Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $E_1 E_2$ and $F_1 F_2$ be the chords of $S$ passing through the point $P_0 \left(\mathrm{1,1}\right)$ and parallel to the $x$ – axis and the $y$ – axis, respectively. Let $G_1 G_2$ be the chord of $S$ passing through $P_0$ and having slope $-1.$ Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3,$ and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3, F_3,$ and $G_3$ lie on the curve

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Let \(A, B, C\) be three sets of complex numbers as defined below.
\(A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ;\) \(C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}\).
Question:
The number of elements in the set \(A \cap B \cap C\) is

The maximum value of the expression \(\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta}\) is

The equation of a plane passing through the line of intersection of the planes \(x+2 y+3 z=2\) and \(x-y+z=3\)

and at a distance \(\frac{2}{\sqrt{3}}\) from the point \((3,1,-1)\) is

Let $f\left(\text{x}\right)=\text{x }\text{sin }\pi \text{x}$, $\text{x}>0$. Then for all natural numbers $\text{n}\text{, }{f}^{\prime }\left(\text{x}\right)$ vanishes at

Let $p,q,r$ be non-zero real numbers that are, respectively, the ${10}^{\mathrm{th} },{100}^{\mathrm{th} }$ and ${1000}^{\mathrm{th} }$ terms of a harmonic progression. Consider the system of linear equations
$x+y+z=1$ $10x+100y+1000z=0$ $qrx+pry+pqz=0$

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

The correct option is:

For a real number $\alpha ,$ if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right] \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=$

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

Let \(a\) and \(b\) be non-zero and real numbers. Then, the equation \(\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0\) represents

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The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
Question:
\(\mathrm{XeF}_4\) and \(\mathrm{XeF}_6\) are expected to be

$2 \mathrm{mol}$ of $\mathrm{Hg}\left(\mathrm{g}\right)$ is combusted in a fixed volume bomb calorimeter with excess of ${\mathrm{O}}_2$ at $298 \mathrm{K}$ and $1 \mathrm{atm}$ into $\mathrm{HgO}\left(\mathrm{s}\right)$. During the reaction, temperature increases from $298.0 \mathrm{K}$ to $312.8 \mathrm{K}$. If heat capacity of the bomb calorimeter and enthalpy of formation of $\mathrm{Hg}\left(\mathrm{g}\right)$ are $20.00 \mathrm{kJ} {\mathrm{K}}^{-1}$ and $61.32 \mathrm{kJ} {\mathrm{mol}}^{-1}$ at $298 \mathrm{K}$, respectively, the calculated standard molar enthalpy of formation of $\mathrm{HgO}\left(\mathrm{s}\right)$ at $298 \mathrm{K}$ is $\mathrm{X} {\mathrm{kJmol}}^{-1}$. The value of $\left|\mathrm{X}\right|$ is___[Given: Gas constant $\mathrm{R}=8.3 \mathrm{J} {\mathrm{K}}^{-1} {\mathrm{mol}}^{-1}$]

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is