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SECTION 1 (Maximum Marks:24)

• This section contains SIX (06) questions.
• Each question has FOUR options for correct answer(s) . ONE OR MORE THAN ONE of these four option(s) is(are) correct option(s).
• For each question, choose the correct option(s) to answer the question.
• 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: -2 In all other cases.
• For example, If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) ,without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.

1. For any positive integer $n$, define $f_n:(0,\infty) \to R$ as $f_n(x)=\sum \limits_{j=1}^{n} tan^{-1}$ $ \left(\frac{1}{1+(x+j)(x+j-1)}\right)$ for all $x \in (0,\infty)$. (Here, the inverse trigonometric function $tan^{-1}x$ assumes values in $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.) Then, which of the following statement(s) is (are) TRUE?
   1. $\sum \limits_{j=1}^{5}tan^2(f_j(0))$=55
   2. $\sum \limits_{j=1}^{10}(1+f_j'(0))sec^2(f_j(0))$=10
   3. For any fixed positive integer $n$, $\lim \limits_{x \to \infty} tan(f_n(x))$=$\frac{1}{n}$
   4. For any fixed positive integer $n$, $\lim \limits_{x \to \infty} sec^2(f_n(x))$=1
2. Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1,S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1,1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then, which of the following statement(s) is (are) TRUE?
   1. The point (-2,7) lies in $E_1$
   2. The point $\left(\frac{4}{5},\frac{7}{5}\right)$ does NOT lie in $E_2$
   3. The point $\left(\frac{1}{2},1\right)$ lies in $E_2$
   4. The point $\left(0,\frac{3}{2}\right)$ does NOT lie in $E_1$
3. Let $S$ be the set of all column matrices $\begin{equation*} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \end{equation*}$ such that $b_1$, $b_2$ and $b_3 \in R$ and the system of equations (in real variables) $ -x + 2y + 5z = b_1$
   $ 2x - 4y + 3z = b_2$
   $ x - 2y + 2z = b_3$
   has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\begin{equation*} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \in S \end{equation*}$?
   1. $ x + 2y + 3z = b_1$, $4y+5z=b_2$ and $x+2y+6z=b_3$
   2. $ x + y + 3z = b_1$, $5x+2y+6z=b_2$ and $-2x-y-3z=b_3$
   3. $ -x + 2y -5z = b_1$, $2x-4y+10z=b_2$ and $x-2y+5z=b_3$
   4. $ x + 2y +5z = b_1$, $2x+3z=b_2$ and $x+4y-5z=b_3$
4. Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$ .Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$ , then which of the following statement(s) is (are) TRUE?
   1. For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 1
   2. For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
   3. The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x$=1 is $\frac{1}{4\sqrt{2}}(\pi-2)$
   4. The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x$=1 is $\frac{1}{16}(\pi-2)$
5. Let $s$, $t$, $r$ be non-zero complex numbers and $L$ be the set of solutions $z$=$x$ +$iy$ $(x, y\in R, i=\sqrt{-1})$ of the equation $sz+t\bar{z}+r=0$, where $\bar{z}=x-iy$ .Then, which of the following statement(s) is (are) TRUE?
   1. If $L$ has exactly one element, then $|s| \neq |t|$
   2. If $|s|$=$|t|$, then $L$ has infinitely many elements
   3. The number of elements in $L$ $\cap{z:|z-1+i|=5}$ is at most 2
   4. If $L$ has more than one element, then $L$ has infinitely many elements
6. Let $f:(0,\pi) \to R$ be a twice differentiable function such that $\lim \limits_{t \to x} \frac{f(x)sint- f(t)sinx}{t-x}=sin^2x$ for all $x \in (0,\pi)$. If $f\left(\frac{\pi}{6}\right)$=$-\frac{\pi}{12}$, the which of the following statement(s) is (are) TRUE?
   1. $f\left(\frac{\pi}{4}\right)$=$-\frac{\pi}{4\sqrt{2}}$
   2. $f(x)$=$\frac{x^4}{6} -x^2$ for all $x \in (0,\pi)$
   3. There exists $\alpha \in (0, \pi)$ such that $f'(\alpha)$=0
   4. $f"(\frac{\pi}{2})+f(\frac{\pi}{2})=0$

SECTION 2 (Maximum Marks:24)

• This section contains EIGHT (08) questions. The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) 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: +3 ONLY if the correct numerical value is entered as answer;
  • Zero Marks: 0 In all other cases.

