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SECTION 1 (Maximum Marks:32)

• This section contains EIGHT 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: +4 If the bubble corresponding to the answer is darkened
  • Zero Marks: 0 In all other cases.

1. The number of distinct solutions of the equation
   $\frac{5}{4}cos ^2 2x$+$cos^4x$+$sin^4x$+$cos^6x$+$sin^6x$=2 in the interval $[0, 2\pi]$ is
2. Let the curve $C$ be the mirror image of the parabola $y^2$=4x with respect to the line $x$+$y$+4=0. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is
3. The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is
4. Let $n$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is
5. If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3)^2+(y+2)^2$=$r^2$, then the value of $r^2$ is
6. Let $f:R \to R$ be a function defined by $f(x)$=$\left\{\begin{array}{ll}[x], & x \leq 2, \\ 0, & x >2 .\end{array}\right.$,
   where $[x]$ is the greatest integer less than or equal to $x$. If $I$ = $\int \limits_{-1}^{2}\hspace{0.2cm}\frac{xf(x^2)}{2+f(x+1)}dx$, then the value of $(4I-1)$ is
7. A cylindrical container is to be made from certain solid material with the following constraints: it has a fixed inner volume of $V$ $mm^3$, has a 2 $mm$ thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 $mm$ and is of radius equal to the outer radius of the container.
   If the volume of the material used to make the container is minimum when the inner radius of the container is 10 $mm$, then the value of $\frac{V}{250 \pi}$ is
8. Let $F(x)$ =$\int \limits_{x}^{x^2+\frac{\pi}{6}}2cos^2tdt$ for all $x \in R$ and $f$ : $\left[0, \frac{1}{2}\right] \to [0, \infty)$ be a continuous function. For $a \in \left[0, \frac{1}{2}\right]$, if $F'(a)$+2 is the area of the region bounded by $x$=0, $y$=0, $y=f(x)$ and $x=a$, then $f(0)$ is

SECTION 2 (Maximum Marks:40)

• This section contains TEN questions.
• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.
• For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS.
• Marking scheme:
  
  • Full Marks: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened.
  • Zero Marks: 0 If none of the bubbles is darkened;
  • Negative Marks: -2 In all other cases.

