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SECTION 1 (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 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 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: +3 ONLY if the correct numerical value is entered;
  • Zero Marks: 0 In all other cases.

1. Considering only the principal values of the inverse trigonometric functions, the value of $\frac{3}{2} cos^{−1} \sqrt{ \frac{2}{2+ \pi^2}} $+ $\frac{1}{4} sin^{−1} \frac{ 2 \sqrt{2} \pi}{2 + \pi ^2 }$ + $tan^{−1} \frac{ \sqrt{2}}{\pi} $is __________ .
2. Let $ \alpha $ be a positive real number. Let $𝑓: R → R$ and $ 𝑔: (\alpha, \infty) → R$ be the functions defined by $𝑓(𝑥) $= $sin (\frac{\pi 𝑥}{12})$ and $𝑔(𝑥)$ =$ \frac{ 2 log_e ( \sqrt{𝑥}− \sqrt{\alpha} )}{log_e ( 𝑒^{ \sqrt{𝑥 }}− 𝑒^{\sqrt{\alpha} })}$. Then the value of $ \lim \limits_{𝑥 \to \alpha^+} 𝑓(𝑔(𝑥))$ is __________.
3. In a study about a pandemic, data of 900 persons was collected. It was found that
   190 persons had symptom of fever,
   220 persons had symptom of cough,
   220 persons had symptom of breathing problem,
   330 persons had symptom of fever or cough or both,
   350 persons had symptom of cough or breathing problem or both,
   340 persons had symptom of fever or breathing problem or both,
   30 persons had all three symptoms (fever, cough and breathing problem).
   If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____________.
4. Let $𝑧$ be a complex number with non-zero imaginary part. If
   $\frac{2 + 3𝑧 + 4𝑧^2}{2 − 3𝑧 + 4𝑧^2}$ is a real number, then the value of $|𝑧|^2$ is _____________.
5. Let $\bar{z}$ denote the complex conjugate of a complex number $𝑧$ and let $ 𝑖 = \sqrt{−1}$ . In the set of complex numbers, the number of distinct roots of the equation $ \bar{z} − 𝑧^2$ =$ 𝑖(\bar{z}+ 𝑧^2)$ is _____________.
6. Let $𝑙_1$,$ 𝑙_2$, … , $𝑙_{100} $ be consecutive terms of an arithmetic progression with common difference $𝑑_1$, and let $𝑤_1$, $𝑤_2$, … ,$ 𝑤_{100} $ be consecutive terms of another arithmetic progression with common difference $𝑑_2$ , where $𝑑_1 𝑑_2$ = 10. For each 𝑖 = 1, 2, … , 100, let $𝑅_𝑖$ be a rectangle with length $𝑙_𝑖$, width $𝑤_𝑖$ and area $𝐴_𝑖$. If $𝐴_{51} −$$ 𝐴_{50}$ = 1000, then the value of $𝐴_{100} − 𝐴_{90} $ is ____________.
7. The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ____________.
8. Let $𝐴𝐵𝐶$ be the triangle with $𝐴𝐵$ = 1, $𝐴𝐶$ = 3 and $\angle 𝐵𝐴𝐶 = \frac{\pi}{2}$. If a circle of radius $𝑟$ > 0 touches the sides $𝐴𝐵$, $𝐴𝐶$ and also touches internally the circumcircle of the triangle $𝐴𝐵𝐶$, then the value of $𝑟$ is _____________.

SECTION 2 (Maximum Marks:24)

• 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 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 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.

