A cubical solid aluminum (bulk modulus \(=-\text V \frac{\text{dP}}{\text{dV}}=70\text{ GPa}\)) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be \(10^3\text{ kg m} ^{-3}\) and \(10\text{ ms} ^{-2}\), respectively, the change in the edge length of the block in mm is _____________ .

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A small block of mass \(1 \mathrm{~kg}\) is released from rest at the top of a rough track. The track is a circular arc of radius \(40 \mathrm{~m}\). The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point \(Q\), as shown in the figure below, is \(150 \mathrm{~J}\). (Take the acceleration due to gravity, \(g=10 \mathrm{~m} \mathrm{~s}^{-2}\) ).





Question:

The speed of the block when it reaches the point \(Q\) is

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude \(p\), oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance \(2 r\) apart along the \(x\) direction. The midpoint of the line joining the two dipoles is \(X\). The possible resultant electric fields \(\vec{E}\) at \(X\) are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)	\(\vec{E}=0\)
\((Q)\)		\((2)\)	\(\vec{E}=-\frac{p}{2 \pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{j}}\)
\((R)\)		\((3)\)	\(\vec{E}=-\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(\hat{\mathrm{i}}-\hat{\mathrm{j}})\)
\((S)\)		\((4)\)	\(\vec{E}=\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})\)
		\((5)\)	\(\vec{E}=\frac{p}{\pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{i}}\)

Let ${\mathrm{E}}_1\left(\mathrm{r}\right),\mathrm{ }{\mathrm{E}}_2\left(\mathrm{r}\right)$ and ${\mathrm{E}}_3\left(\mathrm{r}\right)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $\mathrm{\lambda }$ , and an infinite plane with uniform surface charge density $\mathrm{\sigma }$ . If ${\mathrm{E}}_1\left({\mathrm{r}}_0\right)={\mathrm{E}}_2\left({\mathrm{r}}_0\right)={\mathrm{E}}_3\left({\mathrm{r}}_0\right)$ at a given distance ${\mathrm{r}}_0$ , then

An infinitely long solid cylinder of radius \(R\) has a uniform volume charge density \(\rho\). It has a spherical cavity of radius \(R / 2\) with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point \(P\), which is at a distance \(2 R\) from the axis of the cylinder, is given by the expression \(\frac{23 \rho R}{16 K \varepsilon_{0}}\). The value of \(k\) is

A student performs an experiment to determine the Young's modulus of a wire, exactly \(2 \mathrm{~m}\) long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be \(0.8 \mathrm{~mm}\) with an uncertainty of \(\pm 0.05 \mathrm{~mm}\) at a load of exactly \(1.0 \mathrm{~kg}\). The student also measures the diameter of the wire to be \(0.4 \mathrm{~mm}\) with an uncertainty of \(\pm 0.01 \mathrm{~mm}\). Take \(g=9.8 \mathrm{~m} / \mathrm{s}^2\) (exact). The Young's modulus obtained from the reading is

Paragraph: When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex \(\mathrm{P}\) is formed. In a strong acidic medium, the equilibrium shifts completely towards \(\mathrm{P}\). Addition of zinc chloride to \(\mathrm{P}\) in a slightly acidic medium results in a sparingly soluble complex Q.
Question: The number of moles of potassium iodide required to produce two moles of \(\mathbf{P}\) is

An extended object is placed at point $O$, $10 \mathrm{cm}$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10 \mathrm{cm}$ behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20 \mathrm{cm}$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is:

Let $L_1$ and $L_2$ be the following straight lines.
$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line $L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$ lies in the plane containing $L_1$ and $L_2$, and passes through the point of intersection of $L_1$ and $L_2$. If the line $L$ bisects the acute angle between the lines $L_1$ and $L_2$, then which of the following statements is/are TRUE?

Let \(k \in \mathbb{R}\). If \(\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of \(k\) is

The correct statement(s) about surface properties is(are)

For a double strand DNA, one strand is given below:

The amount of energy required to split the double strand DNA into two single strands is _______ kcal \(\mathrm{mol}^{-1}.\)
[Given: Average energy per \(\mathrm{H}\)-bond for A-T base pair \(=1.0 \mathrm{kcal} \mathrm{mol}^{-1}, \mathrm{G}-\mathrm{C}\) base pair \(=1.5 \mathrm{kcal}\) \(\mathrm{mol}^{-1}\), and A-U base pair \(=1.25 \mathrm{kcal} \mathrm{mol}^{-1}.\) Ignore electrostatic repulsion between the phosphate groups.]