9. Let $X$ and $Y$ be two arbitrary, 3×3, non-zero, skew-symmetric matrices and $Z$ be an arbitrary 3×3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?
   1. $Y^3Z^4-Z^4Y^3$
   2. $X^{44}+Y^{44}$
   3. $X^4Z^3-Z^3X^4$
   4. $X^{23}+Y^{23}$
10. Which of the following values of $\alpha$ satisfy the equation
    $\left|\begin{array}{ccc}(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2 \\ (2+\alpha)^2 & (2+2\alpha)^2 & (2+3\alpha)^2 \\ (3+\alpha)^2 & (3+2\alpha)^2 & (3+3\alpha)^2\end{array}\right|$=$-648 \alpha$.
    1. -4
    2. 9
    3. -9
    4. 4
11. In $R^3$, consider the planes $P_1:y=0$ and $P_2$: $x$+$z$=1. Let $P_3$ be a plane, different from $P_1$ and $ P_2$, which passes through the intersection of $P_1$ and $P_2$. If the distance of the point (0, 1, 0) from $P_3$ is 1 and the distance of a point $(\alpha, \beta, \gamma)$ from $P_3$ is 2, then which of the following relations is(are) true?
    1. $2\alpha$+$\beta$+$2\gamma$+2=0
    2. $2\alpha$-$\beta$+$2\gamma$+4=0
    3. $2\alpha$+$\beta$-$2\gamma$-10=0
    4. $2\alpha$-$\beta$+$2\gamma$-8=0
12. In $R^3$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1$: $x$+ $2y$$ -z$ +1=0 and $P_2$: $2x$$- y$ + $z$ -1=0. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P_1$. Which of the following points lie(s) on $M$?
    1. $\left(0, -\frac{5}{6}, - \frac{2}{3} \right)$
    2. $\left(-\frac{1}{6}, -\frac{1}{3}, \frac{1}{6} \right)$
    3. $\left(-\frac{5}{6}, 0, \frac{1}{6} \right)$
    4. $\left(-\frac{1}{3}, 0, \frac{2}{3} \right)$
13. Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta$OPQ is $3\sqrt{2}$, then which of the following is (are) the coordinates of $P$?
    1. $(4, 2\sqrt{2})$
    2. $(9, 3\sqrt{2})$
    3. $\left(-\frac{1}{4}, \frac{1}{\sqrt{2}} \right)$
    4. $(1, \sqrt{2})$
14. Let $y(x)$ be a solution of the differential equation $(1+e^x)y'$+$ye^x$=1. If $y(0)$=2, then which of the following statements is(are) true?
    1. $y(-4)$=0
    2. $y(-2)$=0
    3. $y(x)$ has a critical point in the interval (-1, 0)
    4. $y(x)$ has no critical point in the interval (-1, 0)
15. Consider the family of all circles whose centers lies on the straight line $y=x$. If this family of circles is represented by the differential equation $Py"$ + $Qy'$ +1=0, where $P$, $Q$ are functions of $x$, $y$ and $y'$ ( here $y'$=$\frac{dy}{dx}$, $y"$=$\frac{d^2y}{dx^2}$), then which of the following statements is(are) true?
    1. $P$=$y$+$x$
    2. $P$=$y$-$x$
    3. $P$+$Q$=1-$x$+$y$+$y'$+$(y')^2$
    4. $P$-$Q$=$x$+$y$-$y'$-$(y')^2$
16. Let $g : R \to R$ be a differential function with $g(0)$=0, $g'(0)$=0 and $g'(1) \neq 1$. Let
    $f(x)$=$\left\{\begin{array}{ll}\frac{x}{|x|}g(x), & x \neq 0, \\ 0, & x =0 .\end{array}\right.$
    and
    $h(x)$=$e^{|x|}$ for all $x \in R$. Let $(f ◦ h)(x)$ denote $f(h(x))$ and $(h ◦ f)(x)$ denote $h(f(x))$. Then which of the following is(are) true?
    1. $f$ is differentiable at $x$=0
    2. $h$ is differentiable at $x$=0
    3. $f ◦ h$ is differentiable at x=0
    4. $h ◦ f$ is differentiable at $x$=0
17. Let $f(x)$=sin $\left(\frac{\pi}{6} \sin \left(\frac{\pi}{2} sinx \right) \right)$ for all $x \in R$ and $g(x)$ =$\frac{\pi}{2}sinx$ for all $x \in R$. Let $(f ◦ g)(x)$ denote $f(g(x))$ and $(g ◦ f)(x)$ denote $g(f(x))$. Then which of the following is (are) true?
    1. Range of $f$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
    2. Range of $f ◦ g$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
    3. $\lim \limits_{x \to 0}\frac{f(x)}{g(x)}=\frac{\pi}{6}$
    4. There is an $x \in R$ such that $(g ◦ f)(x)$=1
18. Let $\Delta PQR$ be a triangle. Let $\vec{a}=\vec{QR}$, $\vec{b}=\vec{RP}$ and $\vec{c}=\vec{PQ}$. If $|\vec{a}|$=12, $|\vec{b}|=4\sqrt{3}$ and $\vec{b}•\vec{c}$=24, then which of the following is(are) true?
    1. $\frac{|\vec{c}|^2}{2}-|\vec{a}|$=12
    2. $\frac{|\vec{c}|^2}{2}+|\vec{a}|$=30
    3. $|\vec{a}×\vec{b}+\vec{c}×\vec{a}|$=$48\sqrt{3}$
    4. $\vec{a}•\vec{b}$=-72

SECTION 3 (Maximum Marks:16)

• This section contains TWO questions.
• Each question contains two columns, Column I and Column II .
• Column I has four entries (A), (B), (C) and (D).
• Column I has five entries (P), (Q), (R), (S) and (T).
• Match the entries in Column I with the entries in Column II.
• One or more entries in Column I may match with one or more entries in Column II .
• The ORS contains 4×5 matrix whose layout will be similar to the one shown below.
  

