Mathematics · Sample Paper 2

General instructions

Section A · Objective

1. Question 11 mark

   Which of the following is the empty set?

   1. A.$\{x:x\in \mathbb{N},x^2=4\}$
   2. B.$\{x:x\in \mathbb{R},x^2+1=0\}$
   3. C.$\{0\}$
   4. D.$\{x:x\text{ is even prime}\}$
2. Question 21 mark

   If $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$, then $A-B$ is

   1. A.$\{1,2\}$
   2. B.$\{5\}$
   3. C.$\{3,4\}$
   4. D.$\{1,2,5\}$
3. Question 31 mark

   If $f(x)=x^2+1$, then $f(-2)$ is

   1. A.3
   2. B.5
   3. C.−3
   4. D.4
4. Question 41 mark

   The domain of $f(x)=\sqrt{x-3}$ is

   1. A.$(3,\infty )$
   2. B.$[3,\infty )$
   3. C.$\mathbb{R}$
   4. D.$(-\infty ,3]$
5. Question 51 mark

   The value of $\cos 120^{\circ }$ is

   1. A.$\frac{1}{2}$
   2. B.$-\frac{1}{2}$
   3. C.$\frac{\sqrt{3}}{2}$
   4. D.$-\frac{\sqrt{3}}{2}$
6. Question 61 mark

   The period of $\sin 2x$ is

   1. A.$\pi$
   2. B.$2\pi$
   3. C.$\frac{\pi }{2}$
   4. D.$4\pi$
7. Question 71 mark

   $(2+3i)(2-3i)$ equals

   1. A.13
   2. B.−5
   3. C.4 − 9i
   4. D.5
8. Question 81 mark

   The conjugate of $\frac{1}{1+i}$ is

   1. A.$\frac{1+i}{2}$
   2. B.$\frac{1-i}{2}$
   3. C.$1+i$
   4. D.$1-i$
9. Question 91 mark

   The solution of $3x-5<7$, $x\in \mathbb{N}$, is

   1. A.$\{1,2,3\}$
   2. B.$\{1,2,3,4\}$
   3. C.$\{4\}$
   4. D.$\{0,1,2,3\}$
10. Question 101 mark

    The value of ${}^6{P}_2$ is

    1. A.12
    2. B.15
    3. C.30
    4. D.36
11. Question 111 mark

    The number of diagonals of a hexagon is

    1. A.6
    2. B.9
    3. C.15
    4. D.12
12. Question 121 mark

    The coefficient of $x^2$ in $(1+x{)}^5$ is

    1. A.5
    2. B.10
    3. C.15
    4. D.20
13. Question 131 mark

    The 6th term of the GP $3,6,12,\dots$ is

    1. A.48
    2. B.96
    3. C.192
    4. D.64
14. Question 141 mark

    The equation of a line with slope 2 and y-intercept −3 is

    1. A.$y=2x-3$
    2. B.$y=-3x+2$
    3. C.$y=2x+3$
    4. D.$2y=x-3$
15. Question 151 mark

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

    1. A.$\frac{5}{4}$
    2. B.$\frac{4}{5}$
    3. C.$\frac{5}{3}$
    4. D.$\frac{3}{4}$
16. Question 161 mark

    The point $(-2,3,-5)$ lies in the octant

    1. A.II
    2. B.III
    3. C.VI
    4. D.VIII
17. Question 171 mark

    ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$ equals

    1. A.0
    2. B.3
    3. C.6
    4. D.9
18. Question 181 mark

    If $P(A)=0.35$, then $P(\text{not }A)$ is

    1. A.0.35
    2. B.0.65
    3. C.0.5
    4. D.1.35
19. Question 191 mark

    Assertion (A): If $n(A)=3$ and $n(B)=2$, then $n(A\times B)=6$.

    Reason (R): $n(A\times B)=n(A)\cdot n(B)$.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
20. Question 201 mark

    Assertion (A): The standard deviation of 5, 5, 5, 5 is 0.

    Reason (R): Standard deviation measures how spread out the data is from the mean.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.

Section B · Very short answer

1. Question 212 marks

   If $A=\{1,2\}$ and $B=\{3,4\}$, write $A\times B$ and find the number of relations from A to B.

2. Question 222 marks

   Find the radian measure of $40^{\circ }20^{\prime }$.

3. Question 232 marks

   Express $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$ in the form $a+ib$.

