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SECTION 1 (Maximum Marks:12)

• This section contains FOUR (04) questions.
• Each question has FOUR options. 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 ONLY if 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.

1. Let $S$ be the set of all complex numbers $z$ satisfying $|z - 2 + i |$ $\geq \sqrt{5}$. If the complex number $z_0$ is such that $\frac{1} {|z_0 - 1|} $ is the maximum of the set ${\frac{1}{|z - 1|}:z\in S} $, then the principle argument of $\frac{4 - z_0 - \bar{z_0}} {z_0 - \bar{z_0} + 2i}$ is
   1. $\frac{- \pi} {2}$
   2. $\frac{\pi} {4}$
   3. $\frac{\pi} {2}$
   4. $\frac{3\pi} {4}$
2. Let
   $\begin{equation*} M = \begin{bmatrix} sin^4 \theta & -1-sin^2 \theta \\ 1+cos^2 \theta & cos^4 \theta \end{bmatrix} \end{equation*}$ = $\alpha I$ + $\beta M^{-1}$
   where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real numbers, and $I$ is 2x2 identity matrix. If
   $\alpha$* is the minimum of the set { $\alpha(\theta)$:$\theta \in [0,2\pi)$}
   $\beta $* is the minimum of the set { $\beta(\theta)$:$\theta \in [0,2\pi)$},
   then the value of $\alpha^ * + \beta^ *$ is
   1. $\frac{- 37} {16}$
   2. $\frac{-31} {16}$
   3. $\frac{-29} {16}$
   4. $\frac{-17} {16}$
3. A line $y=mx+1$ intersects the circle $(x-3)^2+(y+2)^2=25$ at the points $P$ and $Q$. If the midpoints of the linesegment $PQ$ has $x-$coordinate $\frac{-3}{5}$, then which one of the following option is correct?
   1. $-3 \leq m < -1 $
   2. $2 \leq m < 4$
   3. $4 \leq m < 6 $
   4. $-6 \leq m < 8 $
4. The area of the region {$(x, y)$ : $xy \leq 8$, $1 \leq$ y $\leq x^2$ is
   1. $16 log_e 2 - \frac{14} {3} $
   2. $8 log_e 2 - \frac{14} {3} $
   3. $16 log_e 2 - 6$
   4. $8 log_e 2 - \frac{7} {3} $

SECTION 2 (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 (i.e. the question is unanswered) will get 0 marks and choosing any other combination of options will get −1 mark.

