$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in $\mathrm{Hz}$ is _____.

The work done on a particle of mass m by a force $\text{K}\left[\frac{\text{x}}{{\left({\text{x}}^2+{\text{y}}^2\right)}^{3/2}}\widehat{\text{i}}+\frac{\text{y}}{{\left({\text{x}}^2+{\text{y}}^2\right)}^{3/2}}\widehat{\text{j}}\right]$ (K being a constant of appropriate dimensions, when the particle is taken from the point (a,0) to the point (0,a) along a circular path of radius a about the origin in the x-y plane is

A diatomic ideal gas is compressed adiabatically to \(\frac{1}{32}\) of its initial volume. If the initial temperature of the gas is \(T_i\) \(T_i\) (in kelvin) and the final temperature is \(a T_i\), the value of \(a\) is

A monochromatic light wave is incident normally on a glass slab of thickness $d$, as shown in the figure. The refractive index of the slab increases linearly from $n_1$ to $n_2$ over the height $h$. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

A small block is connected to one end of a massless spring of un-stretched length \(4.9 \mathrm{~m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2 \mathrm{~m}\) and released from rest at \(t=0\). It then executes simple harmonic motion with angular frequency \(\omega=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously at \(t=0\), a small pebble is projected with speed \(v\) form point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. Point \(P\) is at a horizontal distance of \(10 \mathrm{~m}\) from \(O\). If the pebble hits the block at \(t=1 \mathrm{~s}\), the value of \(v\) is (take \(g\) \(=10 \mathrm{~m} / \mathrm{s}^{2}\) )

While conducting the Young's double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources $\left(S_1,S_2\right)$ emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3m from the mid-point of $\left(S_1,S_2\right)$ , as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining $\left(S_1,S_2\right)$ . Which of the following is (are) true of the intensity pattern on the screen?

A particle of mass \(m\) is moving in a circular orbit under the influence of the central force \(F(r)=-k r\), corresponding to the potential energy \(V(r)=k r^2 / 2\), where \(k\) is a positive force constant and \(r\) is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by \(L=n \hbar\), where \(\hbar=h /(2 \pi), h\) is the Planck's constant, and \(n\) a positive integer. If \(v\) and \(E\) are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

A large square container with thin transparent vertical walls and filled with water $\left(\text{refractive index}\frac{4}{3}\right)$ is kept on a horizontal table. A student holds a thin straight wire vertically inside the water $12\mathrm{cm}$ from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in $\mathrm{cm}$) between these images is ________.

If \(z\) is any complex number satisfying \(|z-3-2 i| \leq 2\), then the maximum value of \(|2 z-6+5 i|\) is

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y=x+8$ and its point of contact is $(8,16)$, then which of the following options is the only Correct combination?

For a first order reaction, \(A \rightarrow P\), the temperature \((T)\) dependent rate constant \((k)\) was found to follow the equation, \(\log k=-(2000) / T+6.0\)
The pre-exponential factor \(A\) and the activation energy \(\left(E_a\right)\), respectively, are

The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If ${\mathrm{\lambda }}_{{\mathrm{X}}^{-}}^0\approx {\mathrm{\lambda }}_{{\mathrm{Y}}^{-}}^0$ , the difference in their $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}$ values, $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{X}\right)-\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{Y}\right)$ , is (Consider degree of ionization of both acids to << 1)

An electron in a hydrogen atom undergoes a transition from an orbit with a quantum number $n_i$ to another quantum number $n_f$ . $V_i \text{and} V_f$ are the initial and final potential energies of the electron. If $\frac{V_i}{V_f}=6.25$ then the smallest $n_f$ is