The principle quantum number ' \(n\) ' corresponding to the exited state of \(\mathrm{He}^{+}\)ion, if on transition to the ground state two photons in succession with wavelength \(1026 \mathrm{~A}^{\circ}\) and \(304 \mathrm{~A}^{\circ}\) are emitted ( \(\mathrm{R}=1.097 \times 10^7 \mathrm{~m}^{-1}\) )

A microscope consists of an objective of focal length \(1.9 \mathrm{~cm}\) and eye piece of focal length \(5 \mathrm{~cm}\). The two lenses are kept at a distance of \(10.5 \mathrm{~cm}\). If the image is to be formed at the least distance of distinct vision, the distance at which the object is to be placed before the objective is (least distance of distinct vision is \(25 \mathrm{~cm}\) )

For a material, the ratio of Young's modulus to rigidity modulus is 2.8 . If some force is applied on a wire made of this material, its cross-sectional area decreases by \(2 \%\). The percentage change in its length is

A fully loaded Boeing air craft has a mass of \(3.3 \times 10^5 \mathrm{~kg}\). Its total wing area is \(500 \mathrm{~m}^2\). It is in level flight with a speed of \(960 \mathrm{~km} / \mathrm{h}\). Estimate the pressure difference between the lower and upper surfaces of the wings.

When a liquid is heated in copper vessel its coefficient of apparent expansion is \(6 \times 10^{-6} /{ }^{\circ} \mathrm{C}\). When the same liquid is heated in a steel vessel its coefficient of apparent expansion is \(24 \times 10^{-6} /{ }^{\circ} \mathrm{C}\). If coefficient of linear expansion for copper is \(18 \times 10^{-6} /{ }^{\circ} \mathrm{C}\), the coefficient of linear expansion for steel is

A wire of length 2.5 m is fixed at one end and a box of mass 4 kg is tied at the other end. If the wire rotates in a horizontal circle about the fixed end with \(\frac{2}{\pi}\) rotations per second, then the tension in the wire is

Statement (A) Two artificial satellites revolving in the same circular orbit have same period of revolution.
Statement (B) The orbital velocity is inversely proportional to the square root of radius of the orbit.
Statement (C) The escape velocity of the body is independent of the altitude of the point of projection.

The direction cosines of the line joining the points $(-2,4,-5)$ and $(1,2,3)$ are

How many chords can be drawn through $21$ points on a circle?

Two balls of charge \(q_1\) and \(q_2\) initially have a velocity of the same magnitude and direction. After a uniform electric field has been applied during a certain time, the direction of the velocity of the first ball changes by \(60^{\circ}\), and the velocity magnitude is reduced by half. The direction of the velocity of the second ball changes thereby by \(90^{\circ}\).
In what proportion will the velocity of the second ball change? Determine the magnitude of the charge-to-mass ratio for the second ball if it is equal to \(k_1\) for the first ball. The electrostatic interaction between the balls should be neglected.

The equation of any in the complex plane is of the form \(z \bar{z}+b \bar{z}+\bar{b} z+c=0\) where \((b \in \mathbf{C}, c \in \mathbf{R})\)

Let \(\bar{a}=2 \bar{i}+\bar{j}-2 \bar{k}\) and \(\bar{b}=\bar{i}+\bar{j}\) be two vectors. If \(\bar{c}\) is a vector such that \(\bar{a} \cdot \bar{c}=|\bar{c}|,|\bar{c}-\bar{a}|=2 \sqrt{2}\) and the angle between \(\bar{a} \times \bar{b}\) and \(\bar{c}\) is \(30^{\circ}\), then \(|(\bar{a} \times \bar{b}) \times \bar{c}|=\)

For a non real root \(z\) of the equation \(z^4+z^2+1=0\), the value of \(\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^3\)