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SECTION 1

• 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.

1. Let
   $𝑆_1$={$(𝑖,𝑗, 𝑘) ∶ 𝑖,𝑗, 𝑘 \in \text{{1,2, … ,10}}$},
   $𝑆_2$={$(𝑖,𝑗) ∶ 1 \leq 𝑖 < 𝑗 + 2 \leq 10$, $ 𝑖,𝑗 \leq \text{{1,2, … , 10}}$},
   $𝑆_3$={$(𝑖,𝑗, 𝑘, 𝑙) ∶ 1 \leq 𝑖 < 𝑗 < 𝑘 < 𝑙$, $ 𝑖,𝑗, 𝑘, 𝑙 \in \text{{1,2, … ,10}}$}
   and
   $𝑆_4$ = {$(𝑖,𝑗, 𝑘, 𝑙) ∶ 𝑖,𝑗, 𝑘$ and $𝑙$ are distinct elements in {1,2, … ,10}}.
   If the total number of elements in the set $𝑆_𝑟$ is $𝑛_𝑟$, $𝑟$ = 1,2,3,4, then which of the following statements is (are) TRUE ?
   1. $n_1 = 1000$
   2. $n_2 = 44$
   3. $n_3 = 220$
   4. $\frac{n_4}{12} = 420$
2. Consider a triangle $𝑃𝑄𝑅$ having sides of lengths $𝑝$, $𝑞$ and $𝑟$ opposite to the angles $𝑃$,$𝑄$ and $𝑅$, respectively. Then which of the following statements is (are) TRUE ?
   1. $cos P \geq 1 - \frac{p^2}{2qr}$
   2. $\cos R \geq\left(\frac{q-r}{p+q}\right) \cos P$+$\left(\frac{p-r}{p+q}\right) \cos Q$
   3. $\frac{q + r} {p} < 2 \frac{sinQ sinR} {sin P} $
   4. If $𝑝$ < $𝑞$ and $𝑝$ < $𝑟$, then $cos 𝑄$ > $\frac{𝑝} {𝑟}$ and $cos 𝑅$ > $\frac{𝑝} {𝑞} $
3. Let $𝑓:[−\frac{\pi} {2}, \frac{ \pi} {2}]$ → $ℝ$ be a continuous function such that $𝑓(0)$ = 1 and $\int \limits_0^{\frac{\pi} {3}}𝑓(𝑡)𝑑𝑡 = 0$ Then which of the following statements is (are) TRUE ?
   1. The equation $𝑓(𝑥) − 3 cos 3𝑥$ = 0 has at least one solution in $(0, \frac{\pi} {3})$
   2. The equation $𝑓(𝑥) − 3 sin 3𝑥$ = −$\frac{6} {\pi}$ has at least one solution in $(0, \frac{\pi} {3})$
   3. $\lim \limits_{x \to 0} \frac{x \int \limits_{0}^{x} f(t) dt} {1 - e^{x^2}} = - 1$
   4. $\lim \limits_{x \to 0} \frac{sinx \int \limits_{0}^{x} f(t) dt} { x^2} = - 1$
4. For any real numbers $\alpha$ and $\beta$, let $𝑦_{\alpha, \beta} (𝑥), 𝑥 \in ℝ$, be the solution of the differential equation $\frac{𝑑𝑦} {𝑑𝑥}$ + $\alpha 𝑦$ = $𝑥𝑒^{\beta𝑥}$, $𝑦(1)$ = 1. Let $𝑆$ = ${𝑦_{\alpha,\beta} (𝑥) ∶ \alpha, \beta \in ℝ}$. Then which of the following functions belong(s) to the set $𝑆$ ?
   1. $𝑓(𝑥) =\frac{𝑥^2} {2}𝑒^{−𝑥}$ + $(𝑒 − \frac{1} {2})𝑒^{−𝑥} $
   2. $𝑓(𝑥) =\frac{𝑥^2} {2}𝑒^{−𝑥}$ + $(𝑒 + \frac{1} {2})𝑒^{−𝑥} $
   3. $𝑓(𝑥) =\frac{e^x} {2}(x - \frac{1}{2})$+ $(𝑒 − \frac{e^2} {4})𝑒^{−𝑥} $
   4. $𝑓(𝑥) =\frac{e^x} {2}(\frac{1}{2} - x)$+ $(𝑒 + \frac{e^2} {4})𝑒^{−𝑥} $
5. Let $𝑂$ be the origin and $\vec{𝑂A}$ = $2 \hat{i}$ + $2 \hat{j}$ + $\hat{k}$, $\vec{𝑂B}$ = $\hat{i} −$$ 2\hat{j}$ + $2 \hat{k} $ and $\vec{𝑂𝐶}$ = $\frac{1} {2} $$(\vec{𝑂𝐵} −\lambda \vec{𝑂𝐴})$ for some $\lambda$ > 0. If $|\vec{𝑂𝐵} × \vec{𝑂𝐶}|$ = $\frac{9} {2} $, then which of the following statements is (are) TRUE ?
   1. Projection of $\vec{𝑂𝐶}$ on $\vec{𝑂𝐴}$ is $\frac{−3} {2}$
   2. Area of the triangle $𝑂𝐴𝐵$ is $\frac{9} {2}$
   3. Area of the triangle $𝐴𝐵𝐶$ is $\frac{9} {2}$
   4. The acute angle between the diagonals of the parallelogram with adjacent sides $\vec{𝑂A}$ and $\vec{𝑂C}$ is $\frac{\pi} {3} $
6. Let $𝐸$ denote the parabola $𝑦^2 = 8𝑥$. Let $𝑃$ = (−2, 4), and let $𝑄$ and $𝑄′$ be two distinct points on $𝐸$ such that the lines $𝑃𝑄$ and $𝑃𝑄′$ are tangents to $𝐸$. Let $𝐹$ be the focus of $𝐸$. Then which of the following statements is (are) TRUE ?
   1. The triangle $𝑃𝐹𝑄$ is a right-angled triangle
   2. The triangle $𝑄𝑃𝑄′$ is a right-angled triangle
   3. The distance between $𝑃$ and $𝐹$ is $5\sqrt{2}$
   4. $𝐹$ lies on the line joining $𝑄$ and $𝑄′$

