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SECTION 1 (Maximum Marks:28)

• This section contains SEVEN 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, darken the bubble(s) corresponding to all the correct option(s) in the ORS.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened;
  • Partial Marks: +1 For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened;
  • Zero Marks: 0 If none of the bubbles is darkened;
  • Negative Marks: -2 In all other cases.
• For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in +4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) result in -2 marks, as a wrong option is also darkened.

1. If $2x − $$y$ + 1 = 0 is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}$= 1, then which of the following CANNOT be sides of a right angled triangle?
   1. $a$, 4, 1
   2. $a$, 4, 2
   3. $2a$, 8, 1
   4. $2a$, 4, 1
2. If a chord, which is not a tangent, of the parabola $y^2 = 16x$ has the equation $2x$ + $y$ = $p$, and midpoint $(ℎ, k)$, then which of the following is(are) possible value(s) of $p$, $ℎ$ and $k$?
   1. $p$ = −2, $ℎ$ = 2, $k$ = −4
   2. $p$ = −1, $ℎ$ = 1, $k$ = −3
   3. $p$ = 2, $ℎ$ = 3, $k$ = −4
   4. $p$ = 5, $ℎ$ = 4, $k$ = −3
3. Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x) = x cos(\pi(x + [x]))$ is discontinuous?
   1. $x$ = −1
   2. $x$ = 0
   3. $x$ = 1
   4. $x$ = 2
4. Let $f: ℝ$ → (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1)?
   1. $x^9 - f(x)$
   2. $x -$ $\int \limits_0^{\frac{\pi}{2}-x}f(t)costdt$
   3. $e^x$-$\int \limits_0^xf(t)sintdt$
   4. $f(x)$ + $\int \limits_0^xf(t)sintdt$
5. Which of the following is(are) NOT the square of a 3×3 matrix with real entries?
   1. $\begin{equation*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{equation*}$
   2. $\begin{equation*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \end{equation*}$
   3. $\begin{equation*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \end{equation*}$
   4. $ \begin{equation*} \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \end{equation*}$
6. Let $a$, $b$, $x$ and $y$ be real numbers such that $a − b$ = 1 and $y$ ≠ 0. If the complex number $z$ = $x$ + $iy$ satisfies $Im \left(\frac{az+b}{z+1} \right)$= $y$, then which of the following is(are) possible value(s) of $x$?
   1. -1+$\sqrt{1-y^2}$
   2. -1-$\sqrt{1-y^2}$
   3. 1+$\sqrt{1+y^2}$
   4. 1-$\sqrt{1+y^2}$
7. Let $X$ and $Y$ be two events such that $P(X) = \frac{1}{3}$ , $P(X|Y) = \frac{1}{2}$ and $P(Y|X) = \frac{3}{5}$. Then
   1. $P(Y) = \frac{4}{15}$
   2. $P(X'|Y) = \frac{1}{2}$
   3. $P(X \cap Y) = \frac{1}{5}$
   4. $P(Y|X) = \frac{2}{5}$

SECTION 2 (Maximum Marks:15)

• This section contains FIVE questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE.
• For each question, darken the bubble corresponding to the correct integer in the ORS.
• Marking scheme:
  
  • Full Marks: +3 If the bubble corresponding to the answer is darkened
  • Zero Marks: 0 In all other cases.

8. If the bubble corresponding to the answer is darkened For how many values of $p$, the circle $x^2$ + $y^2$ + $2x$ + $4y − p$ = 0 and the coordinate axes have exactly three common points?
9. Let $f:R \to R$ be a differentiable function such that $f(0)$=0, $f\left(\frac{\pi}{2} \right)$=3 and $f'(0)$=1. If $g(x)$=$\int \limits_x^{\frac{\pi}{2}}[f'(t) \text{cosec t − cot t cosec t} f(t)]dt$ for $x \in (0, \frac{\pi}{2}]$, then $\lim \limits_{x \to 0}$ $g(x)$=
10. For a real number $\alpha$, if the system $\begin{equation*} \begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix} \end{equation*}$ $\begin{equation*} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{equation*}$ = $\begin{equation*} \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \end{equation*}$ of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2$ =
11. Words of length 10 are formed using the letters $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$, $J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x}$ =
12. The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

