A bob of mass \(M\) is suspended by a massless string of length \(L\). The horizontal velocity \(v\) at position \(A\) is just sufficient to make it reach the point \(B\). The angle \(\theta\) at which the speed of the bob is half of that at \(A\), satisfies

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then

A circuit is connected as shown in the figure with the switch \(S\) open. When the switch is closed, the total amount of charge that flows from \(Y\) to \(X\) is

A ring of mass $\mathrm{M}$ and radius $\mathrm{R}$ is rotating with angular speed $\mathrm{\omega }$ about a fixed vertical axis passing through its centre O with two point masses each of mass $\frac{\mathrm{M}}{8}$ at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9}\mathrm{\omega }$ and one of the masses is at a distance of $\frac{3}{5}\mathrm{R}$ from O. At this instant the distance of the other mass from O is

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Consider a simple \(R C\) circuit as shown in Figure \(1\).

Process 1: In the circuit the switch \(S\) is closed at \(t=0\) and the capacitor is fully charged to voltage \(V_{0}\) (i.e., charging continues for time \(T>>R C\) ). In the process some dissipation \(\left(E_{D}\right)\) occurs across the resistance \(R\). The amount of energy finally stored in the fully charged capacitor is \(E_{C}\).

Process 2: In a different process the voltage is first set to \(\frac{V_{0}}{3}\) and maintained for a charging time \(T>>R C\). Then the voltage is raised to \(\frac{2 V_{0}}{3}\) without discharging the capacitor and again maintained for a time \(T>>R C\). The process is repeated one more time by raising the voltage to \(V_{0}\) and the capacitor is charged to the same final voltage \(V_{0}\) as in Process 1.

These two processes are depicted in Figure \(2 .\)







Question:

In Process 1 , the energy stored in the capacitor \(E_{C}\) and heat dissipated across resistance \(E_{D}\) are related by:

A metal rod of length \(L\) and mass \(m\) is pivoted at one end. A thin disc of mass \(M\) and radius \(R( < L)\) is attached at its centre to the free end of the rod. Consider two ways the disc is attached. Case A-the disc is not free to rotate about its centre and case \(B\)-the disc is free to rotate about its centre. The rod-disc system performs
SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is/are true?

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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

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(a_{\mathrm{i}}, b_{\mathrm{i}} \in \mathbb{R}\) for \(\mathrm{i} \in\{1,2,3\}\).
Define the functions \(f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}\), and \(h: \mathbb{R} \rightarrow \mathbb{R}\) by
\(\begin{aligned} & f(x)=a_1+10 x+a_2 x^2+a_3 x^3+x^4, \\ & g(x)=b_1+3 x+b_2 x^2+b_3 x^3+x^4, \\ & h(x)=f(x+1)-g(x+2) .\end{aligned}\)
If \(f(x) \neq g(x)\) for every \(x \in \mathbb{R}\), then the coefficient of \(x^3\) in \(h(x)\) is

List-$\mathrm{I}$ shows different radioactive decay processes and List-$\mathrm{II}$ provides possible emitted particles. Match each entry in List-$\mathrm{I}$ with an appropriate entry from List-$\mathrm{II}$, and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	${{}^{238}}_{92}U{{\to }^{234}}_{91}Pa$	$1$	one $\alpha$ particle and one ${\beta }^{+}$ particle
$\mathrm{Q}$	${{}^{214}}_{82}Pb{{\to }^{210}}_{82}Pb$	$2$	three ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{R}$	${{}^{210}}_{81}Tl{{\to }^{206}}_{82}Pb$	$3$	two ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{S}$	${{}^{228}}_{91}Pa{{\to }^{224}}_{88}Ra$	$4$	one $\alpha$ particle and one ${\beta }^{-}$ particle
		$5$	one $\alpha$ particle and two ${\beta }^{+}$ particles

The colour of light absorbed by an aqueous solution of \(\mathrm{CuSO}_{4}\) is:

An optical component and an object \(S\) placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2y}{x+y+z}$, then which of the following statements is/are TRUE?