4. OR

   Question 232 marks

   Solve $\frac{x}{3}>\frac{x}{2}+1$ for real $x$.

5. Question 242 marks

   Show that the points $(-2,3,5)$, $(1,2,3)$ and $(7,0,-1)$ are collinear.

6. Question 252 marks

   Find the derivative of $\frac{x+1}{x-1}$.

Section C · Short answer

1. Question 263 marks

   In a class of 60 students, 25 like tea, 30 like coffee and 10 like both. How many like (i) tea or coffee (ii) neither (iii) only coffee?

2. Question 273 marks

   Prove that $\cos 4x=1-8{\sin }^{2}x{\cos }^{2}x$.

3. Question 283 marks

   In how many ways can the letters of the word PERMUTATIONS be arranged if (i) the words start with P and end with S (ii) the vowels are all together?

4. OR

   Question 283 marks

   Find the middle term in the expansion of ${\left(3-\frac{x^3}{6}\right)}^7$.

5. Question 293 marks

   Find the equation of the line passing through the point of intersection of $x+y=5$ and $x-y=1$ and parallel to $3x+4y=7$.

6. Question 303 marks

   Insert three arithmetic means between 3 and 19. Also find the sum of the resulting AP.

7. Question 313 marks

   A bag has 5 red and 3 black balls. Two balls are drawn at random. Find the probability that (i) both are red (ii) one is red and one is black.

Section D · Long answer

1. Question 325 marks

   Find the mean, variance and standard deviation of the following distribution:

   | x | 2 | 4 | 6 | 8 | 10 | |---|---|---|---|---|---| | f | 3 | 5 | 8 | 5 | 3 |

2. Question 335 marks

   Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola $9y^2-4x^2=36$.

3. OR

   Question 335 marks

   Find the equation of the ellipse with vertices $(\pm 5,0)$ and foci $(\pm 4,0)$. Also find its eccentricity and the length of its latus rectum.

4. Question 345 marks

   (a) Find the derivative of $\sin x$ from first principles. (b) Evaluate ${\lim }_{x\to 0}\frac{1-\cos 2x}{x^2}$.

5. Question 355 marks

   The sum of two numbers is 6 times their geometric mean. Show that the numbers are in the ratio $(3+2\sqrt{2}):(3-2\sqrt{2})$.

Section E · Case study

1. Question 364 marks

   Case study — Planning a diet. A dietician says a student's breakfast must provide at least 8 units of protein. Milk gives 2 units per glass and eggs give 3 units each. The student can have at most 3 glasses of milk. Let $x$ = glasses of milk and $y$ = number of eggs.

   (i) Write the inequality for the protein requirement. (1 mark)

   (ii) Write the inequality for the limit on milk. (1 mark)

   (iii) Is (2, 1) a solution of the system $2x+3y\ge 8$, $x\le 3$, $x,y\ge 0$? Find the least number of eggs needed if the student drinks 1 glass of milk. (2 marks)

   OR

   (iii) Describe the solution region of the system on a graph and name one corner point. (2 marks)

2. Question 374 marks

   Case study — Bacteria growth. A lab finds that a bacterial culture grows by 2% every hour. Starting with $N$ bacteria, after $n$ hours there are $N(1.02{)}^n$ bacteria.

   (i) Write the first three terms of the expansion of $(1+0.02{)}^n$. (1 mark)

   (ii) Using the binomial theorem, approximate $(1.02{)}^5$ to three decimal places. (1 mark)

   (iii) Using the binomial theorem, evaluate $(1.02{)}^{10}$ approximately (first three terms) and estimate the population after 10 hours if $N=5000$. (2 marks)

   OR

   (iii) Find the general term $T_{r+1}$ of $(1+0.02{)}^n$ and write the 4th term for $n=10$. (2 marks)

3. Question 384 marks

   Case study — Two batsmen. The scores of two batsmen in 5 matches are:

   | Batsman A | 30 | 40 | 50 | 60 | 70 | |---|---|---|---|---|---| | Batsman B | 45 | 50 | 50 | 50 | 55 |

   (i) Find the mean score of each batsman. (1 mark)

   (ii) Find the variance of batsman B's scores. (1 mark)

   (iii) Find the coefficient of variation of each batsman and say who is more consistent. (2 marks)

   OR

   (iii) Find the standard deviation of batsman A's scores and the mean deviation about the mean of A's scores. (2 marks)