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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

In a study about a pandemic, data of $900$ persons was collected. It was found that $190$ persons had symptom of fever, $220$ persons had symptom of cough, $220$ persons had symptom of breathing problem, $330$ persons had symptom of fever or cough or both, $350$ persons had symptom of cough or breathing problem or both, $340$ persons had symptom of fever or breathing problem or both, $30$ persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

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Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g^{\prime}\left(\frac{1}{2}\right)\) is

If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right) and \beta =3{\cos }^{-1}\left(\frac{4}{9}\right),$ where the inverse trigonometric functions take only the principal values, then the correct options(s) is(are)

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 _____ .

Consider the set of eight vectors $\text{V}=\left\{\text{a}\widehat{i}+\text{b}\widehat{\text{j}}+\text{c}\widehat{\text{k}}:\; a,\; b,\; c \in \left\{-1,\; 1\right\}\right\}$. Three non-coplanar vectors can be chosen from $\text{V}$ in $2^{\text{p}}$ ways. Then $\text{p}$ is

Let \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\). A vector coplanar to \(\mathbf{a}\) and \(\mathbf{b}\) has a projection along c of magnitude \(\frac{1}{\sqrt{3}}\), then the vector is

Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is (are)

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

In a radioactive decay process, the activity is defined as $A=-\frac{dN}{dt}$, where $N\left(t\right)$ is the number of radioactive nuclei at time $\left(t\right)$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\mathrm{rd}}$ and $7^{\mathrm{th}}$ half-lives, respectively, the ratio $\frac{A_1}{A_2}$ is _____.

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Football teams \(T_{1}\) and \(T_{2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_{1}\) winning, drawing and losing a game against \(T_{2}\) are \(\frac{1}{2}, \frac{1}{6}\) and \(\frac{1}{3}\), respectively. Each team gets \(3\) points for a win, \(1\) point for a draw and \(0\) point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_{1}\) and \(T_{2}\), respectively, after two games.


Question:

\(P(X=Y)\) is

The treatment of galena with ${\mathrm{HNO}}_3$ produces a gas that is

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass \(2 m\) and \(m\) just after the string is cut will be