Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
Now level of the liquid starts decreasing slowly. When the level of liquid is at a height \(h_1\) above the cylinder, the block starts moving up. At what value of \(h_1\), will the block rise?

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

A small block is connected to one end of a massless spring of un-stretched length \(4.9 \mathrm{~m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2 \mathrm{~m}\) and released from rest at \(t=0\). It then executes simple harmonic motion with angular frequency \(\omega=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously at \(t=0\), a small pebble is projected with speed \(v\) form point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. Point \(P\) is at a horizontal distance of \(10 \mathrm{~m}\) from \(O\). If the pebble hits the block at \(t=1 \mathrm{~s}\), the value of \(v\) is (take \(g\) \(=10 \mathrm{~m} / \mathrm{s}^{2}\) )

Three glass cylinders of equal height $\mathrm{H}=30\mathrm{}\mathrm{c}\mathrm{m}$ and same refractive index $\mathrm{n}=1.5$ are placed on a horizontal surface shown in figure. Cylinder $\mathrm{I}$ has a flat top, cylinder $\mathrm{I}\mathrm{I}$ has a convex top and cylinder $\mathrm{I}\mathrm{I}\mathrm{I}$ has a concave top. The radii of curvature of the two curved tops are same $\left(\mathrm{R}=3\mathrm{m}\right).$ If ${\mathrm{H}}_1,{\mathrm{H}}_2$ and ${\mathrm{H}}_3$ are the apparent depths of a point $\mathrm{X}$ on the bottom of the three cylinders, respectively, the correct statement(s) is/are

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by $E=\frac{3}{5}\frac{Z\left(Z-1\right)e^2}{4\pi {\epsilon }_{0}R}$ The measured masses of the neutron, ${}_1{}^{1}H, {}_8{}^{15}O$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u respectively. Given that the radii of both the ${}_7{}^{15}N and {}_8{}^{15}O$ nuclei are same, 1u = 931.5 Me V/ $c^2$ (c is the speed of light) and $\frac{e^2}{\left(4\pi {ϵ}_0\right)}=1.44 MeV fm.$ Assuming that the difference between the binding energies of ${}_7{}^{15}N and {}_8{}^{15}O$ is purely due to the electrostatic energy, the radius of either of the nuclei is $\left(1 fm={10}^{-15}m\right)$

A charge is kept at the central point \(\mathrm{P}\) of a cylindrical region. The two edges subtend a half-angle \(\theta\) at \(\mathrm{P}\), as shown in the figure. When \(\theta=30^{\circ}\), then the electric flux through the curved surface of the cylinder is \(\Phi\). If \(\theta=60^{\circ}\), then the electric flux through the curved surface becomes \(\Phi / \sqrt{n}\), where the value of \(n\) is _______ .

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

Paragraph:
The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field at \(r=R\) is

Two identical moving coil galvanometer have $10\mathrm{ }\mathrm{\Omega }$ resistance and full scale deflection at $2\mathrm{\mu }\mathrm{A}$ current. One of them is converted into a voltmeter of $100\mathrm{ }\mathrm{m}\mathrm{V}$ full scale reading and the other into an Ammeter of $1\mathrm{ }\mathrm{m}\mathrm{A}$ full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $\mathrm{R}=1000\mathrm{ }\mathrm{\Omega }$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?

Let \(a=3 \sqrt{2}\) and \(b=\frac{1}{5^{1 / 6} \sqrt{6}}\). If \(x, y \in \mathbb{R}\) are such that
\(\begin{gathered}3 x+2 y=\log _a(18)^{\frac{5}{4}} \\2 x-y=\log _b(\sqrt{1080})\end{gathered}\)
and then \(4 x+5 y\) is equal to . ________.

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

Let $a, b\mathbb{ }\in \mathbb{R}$ and $a^2+b^2\ne 0$ . Suppose $S=\left\{z\in \mathbb{C}\mathbb{ }:z=\frac{1}{a+ibt}, t\mathbb{ }\in \mathbb{R},\mathbb{ }t \ne 0\right\},$ where $i= \sqrt{-1}.$ If $z=x+iy$ and $z\in S,$ then $\left(x, y\right)$ lies on

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?