One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A block of weight $\alpha W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of  $\left(2\sqrt{2}\right)W$. Which of the following statement(s) is(are) correct?

Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.

The value of $b$ is ________ meter.

Two rectangular blocks, having identical dimensions, can be arranged either in configuration$\text{I}$ or in configuration $\text{I}\text{I}$ as shown in the figure. One of the blocks has thermal conductivity $\text{K}$ and the other $\text{2 K}$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes $\text{9 s}$ to transport a certain amount of heat from the hot end to cold end in the configuration $\text{I}$. The time to transport the same amount of heat in the configuration $\text{II}$ is

A steel wire of diameter 0.5 mm and Young's modulus \(2 \times 10^{11} Nm ^{-2}\) carries a load of mass M . The length of the wire with the load is 1.0 m . A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by \(1.2 k g\), the vernier scale division which coincides with a main scale division is _____________ . Take \(g =10 m s^{-1}\) and \(\pi=3.2\).

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?

Paragraph:
\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.Question:
The ratio \(\frac{x_1}{x_2}\) is

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$ .

Statement I p-hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid.
Statement II o-hydroxybenzoic acid has intramolecular hydrogen bonding.

\(2.5 \mathrm{~mL}\) of \(\frac{2}{5} \mathrm{M}\) weak monoacidic base \(\left(K_b=1 \times 10^{-12}\right.\) at \(\left.25^{\circ} \mathrm{C}\right)\) is titrated with \(\frac{2}{15}\) \(\mathrm{M} \mathrm{HCl}\) in water at \(25^{\circ} \mathrm{C}\). The concentration of \(\mathrm{H}^{+}\)at equivalence point is \(\left(K_w=1 \times 10^{-14}\right.\) at \(\left.25^{\circ} \mathrm{C}\right)\)

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$ . The angle of contact between water and the wall of the capillary tube is $\theta$ . Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

The value of \(n\) in the molecular formula \(\mathrm{Be}_n \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18}\) is

Paragraph:
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 circle touches the line \(L\) and the circle \(C_1\) externally such that both the circles are on the same side of the line, then the locus of centre of the circle is

The total number of local maxima and local minima of the function \(f(x)=\left\{\begin{array}{lc}(2+x)^3 ; & -3 < x \leq-1 \\ x^{\frac{2}{3}} ; & -1 < x < 2\end{array}\right.\) is