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SECTION 1 (Maximum Marks:12)

• This section contains THREE (03) questions.
• Each question has FOUR options (A), (B), (C) and (D). 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 ONLY if (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, 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: -2 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 (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -2 marks.

1. Let $S$= $(0,1) \cup (1,2) \cup (3,4)$ and $T$={0,1, 2,3} . Then which of the following statements is(are) true?
   1. There are infinitely many functions from $S$ to $T$
   2. There are infinitely many strictly increasing functions from $S$ to $T$
   3. The number of continuous functions from $S$ to $T$ is at most 120
   4. Every continuous function from $S$ to $T$ is differentiable
2. Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E$:$\frac{x^2}{6}$+$\frac{y^2}{3}$=1 and the parabola $P$:$y^2$=$12x$ . Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
   1. The area of the quadrilateral $A_1$$A_2$$A_3$$A_4$ is 35 square units
   2. The area of the quadrilateral $A_1$$A_2$$A_3$$A_4$ is 36 square units
   3. The tangents $T_1$ and $T_2$ meet the $x$ -axis at the point ( -3,0)
   4. The tangents $T_1$ and $T_2$ meet the $x$ -axis at the point ( -6,0)
3. Let $f :[0,1] \to [0,1]$ be the function defined by $f(x)$=$\frac{x^3}{3}-$$x^2$+$\frac{5}{9}x$+$\frac{17}{36}$ . Consider the square region $S$=[0,1] × [0,1]. Let $G$=$\left\{ (x, y) \in S \right.$ : $\left. y > f(x) \right\}$ be called the green region and $R$=$\left\{ (x, y) \in S \right.$ : $\left. y < f(x) \right\}$ be called the red region. Let $L_h$=$\left\{ (x, h) \in S \right.$ : $\left. x \in [0, 1] \right\}$ be the horizontal line drawn at a height $h \in [0,1]$. Then which of the following statements is(are) true?
   1. There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the green region below the line $L_h$
   2. There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the red region below the line $L_h$
   3. There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the red region below the line $L_h$
   4. There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the green region below the line $L_h$

SECTION 2 (Maximum Marks:12)

• This section contains FOUR (04) questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
• For each question, choose the option corresponding to the correct answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 If ONLY the correct option 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.

4. Let $f:(0, 1) \to R$ be the function defined as $f(x)$=$\sqrt{n}$ if $x \in \left[ \frac{1}{n+1}, \frac{1}{n} \right)$ where $n \in N$. Let $g:(0, 1) \to R$ be a function such that $\int \limits_{x^2}^x \sqrt{\frac{1-t}{t}}dt < g(x) < 2 \sqrt{x}$ for all $x \in (0, 1)$. Then $\lim \limits_{x \to 0} f(x)g(x)$
   
   1. does NOT exist
   2. is equal to 1
   3. is equal to 2
   4. is equal to 3
5. Let $Q$ be the cube with the set of vertices $\left\{(x_1, x_2, x_3) \in R^3 \right.$ : $\left. x_1, x_2, x_3 \in \text{{0, 1}} \right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in $S$ . For lines $l_1$ and $l_2$, let $d(l_1, l_2)$ denote the shortest distance between them. Then the maximum value of $d(l_1, l_2)$, as $l_1$ varies over $F$ and $l_2$ varies over $S$, is
   
   1. $\frac{1}{\sqrt{6}}$
   2. $\frac{1}{\sqrt{8}}$
   3. $\frac{1}{\sqrt{3}}$
   4. $\frac{1}{\sqrt{12}}$
6. Let $X$=$\left\{ (x,y) \in Z×Z : \frac{x^2}{8} + \right.$$\left. \frac{y^2}{20} < 1 \text{ and } y^2 < 5x \right\}$ . Three distinct points $P$, $Q$ and $R$ are randomly chosen from $X$ . Then the probability that $P$, $Q$ and $R$ form a triangle whose area is a positive integer, is
   
