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SECTION 1 (Maximum Marks:24)

• This section contains SIX (06) questions.
• Each question has FOUR options for correct answer(s) . ONE OR MORE THAN ONE of these four option(s) is(are) correct option(s).
• For each question, choose the correct option(s) to answer the question.
• 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, If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) ,without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.

1. For a non-zero complex number $𝑧$, let $arg(𝑧)$ denote the principal argument with $− 𝜋$ < $arg(𝑧)$ ≤ $𝜋$. Then, which of the following statement(s) is (are) FALSE?
   1. $arg(−1 − 𝑖)$ =$\frac{\pi}{4}$, where $𝑖 = \sqrt{−1}$
   2. The function $𝑓: ℝ$ → $(−𝜋, 𝜋]$, defined by $𝑓(𝑡)$ = $arg(−1 + 𝑖𝑡)$ for all $𝑡 ∈ ℝ$, is continuous at all points of $ℝ$, where $𝑖 = \sqrt{−1}$
   3. For any two non-zero complex numbers $𝑧_1$ and $𝑧_2$, $arg \left(\frac{𝑧_1}{𝑧_2}\right)$ − $arg( 𝑧_1)$ + $arg( 𝑧_2)$ is an integer multiple of $2𝜋$
   4. For any three given distinct complex numbers $𝑧_1$, $𝑧_2$ and $𝑧_3$, the locus of the point 𝑧 satisfying the condition arg $\left(\frac{(𝑧−𝑧_1) (𝑧_2−𝑧_3)}{(𝑧−𝑧_3) (𝑧_2−𝑧_1)}\right)$ = $𝜋$, lies on a straight line
2. In a triangle $𝑃𝑄𝑅$, let $\angle{𝑃𝑄𝑅}$ = 30° and the sides $𝑃𝑄$ and $𝑄𝑅$ have lengths $10\sqrt{3}$ and 10, respectively. Then, which of the following statement(s) is (are) TRUE?
   1. $\angle{𝑄𝑃𝑅}$ = 45°
   2. The area of the triangle $𝑃𝑄𝑅$ is $25\sqrt{3}$ and $\angle{𝑄𝑅𝑃}$ = 120°
   3. The radius of the incircle of the triangle $𝑃𝑄𝑅$ is $10\sqrt{3} − 15$
   4. The area of the circumcircle of the triangle $𝑃𝑄𝑅$ is 100 $𝜋$
3. Let $𝑃_1$: $2𝑥$ + $𝑦 −$$ 𝑧$ = 3 and $𝑃_2$: $𝑥$ + $2𝑦$ + $𝑧$ = 2 be two planes. Then, which of the following statement(s) is (are) TRUE?
   1. The line of intersection of $𝑃_1$ and $𝑃_2$ has direction ratios 1, 2, −1
   2. The line $\frac{3𝑥 − 4}{9}$=$\frac{1 − 3𝑦}{9}$=$\frac{𝑧}{3}$ is perpendicular to the line of intersection of $𝑃_1$ and $𝑃_2$
   3. The acute angle between $𝑃_1$ and $𝑃_2$ is 60°
   4. If $𝑃_3$ is the plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of $𝑃_1$ and $𝑃_2$, then the distance of the point (2, 1, 1) from the plane $𝑃_3$ is $2\sqrt{3}$
4. For every twice differentiable function $𝑓: ℝ$ → [−2, 2] with $(𝑓(0))^2 + (𝑓′(0))^2$= 85, which of the following statement(s) is (are) TRUE?
   1. There exist $𝑟$, $𝑠$ ∈ $ℝ$, where $𝑟$ < $𝑠$, such that $𝑓$ is one-one on the open interval $(𝑟, 𝑠)$
   2. There exists $𝑥_0 \in (−4, 0)$ such that $|𝑓′(𝑥_0)| \leq 1$
   3. $\lim \limits_{𝑥\to \infty}$$𝑓(𝑥)$ = 1
   4. There exists $\alpha \in (−4, 4)$ such that $ 𝑓(\alpha)$ +$ 𝑓′′(\alpha)$ = 0 and $ 𝑓′(\alpha)\neq0$
5. Let $𝑓: ℝ → ℝ$ and $𝑔: ℝ → ℝ$ be two non-constant differentiable functions. If $𝑓′(𝑥)$ = $(𝑒^{(𝑓(𝑥)−𝑔(𝑥))})$𝑔′(𝑥) for all $𝑥$ ∈ $ℝ$, and $𝑓(1)$ = $𝑔(2)$ = 1, then which of the following statement(s) is (are) TRUE?
   1. $𝑓(2)$ < 1 − $log_e 2$
   2. $𝑓(2)$ > 1 − $log_e 2$
   3. $g(2)$ > 1 − $log_e 2$
   4. $g(2)$ < 1 − $log_e 2$
6. Let $𝑓:[0, ∞)$ → $ℝ$ be a continuous function such that $𝑓(𝑥)$ = 1 − $2𝑥$ + $\int \limits_0^x 𝑒^{𝑥−𝑡}𝑓(𝑡)𝑑t$ for all $𝑥$ ∈ [0, ∞). Then, which of the following statement(s) is (are) TRUE?
   1. The curve $𝑦$ = $𝑓(𝑥)$ passes through the point (1, 2)
   2. The curve $𝑦$ = $𝑓(𝑥)$ passes through the point (2, −1)
   3. The area of the region {$(𝑥, 𝑦)$ ∈ [0, 1] × $ℝ$ ∶ $𝑓(𝑥)$ ≤ $𝑦$ ≤ $\sqrt{1 − 𝑥^2}$} is $\frac{𝜋−2}{4}$
   4. The area of the region {$(𝑥, 𝑦)$ ∈ [0, 1] × $ℝ$ ∶ $𝑓(𝑥)$ ≤ $𝑦$ ≤ $\sqrt{1 − 𝑥^2}$} is $\frac{𝜋−1}{4}$

