Let $S$ be the set of all complex numbers $z$ satisfying $\left|z-2+i\right|\ge \sqrt{5}.$ If the complex number $z_0$ is such that $\frac{1}{\left|z_0-1\right|}$ is the maximum of the set $\left\{\frac{1}{\left|z-1\right|}:z\in S\right\},$ then the principal argument of $\frac{4-z_0-{\overline{z}}_0}{z_0-{\overline{z}}_0+2i}$ 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

The normal at a point \(P\) on the ellipse \(x^2+4 y^2=16\) meets the \(x\)-axis at \(Q\). If \(M\) is the mid-point of the line segment \(P Q\), then the locus of \(M\) intersects the latusrectum of the given ellipse at the points

Consider the given data with frequency distribution $\begin{array}{lllllll}{x}_i& 3& 8& 11& 10& 5& 4\\ f_i& 5& 2& 3& 2& 4& 4\end{array}$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The mean of the above data is	$\left(1\right)$	$2.5$
$\left(\mathrm{Q}\right)$	The median of the above data is	$\left(2\right)$	$5$
$\left(\mathrm{R}\right)$	The mean deviation about the mean of the above data is	$\left(3\right)$	$6$
$\left(\mathrm{S}\right)$	The mean deviation about the median of the above data is	$\left(4\right)$	$2.7$
		$\left(5\right)$	$2.4$

The correct option is

The function $\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$ is the solution of the differential equation $\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}+\frac{\mathrm{x}\mathrm{y}}{{\mathrm{x}}^2-1}=\frac{{\mathrm{x}}^4+2\mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}$ in (- 1 , 1) satisfying $\mathrm{f}\left(0\right)=0.$ Then $\underset{-\frac{\sqrt{3}}{2}}{\overset{\frac{\sqrt{3}}{2}}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{ }\mathrm{d}\mathrm{x}$ is

Paragraph:

Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Consider the lines given by
\[
L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0
\]
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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

Paragraph:
Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\)\(\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)+2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=\) \(-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ;\) \(E_{\mathrm{red}}^{\circ}=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\) ]
Question:
Ammonia is always added in this reaction. Which of the following must be incorrect?

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 _____ .

A \(2 \mu \mathrm{F}\) capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch \(S\) is turned to position 2 , is

Two identical moving coil galvanometer have $10\mathrm{ }\mathrm{\Omega }$ resistance and full scale deflection at $2\mathrm{\mu }\mathrm{A}$ current. One of them is converted into a voltmeter of $100\mathrm{ }\mathrm{m}\mathrm{V}$ full scale reading and the other into an Ammeter of $1\mathrm{ }\mathrm{m}\mathrm{A}$ full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $\mathrm{R}=1000\mathrm{ }\mathrm{\Omega }$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?

The correct statement(s) regarding,
(i) $\mathrm{H}\mathrm{C}\mathrm{l}\mathrm{O}$
(ii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_2$
(iii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_3$ and
(iv) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_4$ is /are