Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

Let $X=\left\{\left(x, y\right)\in \mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}} : \frac{x^2}{8}+\frac{y^2}{20}<1 \mathrm{and} y^2<5\mathrm{x}\right\}$. Three distinct point $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is

Let \(f(x)=\frac{x}{\left(1+x^n\right)^{1 / n}}\) for \(n \geq 2\) and \(g(x)=\underbrace{(f \circ f o \ldots o f)}_{f \text { occurs } n \text { times }}(x)\). Then, \(\int x^{n-2} g(x) d x\) equals

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Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which one of the following statements is correct?

The following integral $\underset{\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}{\left(2\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{ }\mathrm{x}\right)}^{17}\mathrm{ }\mathrm{d}\mathrm{x}$ is equal to

The equation of the plane passing through the point $(\mathrm{1,1},1)$ and perpendicular to the planes $2x+y-2z=5$ and $3x-6y-2z=7$ is

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A)The number of permutations containing the word ENDEA, is	(p) \(5 !\)
(B) The number of permutations in which the letter E occurs in the first and the last positions, is	(q) \(2 \times 5!\)
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions, is	(r) \(7 \times 5!\)
(D) The number of permutations in which the letters A, E, O occur only in odd positions, is	(s) \(21 \times 5!\)

The line passing through the extremity \(A\) of the major axis and extremity \(B\) of the minor axis of the ellipse \(x^2+9 y^2=9\) meets its auxiliary circle at the point \(M\). Then, the area of the triangle with vertices at \(A, M\) and the origin \(O\) is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	List I		List II
$\left(\mathrm{a}\right)$	Work done by the system in process $1\to 2\to 3$	$\left(\mathrm{p}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
$\left(\mathrm{b}\right)$	Change in internal energy in process $1\to 2\to 3$	$\left(\mathrm{q}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{c}\right)$	Heat absorbed by the system in process $1\to 2\to 3$	$\left(\mathrm{r}\right)$	$\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{d}\right)$	Heat absorbed by the system in process $1\to 2$	$\left(\mathrm{s}\right)$	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		$\left(\mathrm{t}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		$\left(\mathrm{u}\right)$	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the $\mathrm{T}$ - $\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is,

The given graphs/data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and presure. Which of the following

A block of weight $100\mathrm{}\mathrm{N}$ is suspended by copper and steel wires of same cross-sectional area $0.5\mathrm{}\mathrm{c}{\mathrm{m}}^2$ and, length $\sqrt{3}\mathrm{}\mathrm{m}$ and $1\mathrm{}\mathrm{m},$ respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. If elongation in copper wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}\right)$ and elongation in steel wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}\right),$ then the ratio $\frac{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}}{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}}$ is _____.
[Young's modulus for copper and steel are $1\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ and $2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ respectively]

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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 \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is