A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P-V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic process. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas in n.

Column – 1	Column – 2	Column – 3
(I) \(W_{1 \rightarrow 2}=\) \(\frac{1}{\gamma-1}\left(P_2 V_2-P_1 V_1\right)\)	(i) Isothermal	(P)
(II) \(W_{1 \rightarrow 2}=\) \(-P V_2+P V_1\)	(ii) Isochoric	(Q)
(III) \(W_{1 \rightarrow 2}=0\)	(iii) Isobaric	(R)
(IV) \(W_{1 \rightarrow 2}=\)\(-n R T \ln \left(\frac{V_2}{V_1}\right)\)	(iv) Adiabatic	(S)

Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in ideal gas?

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross-sections of different radii. When dipped in water vertically, the water rises in it to a height $h$, where the radius of its cross-section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$, then the value of $h$ will be ($g$ is the acceleration due to gravity)

In a Young's double slit experiment, the separation between the two slits is \(d\) and the wavelength of the light is \(\lambda\). The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2 . Choose the correct choice(s),

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4kg$ is at rest on this surface. An impulse of $1.0 N s$ is applied to the block at time $t=0$ so that it starts moving along the x-axis with a velocity $v\left(t\right)=v_0{e}^{-\frac{t}{\tau }}$ , where $v_0$ is a constant and $\tau =4$ . The displacement of the block, in meter, at $t=\tau$ is __________. Take $e^{-1}=0.37$

In the Young's double slit experiment using a monochromatic light of wavelength $\lambda$, the path difference ( in terms of an integer $n$ ) corresponding to any point having half the peak intensity is

A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v I much less than the average speed u of the gas molecules. Which of the following options is true?

A uniform circular disc of mass $\text{50 kg}$ and radius $\text{0.4 m}$ is rotating with an angular velocity of ${\text{10 rad s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $\text{6.25 kg}$ and radius $\text{0.2 m}$, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity $\left({in\; rad\; s}^{-1}\right)$ of the system is

For the function \(f(x)=x \cos \frac{1}{x}, x \geq 1\),

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identity matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE ?

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The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
Initially, the capacitor was uncharged. Now, switch \(S_1\) is closed and \(S_2\) is kept open. If time constant of this circuit is \(\tau\), then

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Let \(F_{1}\left(x_{1}, 0\right)\) and \(F_{2}\left(x_{2}, 0\right)\), for \(x_{1}<0\) and \(x_{2}>0\), be the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .\) Suppose a parabola having vertex at the origin and focus at \(F_{2}\) intersects the ellipse at point \(M\) in the first quadrant and at point \(N\) in the fourth quadrant.


Question:

If the tangents to the ellipse at \(M\) and \(N\) meet at \(R\) and the normal to the parabola at \(M\) meets the \(x\)-axis at \(Q\), then the ratio of area of the triangle \(M Q R\) to area of the quadrilateral \(M F_{1} N F_{2}\) is

The correct statement(s) about surface properties is(are)