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SECTION 1

• This section contains FOUR (04) questions.
• Each question has FOUR options (A), (B), (C) and (D). 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. Consider a triangle $\Delta$ whose two sides lie on the $x-$ axis and the line $𝑥$ + $𝑦$ + 1 = 0. If the orthocenter of $\Delta$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $\Delta$ is
   1. $ 𝑥^2 + 𝑦^2$$ − 3𝑥 + 𝑦 = 0$
   2. $ 𝑥^2 + 𝑦^2$$ + 𝑥 + 3𝑦 = 0$
   3. $𝑥^2 + 𝑦^2$$ + 2𝑦 − 1 = 0$
   4. $ 𝑥^2 + 𝑦^2 $$+ 𝑥 + 𝑦 = 0$
2. The area of the region
   $\left\{(x, y): 0 \leq x \leq \frac{9}{4}, \right.$ $\left. \quad 0 \leq y \leq 1, \quad x \geq 3 y, \quad x+y \geq 2\right\}$
   is
   1. $\frac{11}{32}$
   2. $ \frac{35}{96}$
   3. $\frac{37}{96} $
   4. $\frac{13}{32}$
3. Consider three sets $𝐸_1$ = {1, 2, 3}, $𝐹_1$ = {1, 3, 4} and $𝐺_1$ = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set $𝐸_1$, and let $𝑆_1$ denote the set of these chosen elements. Let $𝐸_2$ = $𝐸_1 − 𝑆_1$ and $𝐹_2$ = $𝐹_1 ∪ 𝑆_1$. Now two elements are chosen at random, without replacement, from the set $𝐹_2$ and let $𝑆_2$ denote the set of these chosen elements. Let $𝐺_2$ = $𝐺_1 ∪ 𝑆_2.$ Finally, two elements are chosen at random, without replacement, from the set $𝐺_2$ and let $𝑆_3$ denote the set of these chosen elements. Let $𝐸_3$ = $𝐸_2 ∪ 𝑆_3$. Given that $𝐸_1= 𝐸_3$, let p be the conditional probability of the event $𝑆_1 = {1, 2}$. Then the value of $p$ is
   1. $\frac{1}{5}$
   2. $ \frac{3}{5}$
   3. $\frac{1}{2}$
   4. $\frac{2}{5}$
4. Let $𝜃_1$, $𝜃_2$, … , $𝜃_{10}$ be positive valued angles (in radian) such that $𝜃_1$ + $𝜃_2$ + ⋯ + $𝜃_10 = 2 \pi$. Define the complex numbers $𝑧_1$ = $𝑒^{𝑖𝜃_1}$ , $𝑧_𝑘$ = $𝑧_{𝑘−1} 𝑒^{𝑖𝜃_𝑘} $ for $𝑘$ = 2, 3, … , 10, where $𝑖 = \sqrt{−1} $. Consider the statements $𝑃$ and $𝑄$ given below:
   $𝑃$ : $|𝑧_2 − 𝑧_1|$ + $|𝑧_3 − 𝑧_2|$ + ⋯ +$ |𝑧_{10} − 𝑧_9|$ + $|𝑧_1 − 𝑧_{10}|$ $ \leq 2 \pi$
   $𝑄$ : $|𝑧_2^2 − 𝑧_1^2|$ + $|𝑧_3^2 − 𝑧_2^2|$ + ⋯ + $|𝑧_{10} ^2 − 𝑧_9^2|$ + $|𝑧_1^2 − 𝑧_{10}^2|$ $ \leq 4 \pi$.
   Then,
   1. $P$ is TRUE and $𝑄$ is FALSE
   2. $𝑄$ is TRUE and $P$ is FALSE
   3. both $P$ and $𝑄$ are TRUE
   4. both $P$ and $𝑄$ are FALSE

SECTION 2

• This section contains THREE (03) questions.
• There are TWO (02) questions corresponding to each question stem.
• The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value corresponding to the answer in the designated place using the mouse and the on-screen virtual numeric keypad.
• If the numerical value has more than 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: +2 If ONLY the correct numerical value is entered at the designated place;
  • Zero Marks: 0 In all other cases.

Question Stem for Question Nos. 5 and 6

Question Stem
Three numbers are chosen at random, one after another with replacement, from the set $𝑆$ = {1,2,3, … ,100}. Let $𝑝_1$ be the probability that the maximum of chosen numbers is at least 81 and $𝑝_2$ be the probability that the minimum of chosen numbers is at most 40.

