Mathematics · Sample Paper 2

General instructions

Section A · Objective

1. Question 11 mark

   Let $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^2$. Then f is

   1. A.one-one and onto
   2. B.one-one but not onto
   3. C.onto but not one-one
   4. D.neither one-one nor onto
2. Question 21 mark

   The principal value of ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is

   1. A.$\frac{\pi }{6}$
   2. B.$\frac{5\pi }{6}$
   3. C.$-\frac{\pi }{6}$
   4. D.$\frac{2\pi }{3}$
3. Question 31 mark

   If $A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$, then $A^T$ is

   1. A.$\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$
   2. B.$\left[\begin{array}{cc}4& 3\\ 1& 2\end{array}\right]$
   3. C.$\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$
   4. D.$\left[\begin{array}{cc}-2& -3\\ -1& -4\end{array}\right]$
4. Question 41 mark

   If A and B are symmetric matrices of the same order, then $AB-BA$ is

   1. A.symmetric
   2. B.skew-symmetric
   3. C.zero matrix
   4. D.identity
5. Question 51 mark

   The value of $\left|\begin{array}{cc}2& 4\\ -1& 3\end{array}\right|$ is

   1. A.10
   2. B.2
   3. C.−10
   4. D.6
6. Question 61 mark

   If A is a $2\times 2$ matrix with $|A|=3$, then $|A^{-1}|$ is

   1. A.3
   2. B.$\frac{1}{3}$
   3. C.9
   4. D.$\frac{1}{9}$
7. Question 71 mark

   The derivative of $\log (\sin x)$ is

   1. A.$\tan x$
   2. B.$\cot x$
   3. C.$\frac{1}{\sin x}$
   4. D.$-\cot x$
8. Question 81 mark

   For what value of k is $f(x)=\{\begin{array}{ll}kx+1,& x\le 2\\ 3x-1,& x>2\end{array}$ continuous at $x=2$?

   1. A.1
   2. B.2
   3. C.3
   4. D.5
9. Question 91 mark

   The slope of the tangent to $y=x^3-x$ at $x=2$ is

   1. A.11
   2. B.12
   3. C.6
   4. D.8
10. Question 101 mark

    $\int {\sec }^{2}xdx$ equals

    1. A.$\tan x+C$
    2. B.$\sec x+C$
    3. C.$\cot x+C$
    4. D.$-\tan x+C$
11. Question 111 mark

    ${\int }_0^{\pi /2}\cos xdx$ equals

    1. A.0
    2. B.1
    3. C.−1
    4. D.$\pi /2$
12. Question 121 mark

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

    1. A.$e^x$
    2. B.$e^{2x}$
    3. C.$2x$
    4. D.$e^{-2x}$
13. Question 131 mark

    The magnitude of $2\hat{i}-3\hat{j}+6\hat{k}$ is

    1. A.7
    2. B.11
    3. C.5
    4. D.49
14. Question 141 mark

    The projection of $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ on $\vec{b}=\hat{i}+\hat{j}$ is

    1. A.$\frac{3}{\sqrt{2}}$
    2. B.$3$
    3. C.$\frac{3}{\sqrt{6}}$
    4. D.$\sqrt{2}$
15. Question 151 mark

    The direction ratios of the line $\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z}{4}$ are

    1. A.1, −3, 0
    2. B.2, −1, 4
    3. C.−1, 3, 0
    4. D.2, 1, 4
16. Question 161 mark

    The feasible region of an LPP is

    1. A.always unbounded
    2. B.the common region satisfying all constraints
    3. C.any point in the first quadrant
    4. D.the line of the objective function
17. Question 171 mark

    A die is thrown twice. The probability that the sum is 7 is

    1. A.$\frac{1}{6}$
    2. B.$\frac{1}{12}$
    3. C.$\frac{5}{36}$
    4. D.$\frac{7}{36}$
18. Question 181 mark

    If $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$ and A, B are mutually exclusive, $P(A\cup B)$ is

    1. A.$\frac{1}{6}$
    2. B.$\frac{5}{6}$
    3. C.$\frac{2}{3}$
    4. D.1
19. Question 191 mark

    Assertion (A): Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

    Reason (R): $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$.

    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): $f(x)=e^x$ is strictly increasing on $\mathbb{R}$.

