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SECTION 1 (Maximum Marks:12)

• 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 If ONLY 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 $ℝ$ denote the set of all real numbers. Let $𝑎_𝑖$, $𝑏_𝑖 \in ℝ$ for $𝑖 \in$ {$1, 2, 3$}.Define the functions $𝑓: ℝ \to ℝ$, $𝑔: ℝ \to ℝ$, and $ℎ: ℝ \to ℝ$ by
   $𝑓(𝑥)$ = $𝑎_1$ + $10𝑥$ + $𝑎_2𝑥^2$ + $𝑎_3𝑥^3$ + $𝑥^4$,
   $𝑔(𝑥)$ = $𝑏_1$ + $3𝑥$ + $𝑏_2𝑥^2$ + $𝑏_3𝑥^3$ + $𝑥^4$,
   $ℎ(𝑥)$ = $𝑓(𝑥 + 1)$ $− 𝑔(𝑥 + 2)$.
   If $𝑓(𝑥) ≠ 𝑔(𝑥)$ for every $𝑥 ∈ ℝ$, then the coefficient of $𝑥^3$in $ℎ(𝑥)$ is
    
   1. 8
   2. 2
   3. $-4$
   4. $-6$
2. Three students $𝑆_1$, $𝑆_2$, and $𝑆_3$ are given a problem to solve. Consider the following events:
   
   $𝑈$: At least one of $𝑆_1$, $𝑆_2$, and $𝑆_3$ can solve the problem,
   $𝑉$: $𝑆_1$ can solve the problem, given that neither $𝑆_2$ nor $𝑆_3$ can solve the problem,
   $𝑊$: $𝑆_2$ can solve the problem and 𝑆3 cannot solve the problem,
   $𝑇$: $𝑆_3$ can solve the problem.
   
   For any event $𝐸$, let $𝑃(𝐸)$ denote the probability of $𝐸$. If
   $𝑃(𝑈)$ =$\frac{1}{2}$, $𝑃(𝑉)$ =$\frac{1}{10}$ , and $𝑃(𝑊)$ =$\frac{1}{12}$,then $𝑃(𝑇)$ is equal to
    
   1. $\frac{13}{36}$
   2. $\frac{1}{3}$
   3. $\frac{19}{60}$
   4. $\frac{1}{4}$
3. Let $ℝ$ denote the set of all real numbers. Define the function $𝑓: ℝ \to ℝ$ by
   $f(x)$=$\left\{\begin{array}{cc} 2-2x^2-x^2\sin \frac{1}{x} & \text { if } x \neq 0. \\ 2 & \text { if } x=0. \end{array}\right.$
   Then which one of the following statements is TRUE?
    
   1. The function 𝑓 is NOT differentiable at $𝑥 = 0$
   2. There is a positive real number $\delta$, such that $𝑓$ is a decreasing function on the interval $(0, \delta)$
   3. For any positive real number $\delta$, the function $𝑓$ is NOT an increasing function on the interval $(−\delta, 0)$
   4. $𝑥 = 0$ is a point of local minima of $𝑓$
4. Consider the matrix
   $\begin{equation*} \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \end{equation*}$
   Let the transpose of a matrix $𝑋$ be denoted by $𝑋^𝑇$. Then the number of 3 × 3 invertible matrices $𝑄$with integer entries, such that
   $𝑄^{−1}$ = $𝑄^𝑇$and $𝑃𝑄 = 𝑄𝑃$,
   is
    
   1. 32
   2. 8
   3. 16
   4. 24

SECTION 2 (Maximum Marks:12)

• This section contains THREE (03) 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 ONLY if (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, 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 (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get -2 marks.

5. Let $𝐿_1$ be the line of intersection of the planes given by the equations
   $2𝑥$ + $3𝑦$ + $𝑧$ = 4 and $𝑥$ + $2𝑦$ + $𝑧$ = 5 .
   Let $𝐿_2$ be the line passing through the point $𝑃(2, −1, 3)$ and parallel to $𝐿_1$. Let $𝑀$ denote the plane given by the equation
   $2𝑥$ + $𝑦 − 2𝑧$ = 6 .
   Suppose that the line $𝐿_2$ meets the plane $𝑀$ at the point $𝑄$. Let $𝑅$ be the foot of the perpendicular drawn from $𝑃$ to the plane $𝑀$. Then which of the following statements is (are) TRUE?
    