5. The value of $\frac{625} {4} 𝑝_1$ is ___ .
6. The value of $\frac{125} {4} 𝑝_2$ is ___ .

Question Stem for Question Nos. 7 and 8

Question Stem
Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations
$𝑥$ + $2𝑦$ + $3𝑧 = \alpha$
$4𝑥$ + $5𝑦$ + $6𝑧 = \beta$
$7𝑥$ + $8𝑦$ + $9𝑧$ = $\gamma − 1$
is consistent. Let $|𝑀|$ represent the determinant of the matrix

$𝑀$ =$\begin{equation*} \begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ −1 & 0 & 1 \end{bmatrix} \end{equation*}$

Let $𝑃$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $𝐷$ be the square of the distance of the point (0, 1, 0) from the plane $𝑃$.

7. The value of $|M|$ is ___ .
8. The value of $𝐷$ is ___ .

Question Stem for Question Nos. 9 and 10

Question Stem
Consider the lines $𝐿_1$ and $𝐿_2$ defined by $𝐿_1$: $𝑥 \sqrt{2}$ + $𝑦 − 1 = 0$ and $𝐿_2$: $𝑥 \sqrt{2} −$$ 𝑦 + 1 = 0$
For a fixed constant $\lambda$, let $𝐶$ be the locus of a point $𝑃$ such that the product of the distance of $𝑃$ from $𝐿_1$ and the distance of $𝑃$ from $𝐿_2$ is $\lambda^2$. The line $𝑦$ = $2𝑥$ + 1 meets $𝐶$ at two points $𝑅$ and $𝑆$, where the distance between $𝑅$ and $𝑆$ is $\sqrt{270}$. Let the perpendicular bisector of $𝑅𝑆$ meet $𝐶$ at two distinct points $𝑅′$ and $𝑆′$. Let $𝐷$ be the square of the distance between $𝑅′$ and $𝑆′$.

9. The value of $𝜆^2$ is ___ .
10. The value of $𝐷$ is ___ .

SECTION 3

• This section contains SIX (06) 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, 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.
• 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(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks.

11. For any 3 × 3 matrix $𝑀$, let $|𝑀|$ denote the determinant of $𝑀$. Let
    
    $\begin{equation*} E= \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}\end{equation*},$$ \begin{equation*} P=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \end{equation*}$ and $\begin{equation*} F= \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix} \end{equation*}$
    