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. 7 and 8

Question Stem

Consider the region $𝑅$ = {$(𝑥, 𝑦) \in ℝ × ℝ ∶ 𝑥 \geq 0 $ and $𝑦^2 \leq 4 − 𝑥$ }. Let ℱ be the family of all circles that are contained in $𝑅$ and have centers on the $𝑥-$axis. Let $𝐶$ be the circle that has largest radius among the circles in ℱ. Let $(\alpha, \beta)$ be a point where the circle $𝐶$ meets the curve $𝑦^2 = 4 − 𝑥$.

7. The radius of the circle $𝐶$ is ___ .
8. The value of $\alpha$ is ___ .

Question Stem for Question Nos. 9 and 10

Question Stem

Let $𝑓_1: (0, \infty) → ℝ$ and $𝑓_2: (0, \infty ) → ℝ$ be defined by $𝑓_1(𝑥)$ = $\int \limits_0^x ∏ \limits_{j=1}^{21}(𝑡 − 𝑗)^𝑗 𝑑𝑡, 𝑥 > 0$ and $𝑓_2(𝑥)$ = $98(𝑥 − 1)^{50}$ − $600(𝑥 − 1)^{49}$ + 2450, $𝑥$ > 0,where, for any positive integer $𝑛$ and real numbers $𝑎_1$, $𝑎_2$, … , $𝑎_𝑛$, $∏ \limits_{𝑖=1}^{𝑛} a_i$ denotes the product of $𝑎_1$, $𝑎_2$, … , $𝑎_𝑛$. Let $𝑚_𝑖$ and $𝑛_𝑖$, respectively, denote the number of points of local minima and the number of points of local maxima of function $𝑓_𝑖$, $ 𝑖$ = 1, 2, in the interval $(0, \infty)$.

9. The value of $2𝑚_1$ + $3𝑛_1$ + $𝑚_1𝑛_1$ is ___ .
10. The value of $6𝑚_2$ + $4𝑛_2$ + $8𝑚_2 𝑛_2$ is ___.

Question Stem for Question Nos. 11 and 12

Question Stem

Let $𝑔_𝑖:[\frac{\pi} {8}, \frac{3 \pi} {8} ] → ℝ$, $𝑖 = 1, 2,$ and $𝑓:[\frac{\pi}{8},\frac{3 \pi}{8}] → ℝ$ be functions such that $𝑔_1(𝑥)$ = 1, $𝑔_2(𝑥)$ = $|4𝑥 − \pi|$ and $𝑓(𝑥)$ = $sin2 𝑥$, for all $𝑥 \in [\frac{ \pi} {8}, \frac{3 \pi} {8}]$ Define $𝑆_𝑖$ = $\int \limits_{\frac{ \pi} {8}}^{\frac{3 \pi} {8}}𝑓(𝑥) ⋅ 𝑔_𝑖(𝑥) 𝑑𝑥$, $𝑖$ = 1, 2

11. The value of $\frac{16𝑆_1} {\pi}$ is ___ .
12. The value of $\frac{48𝑆_2} {\pi^2}$ is ___.

SECTION 3

• This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) 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.

