Students I, II and III perform an experiment for measuring the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table.
Least count for length \(=0.1 \mathrm{~cm}\)
Least count for time \(=0.1 \mathrm{~s}\)

If \(E_{\mathrm{I}}, E_{\mathrm{II}}\) and \(E_{\mathrm{III}}\) are the percentage errors in \(g\), i.e., \(\left(\frac{\Delta g}{g} \times 100\right)\) for students I, II and III, respectively.

A source of constant voltage $\mathrm{V}$ is connected to a resistance$\mathrm{R}$ and two ideal inductors ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ through a switch $\mathrm{S}$ as shown. There is no mutual inductance between the two inductors. The switch $\mathrm{S}$ is initially open. At $\mathrm{t}=0$ , the switch is closed and current begins to flow. Which of the following options is/are correct?

Using the expression $2d \sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angle $\text{θ}$ in the range $0$ to $90°$. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all the values of $\theta$. As $\text{θ}$ increases from $0°$,

An ideal gas is expanding such that \(p T^2=\) constant. The coefficient of volume expansion of the gas is

The circuit shown in the figure contains an inductor \(L\), a capacitor \(C_0\), a resistor \(R_0\) and an ideal battery. The circuit also contains two keys \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter, \(\mathrm{K}_2\) is closed and simultaneously \(\mathrm{K}_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(V_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A human body has a surface area of approximately $1 m^2$ . The normal body temperature is $10 K$ above the surrounding room temperature $T_0$ . Take the room temperature to be $T_0=300 K$ . For $T_0=300 K$ , the value of $\sigma T^4=460\frac{W}{m^2}$ (where $\sigma$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

A horizontal force$F$is applied at the centre of mass of a cylindrical object of mass$m$and radius$R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is$\mu$. The centre of mass of the object has an acceleration$a$. The acceleration due to gravity is$g$. Given that the object rolls without slipping, which of the following statement(s) is (are) correct?

A small object is placed at the centre of a large evacuated hollow spherical container. Assume the container is maintained at $0 \mathrm{K}$. At time $t=0$, the temperature of the object is $200 \mathrm{K}$. The temperature of the object becomes $100 \mathrm{K}$ at $t=t_1$ and $50 \mathrm{K}$ at $t=t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(\frac{t_2}{t_1}\right)$ _________ .

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Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
A line \(M\) through \(A\) is drawn parallel to \(B D\). Points \(S\) moves such that its distances from the line \(B D\) are the vertex \(A\) are equal. If locus of \(S\) cuts \(M\) at \(T_2\) and \(T_3\) and \(A C\) at \(T_1\), then area of \(\Delta T_1 T_2 T_3\) is

For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $\left[0, \pi \right],$ which of the following options is/are correct?

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

Let \(f:(-1,1) \rightarrow I R\) be such that \(f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}\) for \(\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then the value (s) of \(f\left(\frac{1}{3}\right)\) is (are)

A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement(s) is (are) true?