   1. The length of the line segment $𝑃𝑄$ is $9\sqrt{3}$
   2. The length of the line segment $𝑄𝑅$ is 15
   3. The area of $\Delta 𝑃𝑄𝑅$ is $\frac{3}{2}\sqrt{234}$
   4. The acute angle between the line segments $𝑃𝑄$ and $𝑃𝑅$ is $cos^{−1}(\frac{1}{2\sqrt{3}})$
6. Let $ℕ$ denote the set of all natural numbers, and $ℤ$ denote the set of all integers. Consider the functions $𝑓: ℕ \to ℤ$ and $𝑔: ℤ \to ℕ$ defined by
   $f(n)$=$\left\{\begin{array}{cc} (n+1)/2 & \text { if n is odd } . \\ (4-n)/2 & \text { if n is even }. \end{array}\right.$
   and
   $g(n)$=$\left\{\begin{array}{cc} 3+2n & \text { if } n \geq 0. \\ -2n & \text { if } n < 0. \end{array}\right.$
   Define $(𝑔 ∘ 𝑓)(𝑛) = 𝑔(𝑓(𝑛))$ for all $𝑛 \in ℕ$, and $(𝑓 ∘ 𝑔)(𝑛) = 𝑓(𝑔(𝑛))$ for all $𝑛 \in ℤ$.
   Then which of the following statements is (are) TRUE?
    
   1. $𝑔 ∘ 𝑓$ is NOT one-one and $𝑔 ∘ 𝑓$ is NOT onto
   2. $𝑓 ∘ 𝑔$ is NOT one-one but $𝑓 ∘ 𝑔$ is onto
   3. $𝑔$ is one-one and $𝑔$ is onto
   4. $𝑓$ is NOT one-one but $𝑓$ is onto
7. Let $ℝ$ denote the set of all real numbers. Let $𝑧_1$ = $1 + 2𝑖$ and $𝑧_2 = 3𝑖$ be two complex numbers, where $𝑖 = \sqrt{−1}$. Let $𝑆$ = {$(𝑥, 𝑦) ∈ ℝ × ℝ$ ∶ $|𝑥 + 𝑖𝑦 − 𝑧_1|$ = $2|𝑥 + 𝑖𝑦 − 𝑧_2|$ }. Then which of the following statements is (are) TRUE?
    
   1. $𝑆$ is a circle with centre $\left(\frac{−1}{3}, \frac{10}{3}\right)$
   2. $𝑆$ is a circle with centre $\left(\frac{1}{3}, \frac{8}{3}\right)$
   3. $𝑆$ is a circle with radius $\sqrt{2}$
   4. $𝑆$ is a circle with radius $2\sqrt{2}$

SECTION 3 (Maximum Marks:24)

• This section contains SIX (06) 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.

