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SECTION 1 (Maximum Marks:18)

• This section contains SIX (06) questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE.
• For each question, enter the correct numerical value of 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: +3 ONLY if the correct numerical value is entered;
  • Zero Marks: 0 If the question is unanswered.
  • Negative Marks: -1 In all other cases.

1. For a complex number $𝑧$, let $Re(𝑧)$ denote the real part of $𝑧$. Let $𝑆$ be the set of all complex numbers $𝑧$ satisfying $𝑧^4 − |𝑧|^4$ = $4 𝑖 𝑧^2$, where $𝑖 = \sqrt{−1}$ . Then the minimum possible value of $|𝑧_1 − 𝑧_2|^2$, where $𝑧_1, 𝑧_2 \in 𝑆$ with $Re(𝑧_1 ) > 0$ and $Re(𝑧_2) < 0$, is _____
2. The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is_____
3. Let $𝑂$ be the centre of the circle $𝑥^2 + 𝑦^2 = 𝑟^2$, where $𝑟 > \frac{\sqrt{5}} {2} $. Suppose $𝑃𝑄$ is a chord of this circle and the equation of the line passing through $𝑃$ and $𝑄$ is $2𝑥$ + $4𝑦$ = 5. If the centre of the circumcircle of the triangle $𝑂𝑃𝑄$ lies on the line $𝑥$ + $2𝑦$ = 4, then the value of $𝑟$ is _____
4. The trace of a square matrix is defined to be the sum of its diagonal entries. If $𝐴$ is a 2 × 2 matrix such that the trace of $𝐴$ is 3 and the trace of $𝐴^3$ is −18, then the value of the determinant of $𝐴$ is _____
5. Let the functions $𝑓: (−1, 1)$ → $R$ and $𝑔: (−1, 1)$ → (−1, 1) be defined by $𝑓(𝑥)$ = $|2𝑥 − 1|$ + $|2𝑥 + 1|$ and $𝑔(𝑥)$ = $𝑥 − [𝑥]$, where $[𝑥]$ denotes the greatest integer less than or equal to $𝑥$. Let $𝑓 ∘ 𝑔: (−1, 1)$ → $R$ be the composite function defined by $(𝑓 ∘ 𝑔)(𝑥)$ = $𝑓(𝑔(𝑥))$. Suppose $𝑐$ is the number of points in the interval (−1, 1) at which $𝑓 ∘ 𝑔$ is NOT continuous, and suppose $𝑑$ is the number of points in the interval (−1, 1) at which $𝑓 ∘ 𝑔$ is NOT differentiable. Then the value of $𝑐 + 𝑑$ is _____
6. The value of the limit $\lim \limits_{𝑥 \to \frac{\pi} {2}} \frac{4\sqrt{2}(sin 3𝑥 + sin 𝑥)} {(2 sin 2𝑥 sin \frac{3𝑥} {2} + cos \frac{5𝑥} {2} ) − (\sqrt{2} + \sqrt{2} cos 2𝑥 + cos \frac{3𝑥} {2})}$ is _____

SECTION 2 (Maximum Marks:24)

• This section contains SIX (06) 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: -2 In all other cases.

