Let $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(0\right)=0, f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }\left(0\right)=1.$ If $g\left(x\right)={\int }_x^{\frac{\mathrm{\pi }}{2}} \left[f^{\prime }\left(t\right)cosect-\cot t cosect f\left(t\right)\right]dt$ , for $x\in \left(0,\frac{\pi }{2}\right]$ then $\underset{x\to 0}{\lim }g\left(x\right)=$

The value of  $cot \left(\underset{\text{n}=1}{\overset{23}{\Sigma }}{\text{cot}}^{-1}\left(1+\underset{\text{k}=1}{\overset{\text{n}}{\Sigma }}2\text{k}\right)\right)$  is

For a point P in the plane, let $d_1\left(P\right)$ and $d_2\left(P\right)$ be the distances of the point P from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying $2 \le d_1\left(P\right)+d_2\left(P\right) \le 4,$ is ________

If \(w=\alpha+i \beta\), where \(\beta \neq 0\) and \(z \neq 1\) satisfies the condition that \(\left(\frac{w-\bar{w} z}{1-z}\right)\) is purely real, then the set of values of \(z\) is

Let $\alpha$ and $\beta$ be non zero real numbers such that $2\left(\cos \beta -\cos \alpha \right)+\cos \alpha \cos \beta =1$. Then which of the following is/are true?

Let \(S\) denote the locus of the mid-points of those chords of the parabola \(y^2=x\), such that the area of the region enclosed between the parabola and the chord is \(\frac{4}{3}\). Let \(R\) denote the region lying in the first quadrant, enclosed by the parabola \(y^2=x\), the curve \(S\), and the lines \(x=1\) and \(x=4\).
Then which of the following statements is (are) TRUE?

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SP{S}_1=\alpha$, with $\alpha <\frac{\pi }{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1}P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $S{P}_1$, and $\beta =S_{1}P$. Then the greatest integer less than or equal to $\frac{\beta \delta }{9}\sin \frac{\alpha }{2}$ is _____ .

Paragraph :
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the density of air is \(\rho_{a}\) and that of the liquid \(\rho_{\ell}\), then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

Let complex numbers α and $\frac{1}{\overline{\alpha }}$   lie on circles ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2={\text{r}}^2\text{, }$  and  ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2={4\text{r}}^2\text{, }$respectively. If $z_0=x_0+i{y}_0$ satisfies the equation $2{\left|z_0\right|}^2=r^2+2\text{,}\text{then}\left|\alpha \right|=$

Match the reactions in List-I with the features of their products in List-II and choose the correct option.

List - I	List - II
(P)	1. Inversion of configuration
(Q)	2. Retention of configuration
(R)	3. Mixture of enantiomers
(S)	4. Mixture of structural isomers
	5. Mixture of diastereomers

The complex(es), which can exhibit the type of isomerism shown by $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_2{\mathrm{Br}}_2\right]$, is(are) $\left[\mathrm{en}={\mathrm{H}}_2{\mathrm{NCH}}_2{\mathrm{CH}}_2{\mathrm{NH}}_2\right]$

An electron in a hydrogen atom undergoes a transition from an orbit with a quantum number $n_i$ to another quantum number $n_f$ . $V_i \text{and} V_f$ are the initial and final potential energies of the electron. If $\frac{V_i}{V_f}=6.25$ then the smallest $n_f$ is

Let $X$ and $Y$ be two events such that $P\left(X\right)=\frac{1}{3}$, $P\left(X|Y\right)=\frac{1}{2}$ and $P\left(Y|X\right)=\frac{2}{5}.$ Then