9. Consider the equation
   $\int \limits_1^e \frac{(log_e x) ^{\frac{1}{2}}}{x(a - (log_e x) ^{\frac{3}{2}})^2} dx = 1$, $a \in ( - \infty, 0) \cup (1, \infty) $
   1. No a satisfies the above equation.
   2. An integer a satisfies the above equation.
   3. An irrational number a satisfies the above equation.
   4. More than one a satisfy the above equation.
10. Let $𝑎_1$, $𝑎_2$, $a_3$, … be an arithmetic progression with $𝑎_1$ = 7 and common difference 8. Let $𝑇_1$, $𝑇_2$, $𝑇_3$, … be such that $𝑇_1$ = 3 and $𝑇_{𝑛+1} − 𝑇_𝑛 = 𝑎_𝑛$ for $𝑛 \geq 1$. Then, which of the following is/are TRUE ?
    1. $T_{20} = 1604$
    2. $\sum \limits_{k = 1}^{20} T_k = 10510$
    3. $T_{30} = 3454$
    4. $\sum \limits_{k = 1}^{30} T_k = 35610$
11. Let $𝑃_1$ and $𝑃_2$ be two planes given by
    $𝑃_1$: $10𝑥$ + $15𝑦$ + $12𝑧$ − 60 = 0 ,
    $𝑃_2$ : $− 2𝑥$ + $5𝑦$ + $4𝑧$ − 20 = 0 .
    Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $𝑃_1$and $𝑃_2$ ?
    1. $\frac{x - 1}{0} = \frac{y - 1}{0} = \frac{z - 1}{5}$
    2. $\frac{x - 6}{-5} = \frac{y}{2} = \frac{z}{3}$
    3. $\frac{x}{-2} = \frac{y - 4}{5} = \frac{z}{4}$
    4. $\frac{x}{1} = \frac{y - 4}{-2} = \frac{z}{3}$
12. Let $𝑆$ be the reflection of a point $𝑄$ with respect to the plane given by $\vec{𝑟}$ = $−(𝑡 + 𝑝)\hat{𝑖}$ + $𝑡\hat{𝑗} $+ $(1 + 𝑝)\hat{𝑘}$ where $𝑡$, $𝑝$ are real parameters and $\hat{𝑖}$ , $\hat{𝑗} $, $\hat{𝑘} $ are the unit vectors along the three positive coordinate axes. If the position vectors of $𝑄$ and $𝑆$ are $10\hat{𝑖}$ + $15\hat{𝑗}$ + $20\hat{𝑘}$ and $\alpha \hat{𝑖}$ + $\beta \hat{ 𝑗}$ + $\gamma \hat{𝑘} $ respectively, then which of the following is/are TRUE ?
    1. $ 3(\alpha + \beta) = - 101$
    2. $ 3(\beta + \gamma) = - 71$
    3. $ 3(\gamma + \alpha) = - 86$
    4. $ 3(\alpha + \beta + \gamma) = - 121$
13. Consider the parabola $𝑦^2 = 4𝑥$. Let $𝑆$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $𝑃$ = (−2, 1) meet the parabola at $𝑃_1$ and $𝑃_2$. Let $𝑄_1$ and $𝑄_2$ be points on the lines $𝑆𝑃_1$ and $𝑆𝑃_2$ respectively such that $𝑃𝑄_1$ is perpendicular to $𝑆𝑃_1$ and $𝑃𝑄_2$ is perpendicular to $𝑆𝑃_2$. Then, which of the following is/are TRUE ?
    1. $SQ_1 = 2$
    2. $ Q_1 Q_2 = \frac{3 \sqrt{10}}{5}$
    3. $PQ_1 = 3$
    4. $SQ_2 = 1$
14. Let $|𝑀|$ denote the determinant of a square matrix $𝑀$. Let $𝑔:[0, \frac{\pi}{2}] → ℝ$ be the function defined by $𝑔(\theta)$ = $\sqrt{𝑓(\theta) − 1}$ + $\sqrt{𝑓 (\frac{\pi}{2}-\theta)-1}$ where $f(\theta)$=$\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|$+$\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$. Let $𝑝(𝑥)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $𝑔(\theta)$, and $𝑝(2)$ = 2 − $\sqrt{2}$ . Then, which of the following is/are TRUE ?
    1. $p \left (\frac{3 + \sqrt{2}}{4} \right )< 0$
    2. $p \left (\frac{3 + \sqrt{2}}{4} \right )> 0$
    3. $p \left (\frac{5 \sqrt{2} - 1}{4} \right ) > 0$
    4. $ p \left (\frac{5 \sqrt{2} - 1}{4} \right )< 0$

SECTION 3 (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 (I), (II), (III) and (IV) and List-II has Five entries (P), (Q), (R), (S) and (T).
• 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.

15. Consider the following lists :                              

    	List - I		List - II
    (I)	$\left\{x \in \right.$ $\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right] :$ $ \cos x+\sin x$ $\left.=1\right\}$	(P)	has two elements
    (II)	$\left\{x \in \right.$ $\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right] :$ $ \sqrt{3}tan3x$ $\left.=1\right\}$	(Q)	has three elements
    (III)	$\left\{x \in \right.$ $\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right] :$ $ 2\cos(2 x)$ $\left.=\sqrt{3}\right\}$	(R)	has four elements
    (IV)	$\left\{x \in \right.$ $\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right] :$ $ \cos x-\sin x$ $\left.=1\right\}$	(S)	has five elements
    		(T)	has six elements