7. Let $𝑏$ be a nonzero real number. Suppose $𝑓: R→ R$ is a differentiable function such that $𝑓(0)$ = 1.If the derivative $𝑓′$ of $𝑓$ satisfies the equation $𝑓′(𝑥)=\frac{𝑓(𝑥)} {𝑏^2 + 𝑥^2} $ for all $𝑥 \in R$, then which of the following statements is/are TRUE?
   1. If $𝑏$ > 0, then $𝑓$ is an increasing function
   2. If $𝑏$ < 0, then $𝑓$ is a decreasing function
   3. $𝑓(𝑥)𝑓(−𝑥)$ = 1 for all $𝑥 \in R$
   4. $𝑓(𝑥)𝑓(−𝑥)$ = 1 for all $𝑥 \in R$
8. Let $𝑎$ and $𝑏$ be positive real numbers such that $𝑎$ > 1 and $𝑏$ < $𝑎$. Let $𝑃$ be a point in the first quadrant that lies on the hyperbola $ \frac{𝑥^2} {𝑎^2} − \frac{𝑦^2} {𝑏^2} = 1.$ Suppose the tangent to the hyperbola at $𝑃$ passes through the point (1, 0), and suppose the normal to the hyperbola at $𝑃$ cuts off equal intercepts on the coordinate axes. Let ∆ denote the area of the triangle formed by the tangent at $𝑃$, the normal at $𝑃$ and the $𝑥-$axis. If $𝑒$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
   1. 1 < $e$ <$ \sqrt{2}$
   2. $\sqrt{2}$ < e < 2
   3. $\Delta = a^4$
   4. $\Delta = b^4$
9. Let $𝒇: R → R$ and $𝒈: R → R$ be functions satisfying $𝑓(𝑥 + 𝑦)$ = $𝑓(𝑥)$ + $𝑓(𝑦)$ + $𝑓(𝑥)𝑓(𝑦)$ and $𝑓(𝑥)$ = $𝑥𝑔(𝑥)$ for all $𝑥, 𝑦 \in R$. If $\lim \limits_{𝑥 \to 0} 𝑔(𝑥) = 1$, then which of the following statements is/are TRUE?
   1. $𝑓$ is differentiable at every $𝑥 \in R$
   2. If $ 𝑔(0)$ = 1, then $𝑔$ is differentiable at every $𝑥 \in R$
   3. The derivative $𝑓′(1)$ is equal to 1
   4. The derivative $𝑓′(0)$ is equal to 1
10. Let $\alpha$, $\beta$, $\gamma$, $\delta$ be real numbers such that $\alpha^2$ + $\beta^2$ + $\gamma^2 ≠ 0$ and $\alpha$ + $\gamma = 1$. Suppose the point (3, 2, −1) is the mirror image of the point (1, 0, −1) with respect to the plane $\alpha 𝑥$ + $\beta 𝑦$ + $\gamma 𝑧 $= $\delta$. Then which of the following statements is/are TRUE?
    1. $\alpha + \beta $ = 2
    2. $ \alpha - \gamma$ = 3
    3. $\delta + \beta $ = 4
    4. $ \alpha + \beta + \gamma = \delta$
11. Let $𝑎$ and $𝑏$ be positive real numbers. Suppose $\vec{𝑃Q}$ = $𝑎\hat{i} + 𝑏\hat{j}$ and $\vec{𝑃S}$ = $𝑎\hat{i} − 𝑏\hat{j}$ are adjacent sides of a parallelogram $𝑃𝑄𝑅𝑆$. Let $\vec{u} $ and $\vec{v} $ be the projection vectors of $\vec{w}$ = $\hat{𝑖} + \hat{𝑗}$ along $\vec{𝑃Q} $ and $\vec{𝑃S} $, respectively. If $|\vec{𝑢} |$ + $|\vec{𝑣} |$ = $|\vec{𝑤}|$ and if the area of the parallelogram $𝑃𝑄𝑅𝑆$ is 8, then which of the following statements is/are TRUE?
    1. $a + b$ = 4
    2. $a - b$ = 2
    3. The length of the diagonal $𝑃𝑅$ of the parallelogram $𝑃𝑄𝑅𝑆$ is 4
    4. $\vec{w}$ is an angle bisector of the vectors $\vec{𝑃Q} $ and $\vec{𝑃𝑆} $.
12. For nonnegative integers $𝑠$ and $𝑟$, let
    $\left(\begin{array}{ll} s \\ r \end{array}\right)= \begin{cases}\frac{s !}{r !(s-r) !} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{cases}$
    For positive integers $𝑚$ and $𝑛$, let
    $\begin{equation*} g(m,n)=\sum \limits_{p=0}^{m+n} \frac{f(m,n,p)}{\begin{pmatrix} n+p \\ p \end{pmatrix}} \end{equation*}$
    where for any nonnegative integer $𝑝$,
    $f(m,n,p)$= $\begin{equation*} \sum \limits_{i=0}^{p} \begin{pmatrix} m \\ i \end{pmatrix} \begin{pmatrix} n+i \\ p \end{pmatrix} \begin{pmatrix} p+n \\ p-i \end{pmatrix} \end{equation*}$
    Then which of the following statements is/are TRUE?
    1. $𝑔(𝑚, 𝑛)$ = $𝑔(𝑛, 𝑚)$ for all positive integers $𝑚$, $𝑛$
    2. $𝑔(𝑚, 𝑛 + 1)$ = $𝑔(𝑚 + 1, 𝑛)$ for all positive integers $𝑚$, $𝑛$
    3. $𝑔(2𝑚, 2𝑛)$ = $2 𝑔(𝑚, 𝑛)$ for all positive integers $𝑚$, $𝑛$
    4. $𝑔(2𝑚, 2𝑛)$ = $(𝑔(𝑚, 𝑛)) ^2$ for all positive integers $𝑚$, $𝑛$

SECTION 3 (Maximum Marks:24)

• 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: +4 ONLY if the correct numerical value is entered;
  • Zero Marks: 0 In all other cases.

13. An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is _____
14. In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then thenumber of all possible ways in which this can be done is _____
15. Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $𝑝$ is the probability that this perfect square is an odd number, then the value of $14𝑝$ is _____
16. Let the function $𝑓:[0, 1] → ℝ$ be defined by$𝑓(𝑥)=\frac{4^𝑥} {4^{𝑥 + 2}}$.Then the value of $𝑓 (\frac{1} {40}) + 𝑓 (\frac{2} {40})$+ $𝑓 (\frac{3} {40}) + ⋯ $+ $𝑓 (\frac{39} {40}) − 𝑓({1} {2})$ is ___
17. Let $𝑓: ℝ → ℝ$ be a differentiable function such that its derivative $𝑓′$ is continuous and $𝑓(\pi)$ = −6. If $𝐹:[0, \pi]$ → ℝ is defined by $𝐹(𝑥) = \int \limits_0^x 𝑓(𝑡)𝑑𝑡 $, and if $\int \limits_0^{\pi}( 𝑓′(𝑥) + 𝐹(𝑥)) cos 𝑥 𝑑𝑥 = 2$, then the value of $𝑓(0)$ is _____
18. Let the function $𝑓: (0, \pi)$ → ℝ be defined by $𝑓(\theta)$ = $(sin \theta + cos \theta)^2$ + $(sin \theta − cos \theta)^4$.Suppose the function $𝑓$ has a local minimum at $\theta$ precisely when $\theta \in {\lambda_1 \pi, … , \lambda_𝑟 \pi}$, where $0 <𝜆_1 < ⋯ < 𝜆_𝑟 < 1$. Then the value of $𝜆_1 + ⋯ + 𝜆_𝑟$is _____

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