Assume that the nuclear binding energy per nucleon \((B / A)\) versus mass number \((A)\) is as shown in the figure. Use this plot to choose the correct choice (s) given below.

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $\mathrm{T}$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $\mathrm{n}\mathrm{T}$ years, then the value of $\mathrm{n}$ is

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \(m\) and radius \(r\) and it is in a uniform vertical magnetic field \(B_0\), as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity \(g\), on two conducting supports at \(\mathrm{P}\) and \(\mathrm{Q}\). When a current \(I\) is passed through the loop, the loop turns about the line \(\mathrm{PQ}\) by an angle \(\theta\) given by

A real gas behaves like an ideal gas if its

An object and a concave mirror of focal length $f=10 \mathrm{cm}$ both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_0=15 \mathrm{cm} {\mathrm{s}}^{-1}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$. When $u=30 \mathrm{cm}$, the speed of the mirror $V_m$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_m$ is _______ $\mathrm{cm} {\mathrm{s}}^{-1}$.

A medium having dielectric constant $K>1$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $d$. The capacitor is connected to a battery of voltage $V$. as shown in Figure (a). Now, both the plates are moved by a distance of $\frac{d}{2}$ from their original positions, as shown in Figure (b).

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow{\text{u}}}_1=4\hat{\text{i}}{\text{ms}}^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity ${\overrightarrow{\text{u}}}_2=2\left(\sqrt{3}\hat{\text{i}}+\hat{\text{j}}\right){\text{ms}}^{-1}$. The correct statement (s) is (are)

A tangent \(P T\) is drawn to the circle \(x^{2}+y^{2}=4\) at the point \(P(\sqrt{3}, 1)\). A straight line \(L\), perpendicular to \(P T\) is a tangent to the circle \((x-3)^{2}+y^{2}-1\).

Question: A possible equation of \(L\) is

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The vapour pressure of the solution \(M\) is

Let the function \(f:[1, \infty) \rightarrow \mathbb{R}\) be defined by
\(f(t)=\left\{\begin{array}{cc}(-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \mathbb{N}, \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1 < t < 2 n+1, n \in \mathbb{N} .\end{array}\right.\)
Define \(g(x)=\int_1^x f(t) d t, x \in(1, \infty)\). Let \(\alpha\) denote the number of solutions of the equation \(g(x)=0\) in the interval \((1,8]\) and \(\beta=\lim _{x \rightarrow 1^+} \frac{g(x)}{x-1}\). Then the value of \(\alpha+\beta\) is equal to ________.

${ }^{131}I is$ an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of $8 days$ . A small amount of a serum labelled with ${ }^{131}I$ is injected into the blood of a person. The activity of the amount of ${ }^{131}I$ injected was $2.4\times {10}^5$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of $115 Bq$ . The total volume of blood in the person's body, in litres is approximately (you may use \(\mathrm{e}^{\mathrm{x}} \approx 1+\mathrm{x}\) for \(|\mathrm{x}|<<1\) and \(\ln 2 \approx 0.7\)).

A current carrying wire heats a metal rod. The wire provides a constant power $\left(\mathrm{P}\right)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $\left(\mathrm{T}\right)$ in the metal rod changes with time $\left(\mathrm{t}\right)$ as:
$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{T}}_0\left(1+\mathrm{\beta }{\mathrm{t}}^{1/4}\right)$
where $\mathrm{\beta }$ is a constant with appropriate dimension while ${\mathrm{T}}_0$ is a constant with dimension of temperature. The heat capacity of the metal is:

Let \(\left(x_0, y_0\right)\) be the solution of the following equations \((2 x)^{\log 2}=(3 y)^{\log 3}\), \(3^{\log x}=2^{\log y}\), then \(x_0\) is equal to