Two capacitors with capacitance values $C_1=\left(2000\pm 10\right) \mathrm{pF}$  and $C_2=\left(3000\pm 15\right) \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=\left(5.00\pm 0.02\right) \mathrm{V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is __________.

A ball of mass \(0.2 \mathrm{~kg}\) rests on a vertical post of height \(5 \mathrm{~m}\). A bullet of mass \(0.01 \mathrm{~kg}\), travelling with a velocity \(v \mathrm{~m} / \mathrm{s}\) in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of \(20 \mathrm{~m}\) and the bullet at a distance of \(100 \mathrm{~m}\) from the foot of the post. The initial velocity \(v\) of the bullet is

Four charges ${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ and ${\mathrm{Q}}_4$ of same magnitude are fixed along the x axis at $\mathrm{x}=-2\mathrm{a},\mathrm{}-\mathrm{a},\mathrm{}+\mathrm{a}$ and $+2\mathrm{a}$ , respectively. A positive charge q is placed on the positive y axis at a distance $\mathrm{b}\mathrm{}>\mathrm{}0$. Four options of the signs of these charges are given in List I. The direction of the forces on the charge q is given in List II. Match List I with List II and select the correct answer using the code given below the lists.

	List I		List II
A.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3\mathrm{}{\mathrm{Q}}_4$ all positive	$\mathrm{P}.$	$+\mathrm{x}$
B.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2$ positive; ${\mathrm{Q}}_3,\mathrm{}{\mathrm{Q}}_4$ negative	Q.	$-\mathrm{x}$
C.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_4$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ negative	$\mathrm{R}.$	$+\mathrm{y}$
D.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_3$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_4$ negative	$\mathrm{S}.$	$-\mathrm{y}$

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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 :
For periodic motion of small amplitude \(A\), the time period \(T\) of this particle is proportional to

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

Find the time constant for the given R C circuits in correct order.

\(R_1=1 \Omega, R_2=2 \Omega, C_1=4 \mu \mathrm{F}, C_2=2 \mu \mathrm{F}\)

A heavy nucleus $Q$ of half-life $20$ minutes undergoes alpha-decay with probability of $60\%$  and beta-decay with probability of $40\%$. Initially, the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

In a study about a pandemic, data of $900$ persons was collected. It was found that $190$ persons had symptom of fever, $220$ persons had symptom of cough, $220$ persons had symptom of breathing problem, $330$ persons had symptom of fever or cough or both, $350$ persons had symptom of cough or breathing problem or both, $340$ persons had symptom of fever or breathing problem or both, $30$ persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

Let ${\ell }_1$ and ${\ell }_2$ be the lines ${\overrightarrow{r}}_1=\lambda \left(\hat{i}+\hat{j}+\hat{k}\right)$ and ${\overrightarrow{r}}_2=\left(\hat{j}-\hat{k}\right)+\mu \left(\hat{i}+\hat{k}\right)$, respectively, Let $X$ be the set of all the planes $H$ that contain the line ${\ell }_1$. For a plane $H$, let $d\left(H\right)$ denote the smallest possible distance between the points of ${\ell }_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d\left(H_0\right)$ is the maximum value of $d\left(H\right)$ as $H$ varies over all planes in $X$. Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The value of $d\left(H_0\right)$ is	$\left(1\right)$	$\sqrt{3}$
$\left(\mathrm{Q}\right)$	The distance of the point $\left(0, 1, 2\right)$ from $H_0$ is	$\left(2\right)$	$\frac{1}{\sqrt{3}}$
$\left(\mathrm{R}\right)$	The distance of origin from $H_0$ is	$\left(3\right)$	$0$
$\left(\mathrm{S}\right)$	The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_0$ is	$\left(4\right)$	$\sqrt{2}$
		$\left(5\right)$	$\frac{1}{\sqrt{2}}$

The correct option is

The figure below is the plot of potential energy versus internuclear distance $\left(\mathrm{d}\right)$ of ${\mathrm{H}}_2$ molecule in the electronic ground state. What is the value of the net potential energy ${\mathrm{E}}_0$ (as indicated in the figure) in $\mathrm{kJ}{\mathrm{mol}}^{-1},$ for $\mathrm{d}={\mathrm{d}}_0$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of $\mathrm{H}$ atom is taken as zero when its electron and the nucleus are infinitely far apart.
Use Avogadro constant as $6.023\times {10}^{23}{\mathrm{mol}}^{-1}.$

Mark total potential energy for 2 H-atom (per mole) as answer.

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Consider the statements:

\(P\) : There exists some \(x \in \mathrm{R}\) such that \(f(x)+2 x\)

\(=2\left(1+x^{2}\right)\)

\(Q\) : There exists some \(x \in \mathrm{R}\) such that \(2 f(x)+1\) \(=2 x(1+x)\)

Then