Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

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Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

\(|\overrightarrow{O X} \times \overrightarrow{O Y}|=\)

The number of all possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\((y+z) \cos 3 \theta=(x y z) \sin 3 \theta \) \( x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z}\) and \((x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta\) \(+y \sin 3 \theta\) have a solution \(\left(x_0, y_0, z_0\right)\) with \(\quad y_0 z_0 \neq 0\), is

Let \(f(x)=x^2\) and \(g(x)=\sin x\) for all \(x \in R\). Then, the set of all \(x\) satisfying \((\) fogogof \()(x)=(\operatorname{gogof})(x)\), where \((f \circ g)(x)=f(g(x))\) is

Let $\beta$ be a real number. Consider the matrix $A=\left(\begin{array}{ccc}\beta & 0& 1\\ 2& 1& -2\\ 3& 1& -2\end{array}\right)$. If $A^7-\left(\beta -1\right)A^6-\beta A^5$ is a singular matrix, then the value of $9\beta$ is _____ .

Tangent is drawn at any point \(P\) of a curve which passes through \((1,1)\) cutting \(X\)-axis and \(Y\)-axis at \(A\) and \(B\), respectively. If \(A P: B P=3: 1\), then

The maximum value of the function \(f(x)=2 x^3-15 x^2+36 x-48\) on the set \(A=\left\{x \mid x^2+20 \leq 9 x\right\}\) is

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields \(\vec{E}=E_{0} \hat{j}\) and \(\vec{B}=B_{0} \hat{j}\). At time \(t=0\), this charge has velocity \(\vec{v}\) in the in the \(x\)-y plane, making an angle \(\theta\) with the \(x\)-axis. Which of the following option(s) is (are) correct for time \(t>0\) ?

An infinitely long thin wire, having a uniform charge density per unit length of \(5 \mathrm{nC} / \mathrm{m}\), is passing through a spherical shell of radius \(1 \mathrm{~m}\), as shown in the figure. A \(10 \mathrm{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(\mathrm{P}\) and \(\mathrm{R}\), in Volt, is _______ .
[Given: In SI units \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7\). Ignore the area pierced by the wire.]

Let $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(0\right)=0, f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }\left(0\right)=1.$ If $g\left(x\right)={\int }_x^{\frac{\mathrm{\pi }}{2}} \left[f^{\prime }\left(t\right)cosect-\cot t cosect f\left(t\right)\right]dt$ , for $x\in \left(0,\frac{\pi }{2}\right]$ then $\underset{x\to 0}{\lim }g\left(x\right)=$

For a prism of prism angle $\theta =60°$, the refractive indices of the left half and the right half are, respectively, $n_1$ and $n_2$, $\left(n_2\ge n_1\right)$ as shown in the figure. The angle of incidence $i$ is chosen such that the incident light rays will have minimum deviation if $n_1=n_2=n=1.5.$ For the case of unequal refractive indices, $n_1=n$ and $n_2=n+∆n$ (where $∆n<<n$), the angle of emergence $e=i+∆e$. Which of the following statement(s) is(are) correct?

Considering ideal gas behavior, the expansion work done (in kJ) when 144 g of water is electrolyzed completely under constant pressure at 300 K is _____
Use: Universal gas constant \((\mathrm{R})=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\); Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\) (mark absolute value as answer)

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


Question :
Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be