An engine of a train, moving with uniform acceleration, passes the signal-post with velocity \(u\) and the last compartment with velocity \(v\). The velocity with which middle point of the train passes the signal post is

Two radioactive substances \(A\) and \(B\) have decay constants \(5\lambda \) and \(\lambda \) respectively. At \(t = 0\), a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become \((\frac {1}{e})^2\) will be

Two simple pendulums of length \(1\, m\) and \(4\, m\) respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to

An \(LCR\) circuit contains resistance of \(110 \, \Omega\) and a supply of \(220\, V\) at \(300 \,rad / s\) angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by \(45^{\circ}\). If on the other hand, only inductor is removed the current leads by \(45^{\circ}\) with the applied voltage. The rms current flowing in the circuit will be ...... \(A\)

A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of Green light of wavelength 550 nm . [Assume that the light is incident nearly perpendicular to the glass surface.]

The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ___________ cm.

Match the List \(-I\) (Phenomenon associated with electromagnetic radiation) with List \(-II\) (Part of electromagnetic spectrum) and select the correct code from the choices given below this lists

List \(I\)	List \(II\)
\(I\) Doublet of sodium	\((A)\) Visible radiation
\(II\) Wavelength corresponding to temperature associated with the isotropic radiation filling all space	\((B)\) Microwave
\(III\) Wavelength emitted by atomic hydrogen in interstellar space	\((C)\) Short radio wave
\(IV\) Wavelength of radiation arising from two close energy levels in hydrogen	\((D)\) \(X-\) rays

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is

The differential equation representing the family of ellipse having foci either on the \(x-\) axis or on the \(y-\) axis centre at the origin and passing through the point  \((0,3)\) is

Two integers are selected at random from the set \(\{1, 2, …, 11\}.\) Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is

The magnetic field of an electromagnetic wave is given by \(\vec B = 1.6 \times {10^{ - 6}}\,\cos \,\left( {2 \times {{10}^7}z + 6 \times {{10}^{15}}t} \right)\left( {2\hat i + \hat j} \right)\frac{{Wb}}{{{m^2}}}\) The associated electric field will be

Let a circle \(\mathrm{C}\) of radius \(1\) and closer to the origin be such that the lines passing through the point \((3,2)\) and parallel to the coordinate axes touch it. Then the shortest distance of the circle \(\mathrm{C}\) from the point \((5,5)\) is :

The perpendicular distance from the origin to the plane containing the two lines,\(\frac{{x + 2}}{3} = \frac{{y - 2}}{5} = \frac{{z + 5}}{7}\) and \(\frac{{x - 1}}{1} = \frac{{y - 4}}{4} = \frac{{z + 4}}{7}\), is