A ball is thrown from the location \(\left(x_0, y_0\right)=(0,0)\) of a horizontal playground with an initial speed \(v_0\) at an angle \(\theta_0\) from the \(+x\)-direction. The ball is to be hit by a stone, which is thrown at the same time from the location \(\left(x_1, y_1\right)=(L, 0)\). The stone is thrown at an angle \(\left(180-\theta_1\right)\) from the \(+x\)-direction with a suitable initial speed. For a fixed \(v_0\), when \(\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)\), the stone hits the ball after time \(T_1\), and when \(\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)\), it hits the ball after time \(T_2\). In such a case, \(\left(T_1 / T_2\right)^2\) is _______ .

Statement I In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.
Statement II In an elastic collision, the linear momentum of the system is conserved.

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the \(4 \times 4\) matrix given in the ORS.

$S_1$ 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 _____.

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \(\omega\), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \(z\)-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \(P\) and \(Q\) ). Both these motions have the same angular speed \(\omega\) in this case

Now consider two similar systems as shown in the figure: Case
(a) the disc with its face vertical and parallel to \(x-z\) plane; Case
(b) the disc with its face making an angle of
\(45^{\circ}\) with \(x-y\) plane and its horizontal diameter parallel to \(x\)-axis. In both the cascs, the disc is welded at point \(\mathrm{P}\), and the systems are rotated with constant angular speed \(\omega\) about the \(z\) axis.

Question : Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

One mole of a monatomic ideal gas is taken through a cycle \(A B C D A\) as shown in the \(p\)-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.

The number of points in $\left(-\infty ,\infty \right)$, for which $x^2-x\sin x-\cos x=0$, is:

Let $\alpha =\sum _{k=1}^{\infty }{\sin }^{2k}\left(\frac{\pi }{6}\right)$. Let $g:\left[0,1\right]\to \mathrm{ℝ}$ be the function defined by $g\left(x\right)=2^{ax}+2^{a\left(1-x\right)}$. Then, which of the following statements is/are TRUE?

Area of the region bounded by the curve \(y=e^x\) and lines \(x=0\) and \(y=e\) is

Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=\pi /(2 \omega)\), then the value of \(4 b^2 / a^2\) will be ________ .

In a Young's double slit experiment, the slit separation $\mathrm{d}$ is $0.3\mathrm{}\mathrm{m}\mathrm{m}$ and the screen distance $\mathrm{D}$ is $1\mathrm{m}.$ A parallel beam of light of wavelength $600\mathrm{}\mathrm{n}\mathrm{m}$ is incident on the slits at angle $\alpha$ as shown in figure. On the screen, the point $\mathrm{O}$ is equidistant from the slits and distance $\mathrm{P}\mathrm{O}$ is $11.0\mathrm{}\mathrm{m}\mathrm{m}.$ Which of the following statement(s) is/are correct?

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?