Paragraph

Let $𝑀$ = { $(𝑥, 𝑦) \in ℝ × ℝ ∶ 𝑥^2 + 𝑦^2 \leq 𝑟^2$ },where $𝑟$ > 0. Consider the geometric progression $𝑎_𝑛$ = $\frac{1} {2^{𝑛−1}}$, $𝑛$ = 1, 2, 3, … . Let $𝑆_0 = 0$ and, for $𝑛 \geq 1$, let $𝑆_𝑛$ denote the sum of the first $𝑛$ terms of this progression. For $𝑛 \geq 1$, let $𝐶_𝑛$ denote the circle with center $(𝑆_𝑛−1, 0)$ and radius $𝑎_𝑛$, and $𝐷_𝑛$ denote the circle with center $(𝑆_{𝑛−1} , 𝑆_{𝑛−1})$ and radius $𝑎_𝑛$.

13. Consider $𝑀$ with $𝑟$ = $\frac{1025} {513}$. Let $𝑘$ be the number of all those circles $𝐶_𝑛$ that are inside $𝑀$. Let $𝑙$ be the maximum possible number of circles among these $𝑘$ circles such that no two circles intersect. Then
    1. $𝑘 + 2𝑙$ = 22
    2. $2𝑘 + 𝑙$ = 26
    3. $2𝑘 + 3𝑙$ = 34
    4. $3𝑘 + 2𝑙$ = 40
14. Consider $𝑀$ with $𝑟 $=$\frac{(2^{199} −1)\sqrt{2}} {2^{198}} $. The number of all those circles $𝐷_𝑛$ that are inside $𝑀$ is
    1. 198
    2. 199
    3. 200
    4. 201

Paragraph

Let $𝜓_1$:$[0, \infty) → ℝ$, $𝜓_2$:$[0, \infty) → ℝ$, $𝑓:[0, \infty) → ℝ$ and $𝑔:[0, \infty ) → ℝ$ be functions such that $𝑓(0)$ = $𝑔(0)$ = 0, $𝜓_1(𝑥)$ =$ 𝑒^{−𝑥} + 𝑥$, $𝑥 \geq 0$, $𝜓_2(𝑥)$ = $𝑥^2 − 2𝑥$$ − 2𝑒^{−𝑥} + 2$, $𝑥 \geq 0,$ $𝑓(𝑥)$ = $\int \limits_{ - x} ^{x} (|𝑡| − 𝑡^2)𝑒^{−𝑡^2} 𝑑𝑡$, $𝑥 > 0$ and $ 𝑔(𝑥)$ = $\int \limits_{0}^{x^2} \sqrt{𝑡} 𝑒^{−𝑡} 𝑑𝑡$, $𝑥 > 0$.

15. Which of the following statements is TRUE ?
    1. $𝑓(\sqrt{ln 3})$ + $𝑔(\sqrt{ln 3}) $=$\frac{1} {3} $
    2. For every $𝑥$ > 1, there exists an $𝛼 \in (1, 𝑥)$ such that $𝜓_1 (𝑥) $= 1 + $\alpha 𝑥$
    3. For every $𝑥$ > 0, there exists a $𝛽 \in (0, 𝑥)$ such that $𝜓_2(𝑥)$ = $2𝑥(𝜓_1(\beta) − 1)$
    4. $𝑓$ is an increasing function on the interval $[0, \frac{3} {2} ]$
16. Which of the following statements is TRUE?
    1. $𝜓_1(𝑥) \leq 1$, for all $𝑥$ > 0
    2. $𝜓_2(𝑥) \leq 0$, for all $𝑥$ > 0
    3. $ 𝑓(𝑥) \geq 1 − 𝑒^{−𝑥^2} −$$ \frac{2} {3}𝑥^3$ + $\frac{2}{5}𝑥^5$ , for all $𝑥 \in (0, \frac{1}{2})$
    4. $𝑔(𝑥) \leq \frac{2} {3} 𝑥^3 −$$ \frac{2} {5} 𝑥^5$ + $\frac{1} {7} 𝑥^7$, for all $𝑥 \in (0, \frac{1} {2})$

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. A number is chosen at random from the set {1, 2, 3, … , 2000}. Let $𝑝$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500𝑝 is ___ .
18. Let $E$ be the ellipse $\frac{𝑥^2} {16}$ + $\frac{𝑦^2} {9} = 1$. For any three distinct points $𝑃$, $𝑄$ and $𝑄′$ on $E$, let $𝑀(𝑃, 𝑄)$ be the mid-point of the line segment joining $P$ and $𝑄$, and $𝑀(𝑃, 𝑄′)$ be the mid-point of the line segment joining $P$ and $𝑄′$. Then the maximum possible value of the distance between $𝑀(𝑃,𝑄)$ and $𝑀(𝑃,𝑄′)$, as $𝑃$, $𝑄$ and $𝑄′$ vary on $𝐸$, is ___ .
19. For any real number $𝑥$, let $[𝑥]$ denote the largest integer less than or equal to $𝑥$. If $𝐼$ = $\int \limits_0^{10} \left[\sqrt{\frac{10𝑥} {𝑥 + 1}} \right] 𝑑𝑥$, then the value of $9𝐼$ is ___ .

2021:Paper-1

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