A ray of light, travelling in the direction $\frac{1}{2}\left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}\right)$, is incident on a plane mirror. After reflection, it travels along the direction $\frac{1}{2}\left(\hat{\mathrm{i}}-\sqrt{3}\widehat{ \mathrm{j}}\right)$. The angle of incidence is

Two large, identical water tanks, 1 and 2, kept on the top of a building of height \(H\), are filled with water up to height \(h\) in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are \(t_1\) and \(t_2\), respectively. If \(H=\left(\frac{16}{9}\right) h\), then the ratio \(t_1 / t_2\) is _______ .

A cube of unit volume contains \(35 \times 10^7\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is_______

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 _____________ .

Starting at time $t=0$ from the origin with speed $1 \mathrm{m} {\mathrm{s}}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\frac{x^2}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_x$ and $a_y$, respectively. Then

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 \(\pi=3.2\).

Paragraph :
A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{Q}{R{T}_0}$ is:

Match the conversions in Column I with the type(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Among cellulose, polyvinyl chloride, nylon and natural rubber, the polymer in which the intermolecular force of attraction is weakest in

Let \(g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]\)

Define

\(

S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, \quad i=1,2

\)

The value of $\frac{48S_2}{{\pi }^2}$ is ____.

The species which by definition has zero standard molar enthalpy of formation at \(298 \mathrm{~K}\) is

A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is

Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. \(C _6 H _5 CHO\)	p. Gives precipitate with 2, 4-dinitrophenylhydrazine
B. \(CH _3 \equiv CH\)	q. Gives precipitate with \(AgNO _3\)
C. \(CN ^{-}\)	r. Is a nucleophile
D. \(I ^{-}\)	s. Is involved in cyanohydrin formation