SECTION 3 (Maximum Marks:18)

• This section contains SIX questions of matching type.
• This section contains TWO tables (each having 3 columns and 4 rows).
• Based on each table, there are THREE questions .
• Each question has FOUR options [A], [B], [C] and [D].ONLY ONE of these four options is correct.
• For each question, darken the bubble corresponding to the correct option in the ORS.
• For each question, marks will be awarded in one of the following categories:
  
• Marking scheme:
  
  • Full Marks: +3 If the bubble corresponding to the correct option is darkened
  • Zero Marks: 0 If none of the bubbles is darkened
  • Negative Marks: -1 In all other cases.

                       

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
Column 1	Column 2	Column 3
(I) $x^2$+$y^2$=$a^2$	(i) $my$=$mx^2$+$a$	(P) $\left(\frac{a} {m^2}, \frac{2a}{m} \right) $
(II) $x^2$+$a^2y^2$=$a^2$	(ii) $y$=$mx$+$a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma} {\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}} \right) $
(III) $y^2$=4$a$x	(iii) $y$=$mx$+$\sqrt{a^2m^2-1}$	(R) $\left(\frac{-a^2m} {\sqrt{a^2m^2+1} }, \frac{1}{\sqrt{a^2m^2+1}} \right) $
(IV) $x^2-$ $a^2y^2$=$a^2$	(iv) $y$=$mx$+$\sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m} {\sqrt{a^2m^2-1} }, \frac{-1}{\sqrt{a^2m^2-1}} \right) $

13. For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?
    1. (I) (i) (P)
    2. (I) (ii) (Q)
    3. (II) (ii) (Q)
    4. (III) (i) (P)
       
14. If a tangent to a suitable conic (Column 1) is found to be $y$ = $x$ + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?  
    1. (I) (ii) (Q)
    2. (II) (iv) (R)
    3. (III) (i) (P)
    4. (III) (ii) (Q)
       
15. The tangent to a suitable conic (Column 1) at $(\sqrt{3}, \frac{1}{2})$ is found to be $\sqrt{3}x$+$2y$ = 4, then which of the following options is the only correct combination?  
    1. (IV) (iii) (S)
    2. (IV) (iv) (S)
    3. (II) (iii) (R)
    4. (II) (iv) (R)
       

                       

Let $f(x)$ = $x$ + $log_e x$ - $xlog_e x$, $x \in (0, \infty) $. Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f"(x)$. Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f"(x)$ at infinity. Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.
Column 1	Column 2	Column 3
(I) $f(x)$ = 0 for some $x \in (1, e^2)$	(i) $\lim \limits_{x \to \infty} f(x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f(x)$ = 0 for some $x \in (1, e)$	(ii) $\lim \limits_{x \to \infty} f(x)$ = $- \infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x)$ = 0 for some $x \in (0, 1)$	(iii) $\lim \limits_{x \to \infty} f'(x)$ = $- \infty$	(R) $f'$ is increasing in (0, 1)
(IV) $f"(x)$ = 0 for some $x \in (1, e)$	(iv) $\lim \limits_{x \to \infty} f"(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

16. Which of the following options is the only correct combination?  
    1. (I) (i) (P)
    2. (II) (ii) (Q)
    3. (III) (iii) (R)
    4. (IV) (iv) (S)
       
17. Which of the following options is the only correct combination?  
    1. (I) (ii) (R)
    2. (II) (iii) (S)
    3. (III) (iv) (P)
    4. (IV) (i) (S)
       
18. Which of the following options is the only correct combination?  
    1. (I) (iii) (P)
    2. (II) (iv) (Q)
    3. (III) (i) (R)
    4. (II) (iii) (P)
       

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