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Three identical capacitors, P, Q and S, each of the capacitance C, are connected to a battery of voltage $V,$ as shown in the figure. If the energy stored in the capacitor P and total energy stored in the system are $U_{P}$ and $U_{T}$ respectively, then the ratio $\frac{U_{P}}{U_{T}}$ is: [Image Placeholder]

Two identical inductors are connected in two different configurations P and Q, where a time varying current $l(t)$ is flowing, as shown in the figure. The induced emf between points a and b for configuration P is $E_{P}$ and that for configuration Q is $E_{Q}$ The ratio $E_{P}/E_{Q}$ is: [Neglect the effect of mutual inductance.] [Image Placeholder]

In the measurement of viscosity of liquids using terminal velocity experiment, spherical balls of same radius but having different densities are used. The variation of the terminal velocity (v) with the ratio of density of spherical ball ($\sigma$) to density of the liquid ($\rho$), is best represented by:

An ideal Zener diode with breakdown voltage of -3 V is reverse biased with a negative input voltage $V_{i}=-5$ V. The magnitude of voltage difference between point B and A is: [Image Placeholder]

A beam of light falls on a metal surface such that photo-electrons are generated. If power of the light source starts to decrease linearly with time t, then variation of the photocurrent I and magnitude of the stopping potential V with time is best represented by: [Image Placeholder]

Consider two circuits, (A) and (B), each having two resistors. One of them has a positive temperature coefficient of resistance, +$\alpha$, while the other one has a negative temperature of coefficient, -$\alpha$, as shown in the figure. The current through these circuits are denoted by $I_{A}$ and $I_{B}$. At initial temperature, the resistance of the two resistors is $R_0$. As the temperature is increased, the correct option that describes the variation of current in these circuits is:

Consider a long solenoid of length $l$ and radius $r$. If $n$ is the number of turns per unit length and $\mu_{0}$ is the permeability of free space, the inductance of the solenoid is:

A thin horizontal disc is rotating about a vertical axis passing through its fixed centre O. Its angular momentum is $L_{A}$ and $L_{B}$ computed about points A and B, respectively, with $OB=2\times OA.$ The value of $\frac{L_{A}}{L_{B}}$ is:

Water flows in a streamline motion through a horizontal pipe of circular cross-section as shown in the figure. The pressure difference of water between P and Q is $15~Nm^{-2}.$ The area of cross-section at P and Q are 40 $cm^{2}$ and $20~cm^{2}$, respectively. The rate of flow of water through the pipe, in $cm^{3}s^{-1}$, is: [Take density of water $=1000~kg~m^{-3}]$ [Image Placeholder]

A particle moves with velocity
\[
v(t) = 3t^2 + 2t \ \text{m/s}.
\]
What is its acceleration at \(t = 2\,\text{s}\)?

sove $\sqrt{x}+5+\sin x=\frac{dy}{dx}$

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Find number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\frac{\pi}{6}$ in $(\frac{-1}{2\sqrt{6}},\frac{1}{2\sqrt{6}})$ .

Let $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$ , then $|A^{2025}-3A^{2024}-A^{2023}|$ is equal to .

The number of real solutions of equation $x|x+4|+3|x+2|+10=0$ is equal to .