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SECTION 1 (Maximum Marks : 32)

• This section contains EIGHT (08) questions.
• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).
• For each question, choose the option(s) corresponding to (all) the correct answer(s).
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If only (all) the correct option(s) is(are) chosen;
  • Partial Marks: +3 If all the four options are correct but ONLY three options are chosen;
  • Partial Marks: +2 If three or more options are correct but ONLY two options are chosen and both of which are correct;
  • Partial Marks: +1 If two or more options are correct but ONLY one option is chosen and it is a correct option;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -1 In all other cases.
• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then
  choosing ONLY (A), (B) and (D) will get +4 marks;
  choosing ONLY (A) and (B) will get +2 marks;
  choosing ONLY (A) and (D) will get +2 marks;
  choosing ONLY (B) and (D) will get +2 marks;
  choosing ONLY (A) will get +1 mark;
  choosing ONLY (B) will get +1 mark;
  choosing ONLY (D) will get +1 mark;
  choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −1 marks.

1. Let
   $\begin{equation*} P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}\end{equation*} $$\begin{equation*} P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}\end{equation*}$$\begin{equation*} P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}\end{equation*}$$\begin{equation*} \hspace{0.5cm} and \hspace{0.5cm}X = \sum \limits_{k=1}^{6} P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^T\end{equation*}$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?
   1. If $X \begin{equation*} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \end{equation*}$, then $\alpha $=30
   2. $X$ is a symmetric matrix.
   3. The sum of diagonal entries of $X$ is 18.
   4. $X - 30I$ is an invertible matrix
2. Let $ x \in R$ and let
   $\begin{equation*} P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{bmatrix} Q = \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{bmatrix} \end{equation*}$ and $R = PQP^{ - 1} $. Then which of the following options is/are correct?
   1. There exists a real number $x$ such that $PQ$ = $QP$.
   2. det $R$ = det $\begin{equation*} \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 5 \end{bmatrix} + 8 \end{equation*}$, for all $x \in R $
   3. For $x$ =0, if $R$ $\begin{equation*} \begin{bmatrix} 1 \\ a \\ b \end{bmatrix} = 6 \begin{bmatrix} 1 \\ a \\ b \end{bmatrix} \end{equation*}$, then $a+b$=5.
   4. For $x$ = 1, there exists a unit vector $ \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} $ for which $R$ $\begin{equation*} \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$
3. For non-negative integers $n$, let
   $f(n)=\frac{ \sum \limits_{k=0}^{n} sin \left( \frac{k+1}{n+2} \pi \right) sin \left( \frac{k+2}{n+2} \pi \right)} { \sum \limits_{k=0}^{n} sin^2 \left( \frac{k+1}{n+2} \pi \right)} $
   Assuming $cos^{-1}x$ takes values in $[0, \pi ]$, which of the following options is/are correct?
   1. $f(4)$ = $\frac{\sqrt{3}}{4}$
   2. $\lim \limits_{ n \to \infty} f(n) = \frac{1}{2}$
   3. If $\alpha = tan(cos^{ - 1}f(6)),$$ \alpha^2 + 2 \alpha - 1 = 0$
   4. sin(7 cos$^{ - 1} f(5))$ = 0
4. Let $f : R -> R$ be a function. We say that $f$ has
   PROPERTY 1 if $ \lim \limits_{h \to 0} \frac{f(h) - f(0)}{\sqrt{|h|}} $ exists and is finite, and
   PROPERTY 2 if $ \lim \limits_{h \to 0} \frac{f(h) - f(0)}{h^2}$ exists and is finite. Then which of the following options is/are correct?
   1. $f(x)$ = $|x|$ has PROPERTY 1
   2. $f(x)$ = $x^{2/3}$ has PROPERTY 1
   3. $f(x)$ = $x|x|$ has PROPERTY 2
   4. $f(x)$ = $sin x$ has PROPERTY 2
5. Let
   $ f (x) = \frac{sin \pi x} {x^2}, x >0$.
   Let $ x_1 < x_2 < x_3 <... < x_n <... $ be all points of local maximum of $f$
   and $ y_1 < y_2 < y_3 <... < y_n <...$ be all points of local minimum of $f$.
   Then which of the following options is/are correct?
   1. $ x_1 < x_2$
   2. $x_{n+1} - x_n$ > 2 for every $n$
   3. $x_n \in \left(2n, 2n + \frac{1}{2} \right)$ for every $n$
   4. $|x_n - y_n |$ > 1 for every $n$
6. For $a \in R$, $|a|$ > 1, let
   $\lim \limits_{n \to \infty} \left(\frac{1 + \sqrt[3]{2} +... + \sqrt[3]{n}}{n^{7/3} \left(\frac{1}{(an+1)^2}+\frac{1}{(an+2)^2}+... +\frac{1}{(an+n) ^2} \right)}\right)$ = 54. Then the possible value(s) of $a$ is/are
   1. -9
   2. -6
   3. 7
   4. 8
7. Let $f : R$ - > $R$ be given by $f(x)$ = $(x - 1)$$(x - 2)$$(x - 5)$. Define
   $F(x)$ = $ \int \limits_0^x f(t) dt, x > 0 $. Then which of the following options is/are correct?
   1. $F$ has a local minimum at $x$ = 1
   2. $F$ has a local maximum at $x$ = 2
   3. $F$ has two local maxima and one local minimum in $(0, \infty) $
   4. $F(x)$ $\neq$ 0 for all $x$ $\in$ (0,5)
8. Three lines
   $L_1 : \vec{r} = \lambda \hat{i}, \lambda \in R, $
   $L_2: \vec{r} = \hat{k} + \mu \hat{j}, \mu \in R$ and
   $L_3 : \vec{r} = \hat{i} + \hat{j} + v \hat{k}, v \in R$
   are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $ L_3$ so that $P$, $Q$ and $R$ are collinear?
   1. $\hat{k} - \frac{1}{2} \hat{j} $
   2. $\hat{k}$
   3. $ \hat{k} + \frac{1}{2} \hat{j} $
   4. $ \hat{j} + \hat{k} $

SECTION 2 (Maximum Marks : 18)

• This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 If ONLY the correct numerical value is entered at the designated place;
  • Zero Marks: 0 In all other cases.

