A stationary source emits sound of frequency $f_0=492 Hz$ . The sound is reflected by a large car approaching the source with a speed of $2 m{s}^{-1}$ . The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330 $m{s}^{-1}$ and the car reflects the sound at the frequency it has received).

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.


The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
The acceleration of this particle for \(|x|>X_0\) is

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Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is \(x(t) v s p(t)\) curve in this plane. The arrow on the
curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question :
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and \(E_1\) and \(E_2\) are the total mechanical energies respectively. Then,

Statement I The formula connecting \(u, v\) and \(f\) for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature.
Statement II Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces.

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
When disc \(B\) is brought in contact with disc \(A\), they acquire a common angular velocity in time \(t\). The average frictional torque on one disc by the other during this period is

Consider two solid spheres P and Q each of density 8 gm $c{m}^{-3}$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm $c{m}^{-3}$ and viscosity $\eta =3$ poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm $c{m}^{-\mathrm{3 }}$and viscosity $\eta =2$ poiseulles. the ratio of the terminal velocities of P and Q is.

A wide slab consisting of two media of refractive indices $n_1$ and $n_2$ is placed in air as shown in the figure. A ray of light is incident from medium $n_1$ to $n_2$ at an angle $\theta$, where $\sin \theta$ is slightly larger than $\frac{1}{n_1}$. Take refractive index of air as $1$. Which of the following statement(s) is (are) correct?

A spring - block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0N{m}^{-1}$ and the mass of the block is $2.0kg$ . Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0kg moving with a speed of $2.0m{s}^{-1}$ collides elastically with the first block. The collision is such that the 2.0kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.

Let \(f(x)=2+\cos x\) for all real \(x\).
Statement I For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(c)=0\).
Statement II \(f(t)=f(t+2 \pi)\) for each real \(t\).

For the electrochemical cell, $\text{Mg(s)}\left|{\text{Mg}}^{\text{2+}}\left(\text{aq,1}\text{M}\right)\right|\left|{\text{Cu}}^{\text{2+}}\left(\text{aq,1}\text{M}\right)\right|\text{Cu }\left(\text{s}\right)$ the standard emf of the cell is $\text{2}\text{.70 V}$ at $\text{300 K}\text{.}$ When the concentration of ${\text{Mg}}^{\text{2+}}$ is changed to $\text{x M,}$ the cell potential changes to $\text{2}\text{.67 V}$ at $\text{300 K}\text{.}$ The value of $\text{x}$ is ________.
(given, $\frac{\text{F}}{\text{R}}{\text{=11500 K V}}^{-\text{1}}\text{,}$ where $\text{F}$ is the Faraday constant and $\text{R}$ is the gas constants, $\text{ln}\left(\text{10}\right)\text{=}\text{2}\text{.30}$ )

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If a continuous \(f\) defined on the real line \(R\), assume positive and negative values in \(R\), then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive at some point and its minimum values is negative, then the equation \(f(x)=0\) has a root in \(R\).
Consider \(f(x)=k e^x-x\) for all real \(x\), where \(k\) is real constant.Question:
The line \(y=x\) meets \(y=k e^x\) for \(k \leq 0\) at

Let \(a, b, c, p, q\) be real numbers.
Suppose, \(\alpha, \beta\) are the roots of the equation \(x^2+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^2+2 b x+c=0\), where \(\beta^2 \notin\{-1,0,1\}\).
Statement \(1\left(p^2-q\right)\left(b^2-a c\right) \geq 0\).
Statement \(2 b \neq p a\) or \(c \neq q a\).

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and ${\mathrm{U}}_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

	Column I		Column II
A.	${\mathrm{U}}_1\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left[1-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]}^2$	P.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}\mathrm{a}$
B.	${\mathrm{U}}_2\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2$	Q.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}0$ .
C.	${\mathrm{U}}_3\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\exp \left[-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]$	R.	The force acting on the particle is zero at $\mathrm{x}=\mathrm{}-\mathrm{a}$
D.	${\mathrm{U}}_4\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}\left[\frac{\mathrm{x}}{\mathrm{a}}-\frac{1}{3}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^3\right]$	S.	The particle experiences an attractive force towards $\mathrm{x}\mathrm{}=\mathrm{}0$ in the region $\left|\mathrm{x}\right|<\mathrm{a}$
		T.	The particle with total energy $\frac{{\mathrm{U}}_0}{4}$ can oscillate about the point $\mathrm{x}=\mathrm{}-\mathrm{a}$