8. Let the set of all relations $𝑅$ on the set {$𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓$}, such that $𝑅$ is reflexive and symmetric, and $𝑅$ contains exactly 10 elements, be denoted by $\delta$. Then the number of elements in $\delta$ is _________________.
9. For any two points $𝑀$ and $𝑁$ in the $𝑋𝑌-$plane, let $\vec{𝑀N}$ denote the vector from $𝑀$ to $𝑁$, and $\vec{0}$ denote the zero vector. Let $𝑃$, $𝑄$ and $𝑅$ be three distinct points in the $𝑋𝑌-$plane. Let $𝑆$ be a point inside the triangle $\Delta 𝑃𝑄𝑅$ such that $\vec{𝑆P}$+ $5 \vec{𝑆Q}$ + $6 \vec{𝑆R}$ = $\vec{0}$.Let $𝐸$ and $𝐹$ be the mid-points of the sides $𝑃𝑅$ and $𝑄𝑅$, respectively. Then the value of
   $\frac{\text{length of the line segment 𝐸𝐹}}{\text{ length of the line segment 𝐸𝑆}}$
   is _______________.
10. Let $𝑆$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $𝑆$, but 0210222 is NOT in $𝑆$.
    Then the number of elements  $𝑥$ in $𝑆$ such that at least one of the digits 0 and 1 appears exactly twice in $𝑥$, is equal to ____________.
11. Let $\alpha$ and $\beta$ be the real numbers such that
    $\lim \limits_{x \to 0} \frac{1}{x^3}\left(\frac{\alpha}{2}\int \limits_0^x \frac{1}{1-t^2}dt+\beta x \cos x \right)$=2. Then the value of $\alpha + \beta$ is __________.
12. Let $ℝ$ denote the set of all real numbers. Let $𝑓: ℝ \to ℝ$ be a function such that $𝑓(𝑥) > 0$ for all$𝑥 \in ℝ$, and $𝑓(𝑥 + 𝑦)$ = $𝑓(𝑥)𝑓(𝑦)$ for all $𝑥, 𝑦 \in ℝ$.
    Let the real numbers $𝑎_1$, $𝑎_2$, … , $𝑎_{50}$ be in an arithmetic progression. If $𝑓(𝑎_{31})$ = $64𝑓(𝑎_{25})$, and
    $\sum \limits_{i=1}^{50}f(a_i)$=$3(2^{25}+1)$, then the value of
    $\sum \limits_{i=6}^{30}f(a_i)$
    is _________________.
13. For all $𝑥 > 0$, let $𝑦_1(𝑥)$, $𝑦_2(𝑥)$, and $𝑦_3(𝑥)$ be the functions satisfying
    $\frac{𝑑𝑦_1}{𝑑𝑥}$$− (sin 𝑥)^2 𝑦_1$ = 0,  $𝑦_1(1)$ = 5 ,
    $\frac{𝑑𝑦_2}{𝑑𝑥}$$− (cos 𝑥)^2 𝑦_2$ = 0, $𝑦_2(1)$ =$\frac{1}{3}$,
    $\frac{𝑑𝑦_3}{𝑑𝑥}$$− \left(\frac{2−𝑥^3}{𝑥^3}\right) 𝑦_3$ = 0, $𝑦_3(1)$ = $\frac{3}{5𝑒}$,
    respectively. Then
    $\lim \limits_{𝑥 \to 0^+}\frac{𝑦_1(𝑥)𝑦_2(𝑥)𝑦_3(𝑥) + 2𝑥}{𝑒^{3𝑥} \sin 𝑥}$
    is equal to ______________.

SECTION 4 (Maximum Marks:12)

• This section contains FOUR (04) Matching List Sets.
• Each set has ONE Multiple Choice Question.
• Each set has TWO lists: List-I and List-II.
• List-I has Four entries (P), (Q), (R) and (S) and List-II has Five entries (1), (2), (3), (4) and (5).
• FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.
• Answer to each question will be evaluated according to the following marking scheme:
  
  • Full Marks: +3 ONLY if the option corresponding to the correct combination 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.

14. Consider the following frequency distribution:
    
                                                 

    Value	4	5	8	9	6	12	11
    Frequency	5	$f_1$	$f_2$	2	1	1	3

    
    
    Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.
    For the given frequency distribution, let 𝛼 denote the mean deviation about the mean, 𝛽 denote the mean deviation about the median, and 𝜎 2 denote the variance. Match each entry in List-I to the correct entry in List-II and choose the correct option.                                                                                                     

    	List - I		List - II
    (P)	$7f_1$+$9f_2$ is equal to	(1)	146
    (Q)	$19\alpha$ is equal to	(2)	47
    (R)	$19\beta$ is equal to	(3)	48
    (S)	$19 \sigma^2$ is equal to	(4)	145
    		(5)	55

    
    The correct option is:  
     
    1. (P) ⟶ (5) (Q) ⟶ (3) (R) ⟶ (2) (S) ⟶ (4)
       