7. The value of the integral $\int \limits_0^{\frac{1}{2}} \frac{1+\sqrt{3}}{((x+1)^2(1-x)^6)^{\frac{1}{4}}}$dx is ........
8. Let $P$ be a matrix of order 3 x 3 such that all the entries in $P$ are from the set {-1, 0, 1}. Then, the maximum possible value of the determinant of $P$ is _____ .
9. Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac{1}{5!}(\beta - \alpha)$ is _____ .
10. Let $f:R \to R$ be a differentiable function with $f(0)$=0. If $y$ =$f(x)$ satisfies the differentiable equation $\frac{dy}{dx}=(2+5y)(5y-2)$, then the value of $\lim \limits_{x \to - \infty} f(x)$ is .......
11. Let $f:R \to R$ be a differentiable function with $f(0)$=1 and satisfying the equation $f(x+y)$=$f(x)f'(y)$+$f'(x)f(y)$ for all $x \in R$. Then the value of $log_e(f(4))$ is .......
12. Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y$=3 (that is, the line segment $PQ$ is perpendicular to the plane $x+y$=3 and the mid-point of $PQ$ lies in the plane $x+y$=3) lies on the $z-$ axis. Let the distance of $P$ from the $x-$axis be 5. If $R$ is the image of $P$ in the $xy-$plane, then the length of $PR$ is _____.
13. Consider the cube in the first octant with sides $OP$, $OQ$ and $OR$ of length 1, along the $x-$axis, $y-$axis and $z-$axis, respectively, where $O(0,0,0)$ is the origin. Let $S(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT$. If $\vec{p}=\vec{SP}$, $\vec{q}=\vec{SQ}$, $\vec{r}=\vec{SR}$ and $\vec{t}=\vec{ST}$, then the value of $|(\vec{p}×\vec{q})×(\vec{r}×\vec{t}|$ is _____ .
14. Let $X$= $(^{10}C_1)^2$+$2(^{10}C_2)^2$+$3(^{10}C_3)^2$+...+$19(^{10}C_{10})^2$, where $^{10}C_r, r\in$ {1, 2, ...10} denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is _____ .

SECTION 3 (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 (I), (II), (III) and (IV) and List-II has Five entries (P), (Q), (R), (S) and (T).
• 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.

15. Let $E_1$ = $\left\{x \in R : x\neq 1 \right.$ and $\left.\frac{x}{x-1}>0\right\}$
    and $E_2$ = $\left\{x \in E_1 : sin^{-1} (log_e(\frac{x}{x-1}))\right.$ is a real number }.
    (Here, the inverse trigonometric function $sin^{-1}x$ assumes values in $[-\frac{\pi}{2},\frac{\pi}{2}].$)
    Let $f:E_1 \to R$ be the function defined by $f(x)$= $log_e(\frac{x}{x-1})$
    and $g:E_2 \to R$ be the function defined by $g(x)$= $sin^{-1}(log_e(\frac{x}{x-1}))$                              

    	List - I		List - II
    P.	The range of $f$ is	1.	$(-\infty, \frac{1}{1-e}]\cup$ $[\frac{e}{e-1}, \infty)$
    Q.	The range of $g$ contains	2.	(0,1)
    R.	The domain of $f$ contains	3.	$[ -\frac{1}{2}, \frac{1}{2}]$
    S.	The domain of $g$ is	4.	$(-\infty,0)\cup(0,\infty)$
    		5.	$(-\infty, \frac{e}{e-1}]$
    		6.	$(-\infty, 0)\cup$ $(\frac{1}{2},\frac{e}{e-1}]$

