Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

Let $\alpha$ and $\beta$ be non zero real numbers such that $2\left(\cos \beta -\cos \alpha \right)+\cos \alpha \cos \beta =1$. Then which of the following is/are true?

Match the following.
 

	List – I		List – II
(A)	The number of polynomials f(x) with non – negative integer coefficients of degree $\le 2,$ satisfying f(0) =0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1, is$	(P)	8
(B)	The number of point in the interval $\left[- \sqrt{13}, \sqrt{13}\right]$ at which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is	(Q)	2
(C)	$\underset{-2}{\overset{2}{\int }}\frac{3x^2}{\left(1+e^x\right)}dx$ equals	(R)	4
(D)	$\frac{\left(\underset{-\frac{1}{2}}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}$	(S)	0

If the distance of the point \(P(1,-2,1)\) from the plane \(x+2 y-2 z=\alpha\), where \(\alpha>0\), is 5 , then the foot of the perpendicular from \(P\) to the plane is

For a real number $\alpha ,$ if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right] \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=$

For any positive integer $n$, let $S_n:(0,\infty )\to R$ be defined by
$S_n\left(x\right)=\sum _{k=1}^n{\cot }^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$
where for any $x\in R,{\cot }^{-1}\left(x\right)\in (0,\pi )$ and ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then which of the following statements is (are) TRUE ?

The total number of real solutions of the equation \(\theta=\tan ^{-1}(2 \tan \theta)-\frac{1}{2} \sin ^{-1}\left(\frac{6 \tan \theta}{9+\tan ^2 \theta}\right)\) is (Here, the inverse trigonometric functions \(\sin ^{-1} x\) and \(\tan ^{-1} x\) assume values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), respectively.)

\(\mathrm{NiCl}_{2}\left\{\mathrm{P}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)\right\}_{2}\) exhibits temperature depend-ent magnetic behaviour (paramagnetic/diamagnetic). The coordination geometries of \(\mathrm{Ni}^{2+}\) in the paramagnetic and diamagnetic states are respectively

A physical quantity $\overrightarrow{S}$ is defined as $\overrightarrow{S}=\frac{\left(\overrightarrow{E}\times \overrightarrow{B}\right)}{{\mu }_0}$. where $\overrightarrow{E}$ is electric field, $\overrightarrow{B}$ is magnetic field and ${\mu }_0$ is the permeability of free space. The dimensions of $\overrightarrow{S}$ are the same as the dimensions of which of the following quantity/quantities?

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\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is \(\rho\). In equilibrium, the height \(H\) of the water column in the cylinder satisfies

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

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Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

The value of \(r\) is

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The hydrogen-like species \(\mathrm{Li}^{2+}\) is in a spherically symmetric state \(S_1\) with one radial node. Upon absorbing light the ion undergoes transition to a state \(S_2\). The state \(S_2\) has one radial node and its energy is equal to the ground state energy of the hydrogen atom.
Question:
Energy of the state \(S_1\) in units of the hydrogen atom ground state energy is