The sides of a right-angled triangle are in arithmetic progression. If the triangle has area 24 square units, then what is the length of its smallest side?

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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

The total number of distinct $x\in \left[0, 1\right]$ for which, ${\int }_0^x\frac{t^2}{1+t^4}dt=2x-1$ is

Let \(S_1=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}\)
\(S_2=\{(i, j): 1 \leq i < j+2 \leq 10, i, j\) \(\in\{1,2, \ldots, 10\}\} \)
\(S_3=\{(i, j, k, l): 1 \leq i < j < k < l, i,\) \(j, k, l \in\{1,2, \ldots . ., 10\}\}\)
and \(S_4=\{(i, j, k, l): i, j, k\) and \(l\) are distint elements in \(\{1,2, \ldots, 10\}\}\) If the total number of elements in the set \(S_r\) is \(n_r, r=1,2,3,4\) then which of the following statements is (are) TRUE?

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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:
Let \(z\) be any point in \(A \cap B \cap C\). The \(|z+1-i|^2+|z-5-i|^2\) lies between

If the adjoint of a \(3 \times 3\) matrix \(P\) is \(\left[\begin{array}{lll}1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3\end{array}\right]\), then the possible value(s) of the determinant of \(P\) is (are)

A circle S passes through the point (0, 1) and is orthogonal to the circles ${\left(x-1\right)}^2+y^2=16$ and $x^2+y^2=1$. Then

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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?

Two players, $P_1$ and $P_2$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_1$ and $P_2$, respectively. If $x>y$, then $P_1$ scores $5$ points and $P_2$ scores $0$ point. If $x=y$, then each player scores $2$ points. If $x<y$, then $P_1$ scores $0$ point and $P_2$ scores $5$ points. Let $X_i$ and $Y_i$ be the total scores of $P_1$ and $P_2$, respectively, after playing the $i^{\mathrm{th}}$ round.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Probability of $\left(X_2\ge Y_2\right)$ is	$\left(\mathrm{P}\right)$	$\frac{3}{8}$
$\left(\mathrm{II}\right)$	Probability of $\left(X_2>Y_2\right)$ is	$\left(\mathrm{Q}\right)$	$\frac{11}{16}$
$\left(\mathrm{III}\right)$	Probability of $\left(X_3=Y_3\right)$ is	$\left(\mathrm{R}\right)$	$\frac{5}{16}$
$\left(\mathrm{IV}\right)$	Probability of $\left(X_3>Y_3\right)$ is	$\left(\mathrm{S}\right)$	$\frac{355}{864}$
		$\left(\mathrm{T}\right)$	$\frac{77}{432}$

The correct option is:

The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic \(\mathrm{KOH}\) is

Let $\mathrm{A}\mathrm{P}\mathrm{ }\left(\mathrm{a};\mathrm{d}\right)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0.$ If $\mathrm{A}\mathrm{P}\left(1;3\right)\cap \mathrm{A}\mathrm{P}\left(2;5\right)\cap \mathrm{A}\mathrm{P}\left(3;7\right)=\mathrm{A}\mathrm{P}\left(\mathrm{a};\mathrm{d}\right)$ then $\mathrm{a}+\mathrm{d}$ equals ____

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

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