Consider a \(p-n\) junction as a capacitor, formed with \(p\) and \(n\)-materials acting as thin metal electrodes and depletion layer width acting as separation between them. Basing on this, assume that a \(n-p-n\) transistor is working as an amplifier in CE configuration. If \(C_1\) and \(C_2\) are the base-emitter and collector-emitter junction capacitances, then :

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

Eight drops of mercury, each of same radius and same charge combine to form a bigger drop. The ratio of capacitance of the bigger drop to that of each smaller drop is

Two point sources $S_1$ and $S_2$ separated by a distance $10 \mu \mathrm{m}$ emit light waves of wavelength $4 \mu \mathrm{m}$ in phase. A circular wire of radius $40 \mu \mathrm{m}$ is placed around the sources as shown in figure, then ($O$ is the centre of the circle and $O{S}_1=O{S}_2$)

A 100 kg gun fires a ball of 1 kg horizontally from a cliff of height 500 m. It falls on the ground at a distance of 400 m from the bottom of the cliff. The recoil velocity of the gun is (Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\)).

A body of mass \(2 \mathrm{~kg}\) is on an inclined plane of inclination \(30^{\circ}\) and coefficient of friction is \(\left(\frac{1}{\sqrt{3}}\right)\). The minimum force required to move the body up the inclined plane is
\(\left(\right.\) Acceleration due to gravity \(\left.=10 \mathrm{~ms}^{-2}\right)\)

\(P Q\) represents a wavefront and \(A O\) and \(B P\), the corresponding two rays. Find the condition on \(Q\) for constructive interference at \(P\) between ray \(B P\) and reflected ray \(O P\).

If the equation \(x^4+a x^3+b x^2+c x+d=0\) has three equal roots, then that root is

The solution of the differential equation \(x y^2 d y-\left(x^3+y^3\right) d x=0\) is

\(\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} d x\) is equal to :

The general solution of the differential equation \(\left(x y+y^2\right) d x-\left(x^2-2 x y\right) d y=0\) is

Two weights \(2 \mathrm{~N}\) and \(3 \mathrm{~N}\) are suspended from the ends of an inextensible string passing over a fixed frictionless pulley. If the pulley is pulled up with an acceleration equal to the acceleration due to gravity, then the tension in the string is

Let \(N\) be the set of all natural numbers, \(Z\) be the set of all integers and \(\sigma: N \rightarrow Z\) be defined by
\[
\sigma(n)=\left\{\begin{array}{ccc}
\frac{n}{2}, & \text { if } & n \text { is even } \\
-\frac{n-1}{2}, & \text { if } & n \text { is odd }
\end{array}\right. \text {. Then, }
\]