The focal length of a thin biconvex lens is \(20 \mathrm{~cm}\). When an object is moved from a distance of \(25 \mathrm{~cm}\) in front of it to \(50 \mathrm{~cm}\), the magnification of its image changes from \(m_{25}\) to \(m_{50}\). The ratio \(\frac{m_{25}}{m_{50}}\) is

A thin flexible wire of length \(L\) is connected to two adjacent fixed points and carries a current \(I\) in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength \(B\) going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, 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 $\left(1 fm={10}^{-15}m\right)$

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The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

Consider \(d \gg a\), and the loop is rotated about its diameter parallel to the wires by \(30^{\circ}\) from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

A block of base \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) and height \(15 \mathrm{~cm}\) is kept on an inclined plane. The coefficient of friction between them is \(\sqrt{3}\). The inclination \(\theta\) of this inclined plane from the horizontal plane is gradually increased from \(0^{\circ}\). Then,

A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

Match the following

Column I	Column II
(A) \(CH _3- CHBr - CD _3\) on treatment with alc. KOH gives \(CH _2= CH - CD _3\) as a major product.	(P) E1 reaction
(B) \(Ph - CHBr - CH _3\) reacts faster than \(Ph - CHBr - CD _3\)	(Q) E2 reaction
(C) \(Ph - CH _2- CH _2 Br\) on treatment with \(C _2 H _5 OD / C _2 H _5 O ^{-}\)gives \(Ph - CD = CH _2\) as the major product	(R) El cb reaction
(D) \(PhCH _2 CH _2 Br\) and \(PhCH _2 CH _2 Br\) react with same rate	(S) First order reaction

If $y\left(x\right)$ is the solution of the differential equation $xdy-\left(y^2-4y\right)dx=0$ for $x>0, y\left(1\right)=2$, and the slope of the curve $y=y\left(x\right)$ is never zero, then the value of $10y\left(\sqrt{2}\right)$ is _____ .

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)
Then which of the following statements is (are) TRUE?

The work function ( \(\Phi\) ) of some metals is listed below. The number of metals which will show photoelectric effect when light of \(300 \mathrm{~nm}\) wavelength falls on the metal is

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The conditional probability that \(X \geq 6\) given \(X>3\) equals

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures \(2 T\) and \(3 T\) respectively. The temperature of the middle (i.e. second) plate under steady state condition is