9. Suppose
   det $\begin{equation*} \begin{bmatrix} \sum \limits_{k=0}^{n}k & \sum \limits_{k=0}^{n} {^{n}C_{k}k^2} \\ \sum \limits_{k=0}^{n} {^{n}C_{k}k} & \sum \limits_{k=0}^{n} {^{n}C_{k}3^k} \end{bmatrix}=0 \end{equation*}$
   hold for some positive integer $n$. Then $\sum \limits_{k=0}^{n} \frac{^{n}C_{k}} {k+1}$ equals...
10. Five persons $A$, $B$, $C$, $D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue, green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is...
11. Let $|X|$ denote the number of elements in a set $X$. Let $S$ = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered points $(A, B)$ such that $1 \leq |B| \leq |A|$, equals...
12. The value of
    $sec^{-1} $ $\left( \frac{1}{4} \sum \limits_{k=0}^{10}sec \left( \frac{7 \pi} {12}+ \frac{k \pi} {2} \right) sec \left( \frac{ 7 \pi} {12} + \frac{(k+1) \pi} {2} \right) \right)$
    in the interval $[ - \frac{\pi} {4}, \frac{3 \pi} {4}]$ equals...
13. The value of the integral
    $\int \limits_0^{\pi /2} \frac{3 \sqrt{cos \theta}} {( \sqrt{cos \theta} + \sqrt{sin \theta})^5} d \theta$
    equals...
14. Let $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + \hat{k}$ be two vectors. Consider a vector $ \vec{c} = \alpha \vec{a} + \beta \vec{b}, \alpha, \beta \in R$. If the projection of $ \vec{c} $ on the vector $( \vec{a} + \vec{b}) $ is

SECTION 3 (Maximum Marks:12)

• This section contains TWO (02) Matching List Sets.
• Each set has TWO (02) Multiple Choice Question.
• Each set has TWO lists: List-I and List-II.
• List-I has Four entries (I), (II), (III) and (IV) and List-II has Six entries (P), (Q), (R), (S), (T) and (U).
• FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 ONLY if the option corresponding to the correct combination is chosen;
  • Zero Marks: 0 If none of the options is chosen (i.e. the question is unanswered);
  • Negative Marks: -1 In all other cases.

Answer Q. 15 and Q. 16 by appropriately matching the lists based on the information given in the paragraph.

Let $f(x) = sin( \pi cos x)$ and $g(x) = cos( 2 \pi sinx)$ be two functions defined for $x$ >0. Define the following sets whose elements are written in the increasing order :
$X$ = { $x$ : $f(x)$ = 0}, $Y$ = { $x$ : $f'(x)$ = 0}, $Z$ = { $x$ : $g(x)$ = 0}, $W$ = { $x$ : $g'(x)$ = 0}
List - I contains the sets $X$, $Y$, $Z$ and $W$. List - II contains some informations regarding these sets.
                                                                             

	List - I		List - II
(I)	$X$	(P)	$\supseteq \left \{ \frac{ \pi}{2}, \frac{3 \pi} {2}, 4 \pi, 7 \pi, \right \}$
(II)	$Y$	(Q)	an arithmetic progression
(III)	$Z$	(R)	NOT an arithmetic progression
(IV)	$W$	(S)	$\supseteq \left \{ \frac{ \pi}{6}, \frac{7 \pi} {6}, \frac{13 \pi} {6} \right \}$
		(T)	$\supseteq \left \{ \frac{ \pi}{3}, \frac{2 \pi} {3}, \pi \right \}$
		(U)	$ \supseteq \left \{ \frac{ \pi}{6}, \frac{3 \pi} {4}, \right \}$

15. Which of the following is the only CORRECT combination?
    1. (I), (P), (R)
    2. (II), (Q), (T)
    4. (I), (Q), (U)
    5. (II), (R), (S)
16. Which of the following is the only CORRECT combination?
    1. (III), (R), (U)
    2. (IV), (P), (R), (S)
    3. (III), (P), (Q), (U).
    4. (IV), (Q), (T)

Answer Q. 17 and Q. 18 by appropriately matching the lists based on the information given in the paragraph.

Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x - 3)^2 + (y - 4)^2 = 16$, intersects at the point $X$ and $Y$. Suppose that another circle $C_3 : (x - h) ^2 + (y - k) ^2 = r^2$ satisfies the following conditions :
(i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$.
(ii) $C_1$ and $C_2$ both lie inside $C_3$, and
$C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8ay$. There are some expressions given in the List - I whose values are given in List - II below:

                                                                                                          

	List - I		List - II
(I)	$2h$ + $k$	(P)	6
(II)	$\frac{length of ZW} {length of XY}$	(Q)	$\sqrt{6}$
(III)	$\frac{Area of triangle MZN} {Area of triangle ZMW}$	(R)	$\frac{5}{4}$
(IV)	$\alpha$	(S)	$\frac{21}{5}$
		(T)	$2 \sqrt{6}$
		(U)	$\frac{10}{3}$

15. Which of the following is the only CORRECT combination?
    1. (I), (S)
    2. (I), (U)
    3. (II), (Q)
    4. (II), (T)
16. Which of the following is the only INCORRECT combination?
    1. (I), (P)
    2. (IV), (U)
    3. (III), (R)
    4. (IV), (S)

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