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Marking Scheme

• For each question in Section I, you will be awarded 3 marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubbles are darkened. In all other cases, minus one (−1) mark will be awarded.
• For each question in Section II, you will be awarded 3 marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubbles are darkened. Partial marks will be awarded for partially correct answers. No negative marks will be awarded in this Section.
• For each question in Section III, you will be awarded 3 marks if you darken only the bubble corresponding to the correct answer and zero mark if no bubbles are darkened. In all other cases, minus one (−1) mark will be awarded.
• For each question in Section IV, you will be awarded 3 marks if you darken the bubble corresponding to the correct answer and zero mark if no bubble is darkened. No negative marks will be awarded for in this Section.

SECTION - I (Single Correct Choice Type)

This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

1. If the angles A, B and C of a triangle are in arithmetic progression and if $a$, $b$ and $c$ denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression $\frac{a}{c}$sin2C+$\frac{c}{a}$sin2A is
   1. $\frac{1}{2}$
   2. $\frac{\sqrt{3}}{2}$
   3. 1
   4. $\sqrt{3}$
2. Equation of the plane containing the straight line $\frac{x}{2}$=$\frac{y}{3}$=$\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}$=$\frac{y}{4}$=$\frac{z}{2}$ and $\frac{x}{4}$=$\frac{y}{2}$=$\frac{z}{3}$
   1. $x$+$2y$$-2z$=0
   2. $3x$+$2y$$-2z$=0
   3. $x$$-2y$+$z$=0
   4. $5x$+$2y$$-4z$=0
3. Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r_1$, $r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}$+$\omega^{r_2}$+$\omega^{r_3}$=0 is
   1. $\frac{1}{18}$
   2. $\frac{1}{9}$
   3. $\frac{2}{9}$
   4. $\frac{1}{36}$
4. Let P, Q, R and S be the points on the plane with position vectors $-2\hat{i}-\hat{j}$, $4\hat{i}$, $3\hat{i}+3\hat{j}$ and $-3\hat{i}+2\hat{j}$ respectively. The quadrilateral PQRS must be a
   1. parallelogram, which is neither a rhombus nor a rectangle
   2. square
   3. rectangle but not a square
   4. rhombus but not a square
5. The number of 3×3 matrices $A$ whose entries are either 0 or 1 and for which the system \begin{equation*}A \begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \end{equation*} has exactly two distinct solutions, is
   1. 0
   2. $2^9-1$
   3. 168
   4. 2
6. The value of $\lim \limits_{x \to 0} \frac{1}{x^3} \int \limits_{0}^{x}\frac{t \hspace{0.2mm} ln(1+t)}{t^4+4}dt$ is
   1. 0
   2. $\frac{1}{12}$
   3. $\frac{1}{24}$
   4. $\frac{1}{64}$
7. Let $p$ and $q$ be real numbers such that $p \neq 0$, $p^3 \neq q$ and $p^3 \neq -q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha$+$\beta$=$-p$ and $\alpha^3$+$\beta^3$=$q$, then a quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is
   1. $(p^3+q)x^2$$-(p^3+2q)x$+$(p^3+q)$=0
   2. $(p^3+q)x^2$$-(p^3-2q)x$+$(p^3+q)$=0
   3. $(p^3-q)x^2$$-(5p^3-2q)x$+$(p^3-q)$=0
   4. $(p^3-q)x^2$$-(5p^3+2q)x$+$(p^3-q)$=0
8. Let $f$, $g$ and $h$ be real-valued functions defined on the interval [0, 1] by $f(x)$=$e^{x^2}$+$e^{-x^2}$, $g(x)$=$xe^{x^2}$+$e^{-x^2}$ and $h(x)$=$x^2e^{x^2}$+$e^{-x^2}$. If $a$, $b$ and $c$ denote, respectively, the absolute maximum of $f$, $g$ and $h$ on [0, 1], then
   1. $a=b$ and $c \neq b$
   2. $a=c$ and $a \neq b$
   3. $a \neq b$ and $c \neq b$
   4. $a=b=c$

SECTION - II ((Multiple Correct Choice Type)

This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.

9. Let A and B be two distinct points on the parabola $y^2=4x$. If the axis of the parabola touches a circle of radius $r$ having AB as its diameter, then the slope of the line joining A and B can be
   1. $-\frac{1}{r}$
   2. $\frac{1}{r}$
   3. $\frac{2}{r}$
   4. $-\frac{2}{r}$
10. Let $ABC$ be a triangle such that $\angle{ACB}$ =$\frac{\pi}{6}$ and let $a$, $b$ and $c$ denote the lengths of the sides opposite to A, B and C respectively. The value(s) of $x$ for which $a$=$x^2$+$x$+1, $b$=$x^2-1$ and $c$=$2x$+1 is(are)
    1. $-(2+\sqrt{3})$
    2. $1+\sqrt{3}$
    3. $2+\sqrt{3}$
    4. $4\sqrt{3}$
11. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z$=$(1-t)z_1$+$tz_2$ for some real number $t$ with $0 < t < 1$. If Arg(w) denote the principal argument of a complex number w, then
    1. $|z-z_1|$+$|z-z_2|$=$|z_1-z_2|$
    2. Arg$(z-z_1)$=Arg$(z-z_2)$
    3. $\left|\begin{array}{ccc}z-z_1 & \bar{z}-\bar{z_1} \\ z_2-z_1 & \bar{z_2}-\bar{z_1} \end{array}\right|$.
    4. Arg$(z-z_1)$=Arg$(z_2-z_1)$
12. Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)$=$ln x$+$\int \limits_0^x \sqrt{1+sint}dt$. Then which of the following statement(s) is(are) true?
    1. $f''(x)$ exists for all $x \in (0, \infty)$
    2. $f'(x)$ exists for all $x \in (0, \infty)$ and $f'$ is continuous on $(0, \infty)$ but not differentiable on $(0, \infty)$
    3. there exists $\alpha$ > 1 such that $|f'(x)|$<$|f(x)|$ for all $x \in (\alpha, \infty)$
    4. there exists $\beta$ > 0 such that $|f(x)|$+$|f'(x)|\leq \beta$ for all $x \in (0, \infty)$
13. The value(s) of $\int \limits_0^1 \frac{x^4(1-x)^4}{1+x^2}dx$ is(are)
    1. $\frac{22}{7}-\pi$
    2. $\frac{2}{105}$
    3. 0
    4. $\frac{71}{15}-\frac{3 \pi}{2}$

