{x^4-16}{x-2}$.