5. Let $\alpha$ and $\beta $ be the roots of $x^2-x-1$, with $\alpha > \beta $. For all positive integers $n$, define $\alpha_n = \frac{\alpha^n - \beta^n} {\alpha - \beta} $, $n \geq 1$,
   $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$.
   Then which of the following options is/are correct?
   1. $a_1+a_2+a_3+...+a_n$ $=a_{n+2}-1$ for all $ n \geq 1$
   2. $ \sum \limits _{n=1} ^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
   3. $b_n=\alpha^n+\beta^n$ for all $n \geq 1$
   4. $ \sum \limits _{n=1} ^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
6. Let
   $\begin{equation*} M = \begin{bmatrix} 0 & 1 & a\\ 1 & 2 & 3\\ 3 & b & 1 \end{bmatrix}\end{equation*}$ $\hspace{0.1cm}and \hspace{0.1cm}adj \hspace{0.1cm}$ $M$ = $\begin{equation*} \begin{bmatrix} -1 & 1 & -1\\ 8 & -6 & 2\\ -5 & 3 & -1 \end{bmatrix} \end{equation*}$
   $a$ and $b$ are real numbers. Which of the following options is/are correct?
   1. $a+b$=3
   2. $(adj \hspace{0.1cm}M) ^{-1}+adj \hspace{0.1cm}M^{-1}= - M$
   3. $det(adj \hspace{0.1cm}M^2)=81$
   4. If $\begin{equation*} M \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\end{equation*}$, then $ \alpha -\beta+ \gamma=3$
7. There are three bags $B_1$, $B_2$ and $B_3$. The bag $B_1$ contains 5 red and 5 green balls, $B_2$ contains 3 red and 5 green balls and $B_3$ contains 5 red and 3 green balls. Bags $B_1$, $B_2$ and $B_3$ have probabilities $\frac{3} {10} $, $\frac{3} {10} $ and $\frac{4} {10} $ respectively of being chosen. A bag is selected is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
   1. Probability that the chosen ball is green, given that the selected bag is $B_3$, equals $\frac{3} {8}$
   2. Probability that the chosen ball is green equals $\frac{39}{80}$
   3. Probability that the selected bag is $B_3$, given that the chosen ball is green, equals $\frac{5}{13}$
   4. Probability that the selected bag is $B_3$ and the chosen ball is green equals $\frac{3}{10}$
8. In a non-right-angled triangle $\Delta PQR$, let $p$, $q$, $r$ denotes the length of the sides opposite to the angles $P$, $Q$, $R$ respectively. The median from $R$ meets the side $PQ$ at $S$, the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $O$. If $p$=$\sqrt{3}$, $q$=1, and the radius of the circumcircle of the $\Delta PQR$ equals 1, then which of the following options is/are correct?
   1. Length of $RS$ = $\frac{\sqrt{7}}{2}$
   2. Area of $\Delta SOE$ = $\frac{\sqrt{3}} {12}$
   3. Length of $OE$ = $\frac{1} {6}$
   4. Radius of incircle of $\Delta PQR$ = $\frac{\sqrt{3}} {2}(2-\sqrt{3})$
9. Define the collections {$E_1$, $E_2$, $E_3$,... } of ellipses and {$R_1$, $R_2$, $R_3$,...} of rectangles as follows:
   $E_1 : \frac{x^2}{9} + \frac{y^2}{4}=1$;
   $R_1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;
   $E_n$ : ellipse $\frac{x^2}{a_n^2} + \frac{y^2} {b_n^2}$=1 of largest area inscribed in $R_{n-1}$, $n$>1;
   $R_n$ : rectangle of largest area, with the sides parallel to the axes, inscribed in $E_n$, $n$>1. Then which of the following options is/are correct?
   1. The eccentricities of $E_{18}$ and $E_{19}$ are not equal.
   2. $\sum \limits_{n=1}^ {N}$ (area of $R_n$) < 24, for each positive integer N.
   3. The length of latus rectum of $E_9$ is $\frac{1}{6}$.
   4. The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
10. Let $f : R$ -> $R$ be given by
    $f(x) = \left\{ \begin{aligned} x^5+5 x^4+10 x^3 + \\ 10 x^2+3 x+1, \\ x<0 \\ x^2-x+1, \\ 0 \leq x<1 \\ \frac{2}{3} x^3-4 x^2+7 x-\frac{8}{3}, \\ 1 \leq x<3 \\ (x-2) \log _e(x-2) \\ -x+\frac{10}{3}, \\ x \geq 3 \end{aligned} \right.$
    Then which of the following options is/are correct?
    1. $f$ is increasing on $( - \infty, 0)$.
    2. $f'$ has a local maximum at $x$=1.
    3. $f$ is onto
    4. $f'$ is not differentiable at $x$=1
11. Let $ \Gamma $ denote a curve $y=y(x)$ which is in the first quadrant and let the point (0,1) lie on it. Let the tangent to $ \Gamma $ at a point $P$ intersect the $y-$axis at $ Y_p $. If $PY_p$ has length 1 for each point $P$ on $ \Gamma $, then which of the following options is/are correct?
    1. $y$ = $ log_e ( \frac{1+ \sqrt{1-x^2}}{x}) - \sqrt{1 - x^2}$
    2. $xy'$ + $ \sqrt{1 - x^2}$=0
    3. $ - log_e( \frac{1+ \sqrt{1 - x^2}}{x}) + \sqrt{1 - x^2} $
    4. $xy' - $$\sqrt{1 - x^2}$ = 0
12. Let $ L_1$ and $ L_2$ denote the lines
    $ \vec{r} = \hat{i} + \lambda ( - \hat{i} + 2 \hat{j} + 2 \hat{k}), \lambda \in R $ and
    $ \vec{r} = \mu ( 2 \hat{i} - \hat{j} + 2 \hat{k}), \mu \in R $
    respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?
    1. $ \vec{r} = \frac{2}{9}(4 \hat{i} + \hat{j} + \hat{k})$ + $t(2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R$
    2. $ \vec{r} = \frac{2}{9}(2 \hat{i} - \hat{j} + 2 \hat{k}) $+ $t(2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R$
    3. $ \vec{r} = \frac{1}{3}(2 \hat{i} + \hat{k})$ + $t(2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R$
    4. $t(2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R$

SECTION 3 (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 screenvirtual numeric keypad in the place designated to enter the answer. If the numerical value has morethan 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 ONLY if the correct numerical value is entered;
  • Zero Marks: 0 In all other cases.

13. Let $ \omega \neq 1$ be a cube root of unity. Then the minimum of the set {$| a + b \omega$ + $c$ $\omega ^2|^2 $ : $a$, $b$, $c$ distinct non-zero integers} equals....
14. Let $AP(a;d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d$ >0. If
    $AP(1;3)$ $\cap$ $AP(2;5)$ $\cap$ $AP(3;7)$ = $AP(a;d)$
    then $a+d$ equals...
15. Let $S$ be the sample space of all 3x3 matrices with entries from the set {0,1}. Let the event $E_1$ and $E_2$ be given by
    $E_1$={ $A \in S$ : det $A$ = 0} and
    $E_2$ = { $A \in S$ : sum of entries of $A$ is 7}.
    If a matrix is chosen at random from $S$, then the conditional probability $P(E_1 | E_2)$ equals...
16. Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x - $$6y - $23 = 0. Let $ \Gamma_A $ and $ \Gamma_B $ be the circles of radii of 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $ \Gamma_A $ and $ \Gamma_B $ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is...
17. If
    $I$ = $\frac{2}{\pi}\int \limits_{- \pi /4}^{\pi/4}\frac{dx}{(1+e^{sinx})( 2 - cos2x)}$
    then 27 $I^2$ equals...
18. Three lines are given by
    $ \vec{r} = \lambda \hat{i}, \lambda \in R$
    $ \vec{r} = \mu( \hat{i} + \hat{j}), \mu \in R$ and
    $ \vec{r} = v ( \hat{i} + \hat{j} + \hat{k}), v \in R$.
    Let the lines cut the plane $x$ + $y$ + $z$ = 1 at the points $A$, $B$ and $C$ respectively. If the area of the triangle $ABC$ is $\Delta$ then the value of $(6\Delta)^2$ equals....

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