   1. $\frac{71}{220}$
   2. $\frac{73}{220}$
   3. $\frac{79}{220}$
   4. $\frac{83}{220}$
7. Let $P$ be a point on the parabola $y^2$=$4ax$ , where $a > 0$ . The normal to the parabola at $P$ meets the $x$ -axis at a point $Q$. The area of the triangle $PFQ$ , where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is
   
   1. (2, 3)
   2. (1, 3)
   3. (2, 4)
   4. (3, 4)

SECTION 3 (Maximum Marks:24)

• This section contains SIX (06) questions.
• The answer to each question is a NON-NEGATIVE INTEGER.
• For each question, enter the correct integer corresponding to the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If ONLY the correct integer is entered;
  • Zero Marks: 0 In all other cases.

8. Let $\tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for all $x \in R$. Then the number of real solutions of the equation $\sqrt{1+\cos2x}$=$\sqrt{2} \tan^{-1}(\tan x)$ in the set $\left(-\frac{3\pi}{2}, -\frac{\pi}{2}\right) \cup \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$ is equal to
9. Let $n \geq 2$ be a natural number and $f:[0,1] \to R$ be the function defined by
   $\left\{ \begin{array}{lr}n(1-2nx) & \text { if } 0 \leq x \leq \frac{1}{2n} \\ 2n(2nx-1) & \text { if } \frac{1}{2n} \leq x \leq \frac{3}{4n} \\ 4n(1-nx) & \text { if } \frac{3}{4n} \leq x \leq \frac{1}{n} \\ \frac{n}{n-1}(nx-1) & \text { if } \frac{1}{n} \leq x \leq 1 \end{array} \right.$
   If $n$ is such that the area of the region bounded by the curves $x=0$ , $x=1$ , $y=0$ and $y$=$f(x)$ is 4 , then the maximum value of the function $f$ is
10. Let $7 \overbrace{ 5 \cdots 5 }^{\mathstrut r} 7$ denote the $( r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S$=77+757+7557+...+$7 \overbrace{ 5 \cdots 5 }^{\mathstrut 98} 7$. If $S$=$\frac{7 \overbrace{ 5 \cdots 5 }^{\mathstrut 99} 7+m}{n}$, where $m$ , $n$ are natural numbers less than 3000, then the value of $m+n$ is
11. Let $A$=$\left\{\frac{1967+1686 i sin \theta}{7-3i cos \theta} : \theta \in R \right \}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is
12. Let $P$ be the plane $\sqrt{3}x$+$2y$+$3z$=16 and let $S$=$ \left\{ \alpha \hat{i}+\beta \hat{j}+ \gamma \hat{k} \right.$ : $\alpha^2+ \beta^2+ \gamma^2 =1$ and the distance of $(\alpha, \beta, \gamma)$ form the plane $P$ is $\left. \frac{7}{2} \right\}$. Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u}-\vec{v}|$=$|\vec{v}-\vec{w}|$=$|\vec{w}-\vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}$, $\vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is
13. Let $a$ and $b$ be two non-zero real numbers. If the coefficient of $x^5$ in the expansion of $\left( ax^2 + \frac{70}{27bx} \right)^4$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left( ax - \frac{1}{bx^2} \right)^7$, then the value of $2b$ is

SECTION 4 (Maximum Marks:12)

• This section contains FOUR (04) Matching List Sets.
• Each set has ONE Multiple Choice Question.
• Each set has TWO lists: List-I and List-II.
• List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5).
• 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.