SECTION 2 (Maximum Marks:24)

• This section contains EIGHT (08) questions. The answer to each question is a NUMERICAL VALUE.
• For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) 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 as answer;
  • Zero Marks: 0 In all other cases.

7. The value of $((log_29)^2)^{\frac{1}{log_2(log_29)}}$x$(\sqrt{7})^{\frac{1}{log_47}}$ is ........
8. The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _____ .
9. Let $𝑋$ be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, … , and $𝑌$ be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, … . Then, the number of elements in the set $𝑋$ ∪ $𝑌$ is _____.
10. The number of real solutions of the equation $sin^{−1}\left(\sum \limits_{i=1}^{\infty} 𝑥^{𝑖+1}− 𝑥\sum \limits_{i=1}^{\infty}\left(\frac{𝑥}{2}\right)^𝑖 \right)$ =$\frac{\pi}{2}−cos^{−1}\left(\sum \limits_{i=1}^{\infty}\left(−\frac{𝑥}{2}\right)^i−\sum \limits_{i=1}^{\infty}(−𝑥)^i\right)$ lying in the interval $\left(−\frac{1}{2}, \frac{1}{2}\right)$ is _____ .(Here, the inverse trigonometric functions $sin^{−1}𝑥$ and $cos^{−1}𝑥$ assume values in $[−\frac{\pi}{2}, \frac{\pi}{2}]$ and [0, 𝜋], respectively.)
11. For each positive integer $𝑛$, let $𝑦_𝑛$ =$\frac{1}{𝑛}((𝑛 + 1)(𝑛 + 2) ⋯ (𝑛 + 𝑛))^{\frac{1}{n}}$. For $𝑥$ ∈ $ℝ$, let $[𝑥]$ be the greatest integer less than or equal to $𝑥$. If $\lim \limits_{𝑛 \to \infty}𝑦_𝑛$ = $𝐿$, then the value of $[𝐿]$ is _____ .
12. Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}$ ⋅ $\vec{b}$ = 0. For some $𝑥$, $𝑦$ ∈ $ℝ$, let $\vec{c}$ = $𝑥 \vec{a} + 𝑦 \vec{b} + (\vec{a} × \vec{b})$. If $|\vec{c}|$ = 2 and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then the value of $8 cos^2 \alpha$ is _____ .
13. Let $𝑎$, $𝑏$, $𝑐$ be three non-zero real numbers such that the equation $\sqrt{3} 𝑎 cos 𝑥$ + $2 𝑏 sin 𝑥$ = $𝑐$, $𝑥 \in \left[−\frac{\pi}{2}, \frac{\pi}{2}\right]$, has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3}$. Then, the value of $\frac{𝑏}{𝑎}$ is _____ .
14. A farmer $𝐹_1$ has a land in the shape of a triangle with vertices at $𝑃(0, 0)$, $𝑄(1, 1)$ and $𝑅(2, 0)$. From this land, a neighbouring farmer $𝐹_2$ takes away the region which lies between the side $𝑃𝑄$ and a curve of the form $ 𝑦 = 𝑥^𝑛$ $(𝑛 > 1)$. If the area of the region taken away by the farmer $𝐹_2$ is exactly 30% of the area of $∆𝑃𝑄𝑅$, then the value of $𝑛$ is _____ .

