Paragraph: Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
Question: The value of \(2 \int_0^{\frac{\pi}{2}} f(x) g(x) d x-\int_0^{\frac{\pi}{2}} g(x) d x\) is

Consider the cube in the first octant with sides $OP, OQ$ and OR of length $1$, along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT$. If $\overrightarrow{p}=\overrightarrow{SP}, \overrightarrow{q}=\overrightarrow{SQ}, \overrightarrow{r}=\overrightarrow{SR}$ and $\overrightarrow{t}=\overrightarrow{ST}$, then the value of $\left|\left(\overrightarrow{p}\times \overrightarrow{q}\right)\times \left(\overrightarrow{r}\times \overrightarrow{t}\right)\right|$ is ______.

The value of \(\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^x \frac{t \log (1+t)}{t^4+4} d t\) is

Let ${\ell }_1$ and ${\ell }_2$ be the lines ${\overrightarrow{r}}_1=\lambda \left(\hat{i}+\hat{j}+\hat{k}\right)$ and ${\overrightarrow{r}}_2=\left(\hat{j}-\hat{k}\right)+\mu \left(\hat{i}+\hat{k}\right)$, respectively, Let $X$ be the set of all the planes $H$ that contain the line ${\ell }_1$. For a plane $H$, let $d\left(H\right)$ denote the smallest possible distance between the points of ${\ell }_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d\left(H_0\right)$ is the maximum value of $d\left(H\right)$ as $H$ varies over all planes in $X$. Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The value of $d\left(H_0\right)$ is	$\left(1\right)$	$\sqrt{3}$
$\left(\mathrm{Q}\right)$	The distance of the point $\left(0, 1, 2\right)$ from $H_0$ is	$\left(2\right)$	$\frac{1}{\sqrt{3}}$
$\left(\mathrm{R}\right)$	The distance of origin from $H_0$ is	$\left(3\right)$	$0$
$\left(\mathrm{S}\right)$	The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_0$ is	$\left(4\right)$	$\sqrt{2}$
		$\left(5\right)$	$\frac{1}{\sqrt{2}}$

The correct option is

Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

The boiling point of water in a \(0.1\) molal silver nitrate solution (solution \(\mathbf{A}\) ) is \(\mathbf{x}^{\circ} \mathrm{C}\). To this solution \(\mathbf{A}\), an equal volume of \(0.1\) molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}\).
(Assume: Densities of the solutions \(\mathbf{A}\) and \(\mathbf{B}\) are the same as that of water and the soluble salts dissociate completely.
Use: Molal elevation constant (Ebullioscopic Constant), \(K_{b}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\); Boiling point of pure water as \(100^{\circ} \mathrm{C}\).)
The value of $\left|\mathrm{y}\right|$ is ___.

The cubic unit cell structure of a compound containing cation $\mathrm{M}$ and anion $\mathrm{X}$ is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

A circuit is connected as shown in the figure with the switch \(S\) open. When the switch is closed, the total amount of charge that flows from \(Y\) to \(X\) is

The volume (in \(\mathrm{mL}\) ) of \(0.1 \mathrm{M} \mathrm{AgNO}_3\) required for complete precipitation of chloride ions present in \(30 \mathrm{~mL}\) of \(0.01 \mathrm{M}\) solution of \(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}_{)_5} \mathrm{Cl}\right] \mathrm{Cl}_2\right.\), as silver chloride is close to

Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1, r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0\) is

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be functions defined by
\(f(x)=\left\{\begin{array}{ll}x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0,\end{array} \quad\right.\) and \(g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2} \\ 0, & \text { otherwise }\end{cases}\)
Let \(a, b, c, d \in \mathbb{R}\). Define the function \(h: \mathbb{R} \rightarrow \mathbb{R}\) by
\(h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R}\)
Match each entry in List-I to the correct entry in List-II.

The correct option is :