14. Let $\alpha$, $\beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
    $x$+$2y$+$z$=7
    $x$+$\alpha z$=11
    $2x-$$3y$+$\beta z$=$\gamma$
    
    Match each entry in List-I to the correct entries in List-II
                             

    	List - I		List - II
    (P)	If $\beta$=$\frac{1}{2}(7\alpha-3)$ and $\gamma$=28, then the system has	(1)	a unique solution
    (Q)	If $\beta$=$\frac{1}{2}(7\alpha-3)$ and $\gamma \neq$28, then the system has	(2)	no solution
    (R)	If $\beta \neq$$\frac{1}{2}(7\alpha-3)$ where $\alpha$=1 and $\gamma \neq$28, then the system has	(3)	infinitely many solutions
    (S)	If $\beta \neq$$\frac{1}{2}(7\alpha-3)$ where $\alpha$=1 and $\gamma$=28, then the system has	(4)	$ x $=11, $y$=-2 and $z$ = 0 as a solution
    		(5)	$ x $=-15, $y$=4 and $z$ = 0 as a solution

    
    
    The correct option is:
    
    
    1. (P)->(3) (Q)->(2) (R)->(1) (S)->(4)
    2. (P)->(3) (Q)->(2) (R)->(5) (S)->(4)
    3. (P)->(2) (Q)->(1) (R)->(4) (S)->(5)
    4. (P)->(2) (Q)->(1) (R)->(1) (S)->(3)
15. Consider the given data with frequency distribution

    $x_i$	3	8	11	10	5	4
    $f_i$	5	2	3	2	4	4

    
    Match each entry in List-I to the correct entries in List-II                          

    	List - I		List - II
    (P)	The mean of the above data is	(1)	2.5
    (Q)	The median of the above data is	(2)	5
    (R)	The mean deviation about the mean of the above data is	(3)	6
    (S)	The mean deviation about the median of the above data is	(4)	2.7
    		(5)	2.4

    
    
    The correct option is:
    
    1. (P)->(3) (Q)->(2) (R)->(4) (S)->(5)
    2. (P)->(3) (Q)->(2) (R)->(1) (S)->(5)
    3. (P)->(2) (Q)->(3) (R)->(4) (S)->(1)
    4. (P)->(3) (Q)->(3) (R)->(5) (S)->(5)
16. Let $l_1$ and $l_2$ be the lines $\vec{r}$=$\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r_2}$=$(\hat{j}-\hat{k})$+$\mu(\hat{i}+\hat{k})$ , respectively. Let $X$ be the set of all the planes $H$ that contain the line $l_1$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $l_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d(H_0)$ is the maximum value of $d(H_0)$ as $H$ varies over all planes in $X$.
    
    Match each entry in List-I to the correct entries in List-II                          

    	List - I		List - II
    (P)	The value of $d(H_0)$ is	(1)	$\sqrt{3}$
    (Q)	The distance of the point (0,1,2) from $H_0$ is	(2)	$\frac{1}{\sqrt{3}}$
    (R)	The distance of origin from $H_0$ is	(3)	0
    (S)	The distance of origin from the point of intersection of planes $y$ = $z$, $x = 1$ and $H_0$ is	(4)	$\sqrt{2}$
    		(5)	$\frac{1}{\sqrt{2}}$

    
    The correct option is:
    
    1. (P)->(2) (Q)->(4) (R)->(5) (S)->(1)
    2. (P)->(5) (Q)->(4) (R)->(3) (S)->(1)
    3. (P)->(2) (Q)->(1) (R)->(3) (S)->(2)
    4. (P)->(5) (Q)->(1) (R)->(4) (S)->(2)
17. Let $z$ be a complex number satisfying $|z|^3$+$2z^2$+$4\bar{z}-$8=0, where $z$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be nonzero.
    
    Match each entry in List-I to the correct entries in List-II                          

    	List - I		List - II
    (P)	$|z|^2$ is equal to	(1)	12
    (Q)	$|z-\bar{z}|^2$ is equal to	(2)	4
    (R)	$|z|^2$+$|z+\bar{z}|^2$	(3)	8
    (S)	$|z+1|^2$ is equal to	(4)	10
    		(5)	7

    
    The correct option is:
    
    1. (P)->(1) (Q)->(3) (R)->(5) (S)->(4)
    2. (P)->(2) (Q)->(1) (R)->(3) (S)->(5)
    3. (P)->(2) (Q)->(4) (R)->(5) (S)->(1)
    4. (P)->(2) (Q)->(3) (R)->(5) (S)->(4)

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