    2. (P) ⟶ (5) (Q) ⟶ (2) (R) ⟶ (3) (S) ⟶ (1)
       
    3. (P) ⟶ (5) (Q) ⟶ (3) (R) ⟶ (2) (S) ⟶ (1)
       
    4. (P) ⟶ (3) (Q) ⟶ (2) (R) ⟶ (5) (S) ⟶ (4)
15. Let $ℝ$ denote the set of all real numbers. For a real number $𝑥$, let $[𝑥]$ denote the greatest integer less than or equal to $𝑥$. Let $𝑛$ denote a natural number.
    Match each entry in List-I to the correct entry in List-II and choose the correct option.                                                                                                     

    	List - I		List - II
    (P)	The minimum value of 𝑛 for which the function $f(x)$=$\left[\frac{10x^3-45x^2+60x+35}{n}\right]$ is continuous on the interval [1, 2], is	(1)	8
    (Q)	The minimum value of $𝑛$ for which $𝑔(𝑥)$ = $(2𝑛^2 − 13𝑛 − 15)$$(𝑥^3 + 3𝑥)$,  $𝑥 ∈ ℝ$, is an increasing function on $ℝ$, is	(2)	9
    (R)	The smallest natural number $𝑛$ which is greater than 5, such that $𝑥 = 3$ is a point of local minima of $ℎ(𝑥)$ = $(𝑥^2 − 9)^𝑛$$(𝑥^2 + 2𝑥 + 3)$, is	(3)	5
    (S)	Number of $𝑥_0 ∈ ℝ$ such that $f(x)$=$\sum \limits_{k=0}^{4}\left(\sin|x-k|+\cos|x-k+\frac{1}{2}|\right)$, $x \in R$, is NOT differentiable at $x_0$, is	(4)	6
    		(5)	10

    
    The correct option is:  
     
    1. (P) ⟶ (1) (Q) ⟶ (3) (R) ⟶ (2) (S) ⟶ (5)
       
    2. (P) ⟶ (2) (Q) ⟶ (1) (R) ⟶ (4) (S) ⟶ (3)
       
    3. (P) ⟶ (5) (Q) ⟶ (1) (R) ⟶ (4) (S) ⟶ (3)
       
    4. (P) ⟶ (2) (Q) ⟶ (3) (R) ⟶ (1) (S) ⟶ (5)
16. Let $\vec{w}$=$\hat{i}+\hat{j}-2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u}×\vec{v}$=$\vec{w}$ and $\vec{v}$×$\vec{w}$=$\vec{u}$. Let  $𝛼$, $𝛽$, $𝛾$, and $𝑡$ be real numbers such that
    $\vec{u}$=$\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$, $− 𝑡 𝛼 + 𝛽 + 𝛾$ = 0, $𝛼 − 𝑡 𝛽 + 𝛾$ = 0, and $𝛼 + 𝛽 − 𝑡 𝛾$ = 0.
    Match each entry in List-I to the correct entry in List-II and choose the correct option.                                                                                                     

    	List - I		List - II
    (P)	$|\vec{v}|$ is equal to	(1)	0
    (Q)	If $\alpha$=$\sqrt{3}$, then $\gamma^2$ is equal to	(2)	1
    (R)	If $\alpha=\sqrt{3}$, then $(\beta+\gamma)^2$ is equal to	(3)	2
    (S)	If $\alpha=\sqrt{2}$, then $t+3$ is equal to	(4)	3
    		(5)	5

    
    The correct option is:  
     
    1. (P) ⟶ (2) (Q) ⟶ (1) (R) ⟶ (4) (S) ⟶ (5)
       
    2. (P) ⟶ (2) (Q) ⟶ (4) (R) ⟶ (3) (S) ⟶ (5)
       
    3. (P) ⟶ (2) (Q) ⟶ (1) (R) ⟶ (4) (S) ⟶ (3)
       
    4. (P) ⟶ (5) (Q) ⟶ (4) (R) ⟶ (1) (S) ⟶ (3)

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