Function \(x=A \sin ^2 \omega t+B \cos ^2 \omega t+C\) \(\sin \omega t \cos \omega t\) represents SHM.

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$ . Which of the following statements is (are) true?

Consider a hydrogen atom with its electron in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$

Young’s modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck’s constant $h$ and the speed of light $c$, as $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option?

The figure shows the \(p-V\) plot of an ideal gas taken through a cycle \(A B C D A\). The part \(A B C\) is a semi-circle and \(C D A\) is half of an ellipse. Then,

A container has a base of $50\mathrm{cm}\times 5\mathrm{cm}$ and height $50\mathrm{cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50\mathrm{cm}\times 50\mathrm{cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant $3$at a uniform rate of $250{\mathrm{cm}}^3{\mathrm{s}}^{-1}$. What is the value of the capacitance of the container after $10$seconds?
[Given: Permittivity of free space ${\epsilon }_0=9\times {10}^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]

In a photoelectric experiment, a parallel beam of monochromatic light with the power of $200\text{W}$ is incident on a perfectly absorbing cathode of work function $6.25 \mathrm{eV}$. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of $500\text{V}$ is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F=n\times {10}^{-4}\mathrm{ N}$ due to the impact of the electrons. The value of $n$ is __________. Mass of the electron \(\text{m}_\text e =9 \times 10^{-31} kg\) and \(1.0 eV =1.6 \times\) \(10^{-19}\text{ J}.\)

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $xy$-plane with the center of the loop always on the axis $\hat{\mathrm{n}}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counter clockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{\mathrm{n}}$ direction. List-I gives time dependences of $\hat{\mathrm{n}}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi }{6\omega }$, Let $\alpha =\frac{A^2{\mu }_0^2{m}^2{I}^2\omega }{2R}$.

	List-I		List-II
(i)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{j}}+\cos \omega t\hat{\mathrm{k}}\right)$	(p)	$0$
(ii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{j}}\right)$	(q)	$-\frac{\alpha }{4}\hat{\mathrm{i}}$
(iii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{k}}\right)$	(r)	$\frac{3\alpha }{4}\hat{\mathrm{i}}$
(iv)	$\frac{1}{\sqrt{2}}\left(\cos \omega t\hat{\mathrm{j}}+\sin \omega t\hat{\mathrm{k}}\right)$	(s)	$\frac{\alpha }{4}\hat{\mathrm{j}}$
		(t)	$-\frac{3\alpha }{4}\hat{\mathrm{i}}$

Which one of the following options is correct?

Match the following.

The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in list II. match List I with List II and select the correct answer using the code given below the lists :

List - I	List - II
A. \(PbO _2+ H _2 SO _4 \xrightarrow{?}\) \(PbSO _4 + O _2+\) other product	P. \(\text{NO}\)
B. \(Na _2 S_2 O _3+ H _2 O \xrightarrow{?}\) \(NaHSO _4~+\) other product	Q. \(\text{I}_2\)
C. \(N _2 H _4 \xrightarrow{?} N_2 +\) other product	R. Warm
D. \(XeF _2 \xrightarrow{?} Xe ~+\) other product	S. \(\text{Cl}_2\)

A light beam is travelling from Region I to Region IV (refer figure). The refractive index in Region I, II, III and IV are \(n_0, \frac{n_0}{2}, \frac{n_0}{6}\) and \(\frac{n_0}{8}\), respectively. The angle of incidence \(\theta\) for which the beam just misses entering Region IV is

Consider \(L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0\) where \(p\) is a real number and \(C: x^2+y^2+6 x-10 y+30=0\).
Statement 1 If line \(L_1\) is a chord of circle \(C\), then line \(L_2\) is not always a diameter of circle \(C\).
Statement 2 If line \(L_1\) is a diameter of circle \(C\), then line \(L_2\) is not a chord of circle \(C\).

Paragraph:
Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)\) \(+~2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ; E_{\mathrm{red}}^{\circ}\) \(=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\)]
Question:
\(2 \mathrm{Ag}^{+}+\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Ag}(\mathrm{s})\) \(+~\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_4+2 \mathrm{H}^{+}\). Find \(\ln \mathrm{K}\) of this reaction.