    
    The correct option is:
    1. (I) → (P); (II) → (S); (III) → (P); (IV) → (S)
    2. (I) → (P); (II) → (P); (III) → (T); (IV) → (R)
    3. (I) → (Q); (II) → (P); (III) → (T); (IV) → (S)
    4. (I) → (Q); (II) → (S); (III) → (P); (IV) → (R)
16. Two players, $𝑃_1$ and $𝑃_2$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $𝑥$ and $𝑦$ denote the readings on the die rolled by $𝑃_1$ and $𝑃_2$ , respectively. If $𝑥 > 𝑦$, then $𝑃_1 $ scores 5 points and $𝑃_2$ scores 0 point. If $𝑥 = 𝑦$, then each player scores 2 points. If $𝑥 < 𝑦$, then $𝑃_1$ scores 0 point and $𝑃_2$ scores 5 points. Let $𝑋_𝑖$ and $𝑌_𝑖$ be the total scores of $𝑃_1$ and $𝑃_2$, respectively, after playing the $𝑖^ {𝑡ℎ} $ round.                              

    	List - I		List - II
    (I)	Probability of $(𝑋_2 \geq 𝑌_2)$ is	(P)	$\frac{3}{8}$
    (II)	Probability of $(𝑋_2 > 𝑌_2)$ is	(Q)	$\frac{11}{16}$
    (III)	Probability of $(𝑋_3 = 𝑌_3)$ is	(R)	$\frac{5}{16}$
    (IV)	Probability of $(𝑋_3 > 𝑌_3)$ is	(S)	$\frac{355}{864}$
    		(T)	$\frac{77}{432}$

    
    The correct option is:
    1. (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)
    2. (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)
    3. (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)
    4. (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)
17. Let $𝑝$, $𝑞$, $𝑟$ be nonzero real numbers that are, respectively, the $10^{𝑡ℎ}$, $100^{𝑡ℎ}$ and $1000^{𝑡ℎ} $ terms of a harmonic progression. Consider the system of linear equations
    $𝑥$ + $𝑦$ + $𝑧$ = 1
    $10𝑥$ + $100𝑦$ + $1000𝑧$ = 0
    $𝑞𝑟 𝑥$ + $𝑝𝑟 𝑦$ + $𝑝𝑞 𝑧$ = 0
    .                            

    	List - I		List - II
    (I)	If $\frac{𝑞}{ 𝑟 }$ = 10, then the system of linear equations has	(P)	$ 𝑥$ = 0, $𝑦 $= $\frac{10}{ 9 }$, $𝑧 $= $\frac{− 1}{ 9}$ as a solution
    (II)	If $\frac{p}{ 𝑟 } \neq 100$, then the system of linear equations has	(Q)	$ 𝑥$ = $\frac{10}{ 9}$ , $𝑦$ = $\frac{− 1}{ 9}$ , $𝑧$ = 0 as a solution
    (III)	If $\frac{p}{ q } \neq 10$, then the system of linear equations has	(R)	infinitely many solutions
    (IV)	If $\frac{p}{ q } = 10$, then the system of linear equations has	(S)	no solution
    		(T)	at least one solution

    
    The correct option is:
    1. (I) → (T); (II) → (R); (III) → (S); (IV) → (T)
    2. (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
    3. (I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
    4. (I) → (T); (II) → (S); (III) → (P); (IV) → (T)
18. Consider the ellipse $\frac{𝑥^2}{ 4 }+\frac{ 𝑦^2}{ 3} = 1 $. Let $𝐻(\alpha, 0)$, $0 < \alpha < 2$, be a point. A straight line drawn through $𝐻$ parallel to the $𝑦-$axis crosses the ellipse and its auxiliary circle at points $𝐸$ and $𝐹$ respectively, in the first quadrant. The tangent to the ellipse at the point $𝐸$ intersects the positive $𝑥-$axis at a point $𝐺$. Suppose the straight line joining $𝐹$ and the origin makes an angle $\phi$ with the positive $𝑥-$axis.                            

    	List - I		List - II
    (I)	If $\phi= \frac{\pi}{ 4 }$, then the area of the triangle $𝐹𝐺𝐻$ is	(P)	$\frac{(\sqrt{3}-1)^4}{8}$
    (II)	If $\phi= \frac{\pi}{ 3 }$, then the area of the triangle $𝐹𝐺𝐻$ is	(Q)	1
    (III)	If $\phi= \frac{\pi}{ 6 }$, then the area of the triangle $𝐹𝐺𝐻$ is	(R)	$\frac{3}{4}$
    (IV)	If $\phi= \frac{\pi}{ 12}$, then the area of the triangle $𝐹𝐺𝐻$ is	(S)	$\frac{1}{2\sqrt{3}}$
    		(T)	$\frac{3\sqrt{3}}{2}$

    
    The correct option is:
    
    1. (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
    2. (I) → (R); (II) → (T); (III) → (S); (IV) → (P)
    3. (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
    4. (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

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