SECTION 3 (Maximum Marks:12)

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

Paragraph "X"

Let $𝑆$ be the circle in the $𝑥𝑦-$plane defined by the equation $ 𝑥^2 + 𝑦^2 = 4$.
(There are two questions Q. 15 and Q. 16 based on PARAGRAPH “X")

15. Let $𝐸_1𝐸_2$ and $𝐹_1𝐹_2$ be the chords of $𝑆$ passing through the point $𝑃_0 (1, 1)$ and parallel to the $x-$axis and the $y-$axis, respectively. Let $𝐺_1𝐺_2$ be the chord of $S$ passing through $𝑃_0$ and having slope −1. Let the tangents to $𝑆$ at $𝐸_1$ and $𝐸_2$ meet at $𝐸_3$, the tangents to $𝑆$ at $𝐹_1$ and $𝐹_2$ meet at $𝐹_3$, and the tangents to $𝑆$ at $𝐺_1$ and $𝐺_2$ meet at $𝐺_3$. Then, the points $𝐸_3$, $𝐹_3$, and $𝐺_3$ lie on the curve
    1. $𝑥 + 𝑦$ = 4
    2. $(𝑥 − 4)^2 + (𝑦 − 4)^2$ = 16
    3. $(𝑥 − 4)$$(𝑦 − 4)$ = 4
    4. $𝑥𝑦$ = 4
16. Let $𝑃$ be a point on the circle $𝑆$ with both coordinates being positive. Let the tangent to $𝑆$ at $𝑃$ intersect the coordinate axes at the points $𝑀$ and $𝑁$. Then, the mid-point of the line segment $𝑀𝑁$ must lie on the curve
    1. $(𝑥 + 𝑦)^2 = 3𝑥𝑦$
    2. $𝑥^{2/3} + 𝑦^{2/3} = 2^{4/3}$
    3. $𝑥^2 + 𝑦^2 = 2𝑥𝑦$
    4. $𝑥^2 + 𝑦^2 = 𝑥^2 𝑦^2$

Paragraph "A"

There are five students $𝑆_1, 𝑆_2, 𝑆_3, 𝑆_4$ and $𝑆_5$ in a music class and for them there are five seats $𝑅_1, 𝑅_2, 𝑅_3, 𝑅_4$ and $𝑅_5$ arranged in a row, where initially the seat $𝑅_𝑖$ is allotted to the student $𝑆_𝑖$, $𝑖$ = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.
(There are two questions Q. 17 and Q. 18 based on PARAGRAPH “A")

17. The probability that, on the examination day, the student $𝑆_1$ gets the previously allotted seat $𝑅_1$, and NONE of the remaining students gets the seat previously allotted to him/her is
    1. $\frac{3}{40}$
    2. $\frac{1}{8}$
    3. $\frac{7}{40}$
    4. $\frac{1}{5}$
18. For $𝑖$ = 1, 2, 3, 4, let $𝑇_𝑖$ denote the event that the students $𝑆_𝑖$ and $𝑆_{𝑖+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $𝑇_1 \cap 𝑇_2 \cap 𝑇_3 \cap 𝑇_4$ is
    1. $\frac{1}{15}$
    2. $\frac{1}{10}$
    3. $\frac{7}{60}$
    4. $\frac{1}{5}$

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