    
    The correct option is:
    1. P → 4; Q → 2; R → 1; S → 1
    2. P → 3; Q → 3; R → 6; S → 5
    3. P → 4; Q → 2; R → 1; S → 6
    4. P → 4; Q → 3; R → 6; S → 5
16. In a high school, a committee has to be formed from a group of 6 boys $M_1$, $M_2$, $M_3$, $M_4$, $M_5$, $M_6$ and 5 girls $G_1$, $G_2$, $G_3$, $G_4$, $G_5$.
    1. Let $\alpha_1$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
    2. Let $\alpha_2$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
    3. Let $\alpha_3$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
    4. Let $\alpha_4$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M_1$ and $G_1$ are NOT in the committee together.
                                 

    	List - I		List - II
    P.	The value of $\alpha_1$ is	1.	136
    Q.	The value of $\alpha_2$ is	2.	189
    R.	The value of $\alpha_3$ is	3.	192
    S.	The value of $\alpha_4$ is	4.	200
    		5.	381
    		6.	461

    
    The correct option is:
    1. P → 4; Q → 6; R → 2; S → 1
    2. P → 1; Q → 4; R → 2; S → 3
    3. P → 4; Q → 6; R → 5; S → 2
    4. P → 4; Q → 2; R → 3; S → 1
17. Let $H$:$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a$>$b$ > 0, be a hyperbola in the $xy-$plane whose conjugate axis $LM$ subtends an angle of 60° at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.                          

    	List - I		List - II
    P.	The length of the conjugate axis of $H$ is	1.	8
    Q.	The eccentricity of $H$ is	2.	$\frac{4}{\sqrt{3}}$
    R.	The distance between the foci of $H$ is	3.	$\frac{2}{\sqrt{3}}$
    S.	The length of the latus rectum of $H$ is	4.	4

    
    The correct option is:
    1. P → 4; Q → 2; R → 1; S → 3
    2. P → 4; Q → 3; R → 1; S → 2
    3. P → 4; Q → 1; R → 3; S → 2
    4. P → 3; Q → 4; R → 2; S → 1
18. Let $f_1:R \to R$, $f_2:\left(-\frac{\pi}{2},\frac{\pi}{2} \right) \to R$, $f_3:(1-e^{\frac{\pi}{2}}-2) \to R$ and $f_4: R \to R$ be functions defined by
    1. $f_1(x)$=$sin(\sqrt{1-e^{-x^2}})$.
    2. $f_2(x)$=$\begin{cases}\frac{|sin x|}{tan^{-1}x} & \text { if } x \neq 0 \\ 1 & \text { if } x = 0 \end{cases}$, where the inverse trigonometric function $tan^{-1}x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2} \right)$,
    3. $f_3(x)$=$[sin (log_e(x+2))]$, where, for $t \in R$, $[t]$ denotes the greatest integer less than or equal to $t$,
    4. $f_4(x)$=$\begin{cases}x^2 sin \left(\frac{1}{x} \right)& \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{cases}$.
                             

    	List - I		List - II
    P.	The function $f_1$ is	1.	NOT continuous at $x$=0
    Q.	The function $f_2$ is	2.	continuous at $x$=0 and NOT differentiable at $x$=0
    R.	The function $f_3$ is	3.	differentiable at $x$=0 and its derivative is NOT continuous at $x$=0
    S.	The function $f_4$ is	4.	differentiable at $x$=0 and its derivative is continuous at $x$=0

    
    The correct option is:
    1. P → 2; Q → 3; R → 1; S → 4
    2. P → 4; Q → 1; R → 2; S → 3
    3. P → 4; Q → 2; R → 1; S → 3
    4. P → 2; Q → 1; R → 4; S → 3

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