Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha\)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is

If \(\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}\), then

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Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).

Question:
The probability that \(x_{1}, x_{2}, x_{3}\) are in an arithmetic progression, is

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Let \(S=S_{1} \cap S_{2} \cap S_{3}\), where

\(S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}\) and \(S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}\).


Question:

\(\min _{z \in S}|1-3 i-\mathrm{z}|=\)

Consider the following lists:

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

The correct option is:

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Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{1025}{513}\). Let \(k\) be the number of all those circles \(C_{n}\) that are inside \(M\). Let \(l\) be the maximum possible number of circles among these \(k\) circles such that no two circles intersect. Then

Let \(S\) be the set of all \((\alpha, \beta) \in \mathbb{R} \times \mathbb{R}\) such that
\(\lim _{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0\).
Then which of the following is (are) correct?

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function such that its derivative $f^{\text{'}}$ is continuous and $f\left(\pi \right)=-6$. If $F:\left[0,\pi \right]\to \mathrm{ℝ}$ is defined by $F\left(x\right)={\int }_0^{x}f\left(t\right)dt,$ and if ${\int }_0^{\pi }\left(f^{\text{'}}\left(x\right)+F\left(x\right)\right)\cos x dx=2$, then the value of $f\left(0\right)$ is_______

Match the gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. Hydrogen gas \((P=200 atm,\) \(T=273 K)\)	p. Compressibility factor \(\neq 1\)
B. Hydrogen gas \((P \sim 0,\) \(T=273 K)\)	q. Attractive forces are dominant
C. \(CO _2(P=1 atm,\) \(T=273 K)\)	r. \(P V=n R T\)
D. Real gas with very large molar volume	s. \(P(V-n b)=n R T\)

Plotting $1/{\mathrm{\Lambda }}_{\mathrm{m}}$ against ${\mathrm{c\Lambda }}_{\mathrm{m}}$ for aqueous solutions of a monobasic weak acid $\left(\mathrm{HX}\right)$ resulted in a straight line with $\mathrm{y}-$axis intercept of $\mathrm{P}$ and slope of $\mathrm{S}$. The ratio $\mathrm{P}/\mathrm{S}$ is
$\left[{\mathrm{\Lambda }}_{\mathrm{m}}=\right.\text{ molar conductivity }$ $$\begin{gathered} {\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{o}}= \mathrm{limiting} \mathrm{molar}conductivity \\ \mathrm{c}= \mathrm{molar}concentration \end{gathered}$$ $\left.{\mathrm{K}}_{\mathrm{a}}=\text{ dissociation constant of }\mathrm{HX}\right]$

A point mass is subjected to two simultaneous sinusoidal displacements in \(x\)-direction, \(x_1(t)=A \sin \omega t\) and \(x_2(t)=A \sin \left(\omega t+\frac{2 \pi}{3}\right)\). Adding a third sinusoidal displacement \(x_3(t)=B \sin (\omega t+\phi)\) bring the mass to a complete rest. The values of \(B\) and \(\phi\) are

The bond energy (in \(\mathrm{kcal} \mathrm{mol}^{-1}\) ) of \(\mathrm{C}-\mathrm{C}\) single bond is approximately

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The equation of the locus of the point whose distance from the point \(P\) and the line \(A B\) are equal, is