    If $𝑄$ is a nonsingular matrix of order 3 × 3, then which of the following statementsis (are) TRUE ?
    1. $F$ = $PEP$ and $P^2$ = $\begin{equation*} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{equation*}$
    2. $ |𝐸𝑄 + 𝑃𝐹𝑄^{−1}|$ = $|𝐸𝑄| + |𝑃𝐹𝑄^{−1}|$
    3. $ |(𝐸𝐹)^3|$ > $|𝐸𝐹|^2$
    4. Sum of the diagonal entries of $𝑃^{−1}𝐸𝑃$ + $𝐹$ is equal to the sum of diagonal entries of $𝐸$ + $𝑃^{−1}𝐹𝑃$
12. Let $𝑓: ℝ → ℝ$ be defined by $𝑓(𝑥)$ =$\frac{𝑥^2 − 3𝑥 − 6} {𝑥^2 + 2𝑥 + 4} $ Then which of the following statements is (are) TRUE?
    1. $𝑓$ is decreasing in the interval (−2, −1)
    2. $𝑓$ is increasing in the interval (1, 2)
    3. $𝑓$ is onto
    4. Range of $𝑓$ is $[\frac{−3} {2} , 2]$
13. Let $𝐸$, $𝐹$ and $G$ be three events having probabilities $𝑃(𝐸) = \frac{1} {8}$, $𝑃(𝐹) = \frac{1} {6}$ and $𝑃(𝐺) = \frac{1} {4}$, and let $𝑃(𝐸 ∩ 𝐹 ∩ 𝐺)$ =$ \frac{1} {10} $.For any event $𝐻$, if $𝐻^𝑐$ denotes its complement, then which of the following statements is (are) TRUE ?
    1. $𝑃(𝐸 ∩ 𝐹 ∩ 𝐺^𝑐)$$ \leq\frac{1} {40} $
    2. $𝑃(𝐸^𝑐 ∩ 𝐹 ∩ 𝐺) \leq\frac{1} {15} $
    3. $𝑃(𝐸 ∪ 𝐹 ∪ 𝐺) $$\leq \frac{13} {24} $
    4. $𝑃(𝐸^𝑐 ∩ 𝐹^𝑐 ∩ 𝐺^𝑐) $$\leq\frac{5} {12} $
14. For any 3 × 3 matrix $𝑀$, let $|𝑀|$ denote the determinant of $𝑀$. Let $𝐼$ be the 3 × 3 identity matrix. Let $𝐸$ and $𝐹$ be two 3 × 3 matrices such that $(𝐼 − 𝐸𝐹)$ is invertible. If $𝐺$ = $(𝐼 − 𝐸𝐹)^{−1}$, then which of the following statements is (are) TRUE ?
    1. $|𝐹𝐸|$ = $|𝐼 − 𝐹𝐸|$$|𝐹𝐺𝐸|$
    2. $(𝐼 − 𝐹𝐸)$$(𝐼 + 𝐹𝐺𝐸)$ = $𝐼$
    3. $𝐸𝐹𝐺 $=$𝐺𝐸𝐹$
    4. $(𝐼 − 𝐹𝐸)$$(𝐼 − 𝐹𝐺𝐸)$ = $𝐼$
15. For any positive integer $𝑛$, let $𝑆_𝑛: (0, \infty) → ℝ$ be defined by $𝑆_𝑛(𝑥)=\sum \limits_{k=1}^{n} cot^{−1}\left(\frac{1 + 𝑘(𝑘 + 1)𝑥^2} {𝑥} \right)$ ,where for any $𝑥 \in ℝ, cot^{−1} (𝑥) \in (0, \pi)$ and $tan^{−1} (𝑥) \in (−\frac{\pi} {2}, \frac{\pi} {2})$. Then which of the following statements is (are) TRUE ?
    1. $𝑆_{10}(𝑥)=\frac{\pi} {2} − tan^{−1}\left(\frac{1+11𝑥^2}{10𝑥} \right)$ , for all $𝑥$ > 0
    2. $\lim \limits_{ n \to \infty} cot(𝑆_𝑛(𝑥)) = 𝑥$, for all $𝑥$ > 0
    3. The equation $𝑆_3(𝑥) = \frac{\pi} {4}$has a root in $(0, \infty)$
    4. $tan(𝑆_𝑛(𝑥)) \leq \frac{1} {2}$, for all $𝑛 \geq 1$ and $𝑥$ > 0
16. For any complex number $𝑤$ = $𝑐 + 𝑖𝑑$, let $arg(w) \in (−\pi, \pi]$, where $𝑖 = \sqrt{−1}$ . Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $𝑧$ = $𝑥 + 𝑖𝑦$ satisfying $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, the ordered pair $(𝑥, 𝑦)$ lies on the circle $𝑥^2 + 𝑦^2$ +$ 5𝑥 − 3𝑦 $+ 4 = 0
    Then which of the following statements is (are) TRUE ?
    1. $\alpha = - 1$
    2. $\alpha \beta = 4$
    3. $\alpha \beta = - 4$
    4. $\beta = 4$

SECTION 4

• This section contains THREE (03) questions.
• The answer to each question is a NON-NEGATIVE INTEGER..
• For each question, enter the correct integer corresponding to 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: +4 If ONLY the correct integer is entered;
  • Zero Marks: 0 In all other cases.

17. For $𝑥 \in ℝ$, the number of real roots of the equation $3𝑥^2 − 4|𝑥^2 − 1|$ + $𝑥 − 1$ = 0 is ___ .
18. In a triangle $𝐴𝐵𝐶$, let $𝐴𝐵$ = $\sqrt{23}$, $𝐵𝐶$ = 3 and $𝐶𝐴$ = 4. Then the value of $\frac{cot 𝐴 + cot 𝐶}{cot 𝐵}$ is ___ .
19. Let $\vec{𝑢}, \vec{𝑣}$ and $\vec{𝑤}$ be vectors in three-dimensional space, where $\vec{𝑢}$ and $\vec{𝑣}$ are unit vectors which are not perpendicular to each other and $\vec{𝑢} ⋅ \vec{𝑤}$= 1, $\vec{𝑣} ⋅ \vec{𝑤}$ = 1, $\vec{𝑤} ⋅ \vec{𝑤}$ = 4 If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\vec{𝑢}$, $\vec{𝑣}$ and $\vec{𝑤}$, is $\sqrt{2}$, then the value of $|3 \vec{𝑢} +5 \vec{𝑣}|$ is ___ .

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