    Reason (R): $f^{\prime }(x)>0$ for all $x$ implies f is strictly increasing.

    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

   Check whether the relation $R=\{(a,b):a\le b\}$ on $\mathbb{R}$ is reflexive, symmetric or transitive.

2. Question 222 marks

   Find the value of ${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right)$.

3. OR

   Question 222 marks

   Simplify ${\tan }^{-1}\left(\frac{\cos x}{1-\sin x}\right)$, $-\frac{\pi }{2}<x<\frac{\pi }{2}$.

4. Question 232 marks

   Find $\frac{dy}{dx}$ if $x^2+xy+y^2=100$.

5. Question 242 marks

   Find a unit vector perpendicular to both $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$.

6. Question 252 marks

   Given $P(A)=0.3$, $P(B)=0.4$ and $P(A\cup B)=0.5$, find $P(A\cap B)$ and $P(B|A)$.

Section C · Short answer

1. Question 263 marks

   Evaluate $\int \frac{x}{\sqrt{x+1}}dx$.

2. Question 273 marks

   Evaluate $\int e^x(\sin x+\cos x)dx$.

3. OR

   Question 273 marks

   Evaluate $\int \frac{dx}{x^2+4x+8}$.

4. Question 283 marks

   Find the particular solution of $\frac{dy}{dx}=y\tan x$, given $y=1$ when $x=0$.

5. Question 293 marks

   Minimise $Z=3x+5y$ subject to $x+3y\ge 3$, $x+y\ge 2$, $x,y\ge 0$.

6. Question 303 marks

   Find the intervals in which $f(x)=2x^3-9x^2+12x+15$ is increasing or decreasing.

7. Question 313 marks

   A random variable X has the probability distribution: X = 0, 1, 2, 3 with P = 0.1, 0.3, 0.4, k. Find k and the mean of X.

Section D · Long answer

1. Question 325 marks

   If $A=\left[\begin{array}{ccc}1& 2& -3\\ 2& 3& 2\\ 3& -3& -4\end{array}\right]$, find $A^{-1}$ and hence solve $x+2y-3z=-4$, $2x+3y+2z=2$, $3x-3y-4z=11$.

2. Question 335 marks

   Find the area of the region bounded by the curve $y=x^2$ and the line $y=4$, using integration.

3. Question 345 marks

   Find the vector and Cartesian equations of the line through $(1,2,-4)$ and perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$.

4. OR

   Question 345 marks

   Find the shortest distance between the lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$.

5. Question 355 marks

   Evaluate ${\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx$.

Section E · Case study

1. Question 364 marks

   Case study — A rectangular garden. A gardener has 80 m of fencing to enclose a rectangular garden along a straight wall; the wall forms one side, so only three sides need fencing. Let the side perpendicular to the wall be $x$ m.

   (i) Express the area A of the garden in terms of $x$. (1 mark)

   (ii) Find $\frac{dA}{dx}$. (1 mark)

   (iii) Find the dimensions that give the maximum area and the maximum area. (2 marks)

   OR

   (iii) Use the second derivative test to show that the area found is a maximum. (2 marks)

2. Question 374 marks

   Case study — Quality check. A factory has two machines, M₁ and M₂, producing 60% and 40% of the bolts respectively. 2% of bolts from M₁ and 3% from M₂ are defective. A bolt is picked at random.

   (i) Find the probability that the bolt is from M₁ and defective. (1 mark)

   (ii) Find the probability that the bolt is defective. (1 mark)

   (iii) If the bolt is found defective, find the probability that it came from M₂. (2 marks)

   OR

   (iii) If two bolts are picked independently, find the probability that exactly one is defective. (2 marks)

3. Question 384 marks

   Case study — Cooling tea. A cup of tea at 90 °C is left in a room at 30 °C. By Newton's law of cooling, the temperature $T$ after $t$ minutes satisfies $\frac{dT}{dt}=-k(T-30)$, with $k>0$.

   (i) What is the order and degree of this differential equation? (1 mark)

   (ii) Separate the variables. (1 mark)

   (iii) Solve the equation and find T in terms of t and k. (2 marks)

   OR

   (iii) If the tea cools to 60 °C in 10 minutes, find k. ($\ln 2=0.693$) (2 marks)