The work function ( \(\Phi\) ) of some metals is listed below. The number of metals which will show photoelectric effect when light of \(300 \mathrm{~nm}\) wavelength falls on the metal is

For ${\mathrm{He}}^{+}$, a transition takes place from the orbit of radius $105.8\mathrm{pm}$ to the orbit of radius $26.45\mathrm{pm}$. The wavelength (in $\mathrm{nm}$ ) of the emitted photon during the transition is
Bohr radius, $\mathrm{a}=52.9 \mathrm{pm}$
Rydberg constant, ${\mathrm{R}}_{\mathrm{H}}=2.2\times {10}^{-18} \mathrm{J}$
Planck's constant, $\mathrm{h}=6.6\times {10}^{-34}\mathrm{Js}$
Speed of light, $\mathrm{c}=3\times {10}^8 {\mathrm{ms}}^{-1}$ ]

The smallest ketone and its next homologoue are reacted with \(\mathrm{NH}_2 \mathrm{OH}\) to form oxime

Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. \(\text O _2^{-} \longrightarrow \text{O} _2\) \(+~\text O _2^{2-}\)	P. Redox reaction
B. \(\text{CrO} _4^{2-}+\text H ^{+} \longrightarrow\)	Q. One of the products has trigonal planar structure
C. \(\text{MnO} _4^{-}+ \text{NO} _2^{-}\) \(+~ \text H ^{+} \longrightarrow\)	R. Dimeric bridged tetrahedral metal ion
D. \(\text{NO} _3^{-}+ \text H _2 \text{SO} _4\) \(+~ \text{Fe} ^{2+} \longrightarrow\)	S. Disproportionation

For the reaction:
${\mathrm{I}}^{-}+\mathrm{C}\mathrm{l}{\mathrm{O}}_3^{-}+{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\to \mathrm{C}{\mathrm{l}}^{-}+\mathrm{H}\mathrm{S}{\mathrm{O}}_4^{-}+{\mathrm{I}}_2\mathrm{ }$
The correct statement(s) in the balanced equation is/are:

A positron is emitted from \({ }_{11}^{23} \mathrm{Na}\). The ratio of the atomic mass and atomic number of the resulting nuclide is

A solution of $0.1\mathrm{M}$ weak base $\left(\mathrm{B}\right)$ is titrated with $0.1\mathrm{M}$ of a strong acid $\left(\mathrm{HA}\right)$. The variation of $\mathrm{pH}$ of the solution with the volume of $\mathrm{HA}$ added is shown in the figure below. What is the ${\mathrm{pK}}_{\mathrm{b}}$ of the base? The neutralization reaction is given by$\mathrm{B}+\mathrm{HA}\to {\mathrm{BH}}^{+}+{\mathrm{A}}^{-}$.

A curve passes through the point $\left(1,\frac{\pi }{6}\right)$. Let the slope of the curve at each point $\left(\text{x},\text{y}\right)$ be $\frac{\text{y}}{\text{x}}+\text{sec}\left(\frac{\text{y}}{\text{x}}\right),\; x > 0$. Then the equation of the curve is

Let \(f:(0,1) \rightarrow R\) be defined by \(f(x)=\frac{b-x}{1-b x}\), where \(b\) is a constant such that \(0 < b < 1\). Then,

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 _____ .

To determine the half-life of a radioactive element, a student plots a graph of \(\ln \left|\frac{d N(t)}{d t}\right|\) versust. Here \(\frac{d N(t)}{d t}\) is the rate of radioactive decay at time \(t\). If the number of radioactive nuclei of this element decreases by a factor of \(p\) after \(4.16 \mathrm{yr}\), the value of \(p\) is

Paragraph:

The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:

The correct statement is

If \(\alpha=\int_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x\)
then the value of \(\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)\) is ____________.
(Here, the inverse trigonometric function \(\tan ^{-1} x\) assumes values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).)