Section III (Paragraph Type)

This section contains 2 paragraphs. Based upon the first paragraph 3 multiple choice questions and based upon the second paragraph 2 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.

Paragraph for Questions 14 to 16

Let $p$ be an odd prime number and $T_p$ be the following set of 2×2 matrices :
$T_p$=\begin{equation*} \left \{ A=\begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, 2, ..., p-1\}\right \}\end{equation*}

14. The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $det(A)$ is divisible by $p$ is
    1. $(p-1)^2$
    2. $2(p-1)$
    3. $(p-1)^2$+1
    4. $2p-1$
15. The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but $det(A)$ is divisible by $p$ is
    1. $(p-1)$$(p^2-p+1)$
    2. $p^3-$$(p-1)^2$
    3. $(p-1)^2$
    4. $(p-1)$$(p^2-2)$
16. The number of $A$ in $T_p$ such that the $det(A)$ is not divisible by $p$ is
    1. $2p^2$
    2. $p^3-5p$
    3. $p^3-3p$
    4. $p^3-p^2$

Paragraph for Questions 17 to 18

The circle $x^2$+$y^2-$$8x$=0 and hyperbola $\frac{x^2}{9}-$$\frac{y^2}{4}$=1 intersects at the points A and B.

17. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
    1. $2x-\sqrt{5}y-$20=0
    2. $2x-\sqrt{5}y$+4=0
    3. $3x-4y$+8=0
    4. $4x-3y$+4=0
18. Equation of the circle with AB as its diameter is
    1. $x^2$+$y^2-12x$+24=0
    2. $x^2$+$y^2$+$12x$+24=0
    3. $x^2$+$y^2$+$24x-$12=0
    4. $x^2$+$y^2$$-24x-$12=0

SECTION - IV (Integer Type)

This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9. The correct digit below the question number in the ORS is to be bubbled

19. The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for n=0, ±1, ±2 and $tan \theta$=$cot 5\theta$ as well as $sin 2\theta$=$cos 4\theta$ is
20. The maximum value of the expression $\frac{1}{sin^2\theta+3sin \theta cos \theta +5cos^2 \theta}$ is
21. If $\vec{a}$ and $\vec{b}$ are vectors in space given by $\vec{a}$=$\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\vec{b}$=$\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$, then the value of $\left(2\vec{a}+\vec{b}\right)$•$\left[\left(\vec{a}×\vec{b}\right)\right.$×$\left. \left(\vec{a}-2\vec{b}\right)\right]$ is
22. The line $2x$+$y$ =1 is tangent to the hyperbola $\frac{x^2}{a^2}-$$\frac{y^2}{b^2}$=1. If this line passes through the point of intersection of the nearest directrix and the $x$-axis, the eccentricity of the hyperbola is
23. If the distance between the plane $Ax-$$2y$+$z$=$d$ and the plane containing the lines $\frac{x-1}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{5}$ and $\frac{x-2}{3}$=$\frac{y-3}{4}$=$\frac{z-4}{5}$ is $\sqrt{6}$, then $|d|$ is
24. For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval [-10, 10] by
    $f(x)$=$\left\{\begin{array}{ll}x-[x] & \text { if [x] is odd}, \\ 1+[x]-x & \text { if [x] is even}.\end{array}\right.$ Then the value of $\frac{\pi^2}{10}\int \limits_{-10}^{10}f(x)cos \pi x dx$ is
25. Let $\omega$ be the complex number $\cos \frac{\cos 2 \pi}{3}$+$i \sin \frac{\cos 2 \pi}{3}$. Then the number of distinct complex number $z$ satisfying $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right|$=0 is equal to
26. Let $S_k$, $k$=1, 2,..., 100, denote the sum of the infinite geometric series whose first term is $\frac{k-1}{k!}$ and the common ratio is $\frac{1}{k}$. Then the value of $\frac{100^2}{100!}$+$\sum \limits_{k=1}^{100}|(k^2-3k+1)S_k|$ is
27. The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations
    $(y+z)$$\cos 3\theta$=$(xyz)$ $\sin 3 \theta$
    $x \sin 3 \theta$=$\frac{2 \cos 3 \theta}{y}$+$\frac{2 \sin 3 \theta}{z}$
    $(xyz)$ $\sin 3 \theta$=$(y+2z)$$\cos 3 \theta$+$y$$\sin 3 \theta$
    have a solution $(x_0, y_0, z_0)$ with $y_0z_0 \neq 0$, is
28. Let $f$ be a real valued differentiable function on $R$ (the set of all real numbers) such that $f(1)$=1. If the y-intercept of the tangent at any point $P(x, y)$ on the curve $y=f(x)$ is equal to the cube of the abscissa of $P$, then the value of $f(-3)$ is equal to

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