  (A)	(P)	(Q)	(R)	(S)	(T)
  (B)	(P)	(Q)	(R)	(S)	(T)
  (C)	(P)	(Q)	(R)	(S)	(T)
  (D)	(P)	(Q)	(R)	(S)	(T)

  
  
• For each query in column I , darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T) , then darken these three bubbles in the ORS . Similarly, for entries (B), (C) and (D).
• Marking scheme:
  for each entry inColumn I
  
  • Full Marks: +2 If only the bubble(s) corresponding to all the correct match(es) is (are) darkened.
  • Zero Marks: 0 If none of the bubbles is darkened;
  • Negative Marks: -1 In all other cases.

19.                    

    	Column - I		Column - II
    (A)	projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3}\hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha = 2+\sqrt{3} \beta$, then possible value(s) of $|\alpha|$ is (are)	(P)	1
    (B)	Let $a$ and $b$ be real numbers such that the function $f(x)$=$\left\{\begin{array}{ll}-3ax^2-2, & x < 1, \\ bx+a^2, & x \geq 1 .\end{array}\right.$ is differentiable for all $x \in R$. Then possible value(s) of $a$ is (are)	(Q)	2
    (C)	Let $\omega \neq 1$ be a complex cube root of unity. If (3-3$\omega$+2$\omega^2)^{4n+3}$+(2+3$\omega$-3$\omega^2)^{4n+3}$+(-3+2$\omega$+3$\omega^2)^{4n+3}$=0, then possible value(s) of $n$ is(are)	(R)	3
    (D)	Let the harmonic mean of two positive real numbers $a$ and $b$ be 4. If $q$ is a positive real number such that $a$, 5, $q$, $b$ is an arithmetic progression, then the value(s) of $|q-a|$ is (are)	(S)	4
    		(T)	5

    
    
20.                            

    	List - I		List - II
    (A)	In a triangle $\Delta XYZ$, let $a$, $b$ and $c$ be the length of the sides opposite to the angles $X$, $Y$ and $Z$, respectively. If $2(a^2-b^2)$=$c^2$ and $\lambda$=$\frac{sin(X-Y)}{sinZ}$, then possible values of $n$ for which $cos(n \pi \lambda)$=0 is (are)	(P)	1
    (B)	In a triangle $\Delta XYZ$, let $a$, $b$ and $c$ be the length of the sides opposite to the angles $X$, $Y$ and $Z$, respectively. If 1+$cos2X-$$2cos 2Y$=$2sinXsinY$, then possible value(s) of $\frac{a}{b}$ is (are)	(Q)	2
    (C)	In $R^2$, let $\sqrt{3}\hat{i}$+$\hat{j}$, $\hat{i}$+$\sqrt{3}\hat{j}$ and $\beta \hat{i}$+$(1-\beta)\hat{j}$ be the position vectors of $X$, $Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\vec{OX}$ with $\vec{OY}$ is $\frac{3}{\sqrt{2}}$, then possible value(s) of $|\beta|$ is (are)	(R)	3
    (D)	Suppose that $F(\alpha)$ denotes the area of the region bounded by $x$=0, $x$=2, $y^2=4x$ and $y$=$|\alpha $x-$1|$+$|\alpha x-2|$+$\alpha x$, where $\alpha \in {0,1}$. Then the values of $F(\alpha)$+$\frac{8}{3}\sqrt{2}$, when $\alpha$=0 and $\alpha$=1, is (are)	(S)	5
    		(T)	6

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