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

Let $$\alpha$$ be a positive real number. Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ and $$g:(\alpha, \infty) \rightarrow \mathbb{R}$$ be the functions defined by

$$ f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)} . $$

Then the value of $$\lim \limits_{x \rightarrow \alpha^{+}} f(g(x))$$ is

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 ____________.

Let $$z$$ be a complex number with a non-zero imaginary part. If

$$ \frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}} $$

is a real number, then the value of $$|z|^{2}$$ is _________.

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

$$ \bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right) $$

is _________.

Let $$l_{1}, l_{2}, \ldots, l_{100}$$ be consecutive terms of an arithmetic progression with common difference $$d_{1}$$, and let $$w_{1}, w_{2}, \ldots, w_{100}$$ be consecutive terms of another arithmetic progression with common difference $$d_{2}$$, where $$d_{1} d_{2}=10$$. For each $$i=1,2, \ldots, 100$$, 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 __________.

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $$0,2,3,4,6,7$$ is _________.

Let $$A B C$$ be the triangle with $$A B=1, A C=3$$ and $$\angle B A C=\frac{\pi}{2}$$. If a circle of radius $$r>0$$ touches the sides $$A B, A C$$ and also touches internally the circumcircle of the triangle $$A B C$$, then the value of $$r$$ is __________ .

Consider the equation

$$ \int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) $$

Which of the following statements is/are TRUE?

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an arithmetic progression with $$a_{1}=7$$ and common difference 8. Let $$T_{1}, T_{2}, T_{3}, \ldots$$ 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 ?

Let $$P_{1}$$ and $$P_{2}$$ be two planes given by

$$ \begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\\\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned} $$

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $$P_{1}$$ and $$P_{2}$$ ?

Let $$S$$ be the reflection of a point $$Q$$ with respect to the plane given by

$$ \vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k} $$

where $$t, p$$ are real parameters and $$\hat{\imath}, \hat{\jmath}, \hat{k}$$ are the unit vectors along the three positive coordinate axes. If the position vectors of $$Q$$ and $$S$$ are $$10 \hat{\imath}+15 \hat{\jmath}+20 \hat{k}$$ and $$\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}$$ respectively, then which of the following is/are TRUE ?

Consider the parabola $$y^{2}=4 x$$. 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 $$S P_{1}$$ and $$S P_{2}$$ respectively such that $$P Q_{1}$$ is perpendicular to $$S P_{1}$$ and $$P Q_{2}$$ is perpendicular to $$S P_{2}$$. Then, which of the following is/are TRUE?

Let $$|M|$$ denote the determinant of a square matrix $$M$$. Let $$g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be the function defined by

$$ g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1} $$

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 ?

Consider the following lists :

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

The correct option is:

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 $$i^{\text {th }}$$ round.

List-I	List-II
(I) Probability of $$\left(X_{2} \geq Y_{2}\right)$$ is	(P) $$\frac{3}{8}$$
(II) Probability of $$\left(X_{2}>Y_{2}\right)$$ is	(Q) $$\frac{11}{16}$$
(III) Probability of $$\left(X_{3}=Y_{3}\right)$$ is	(R) $$\frac{5}{16}$$
(IV) Probability of $$\left(X_{3}>Y_{3}\right)$$ is	(S) $$\frac{355}{864}$$
	(T) $$\frac{77}{432}$$

The correct option is:

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$

List-I	List-II
(I) If $$\frac{q}{r}=10$$, then the system of linear equations has	(P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ 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:

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

The correct option is: