The electrostatic potential in a charged spherical region of radius \(r\) varies as \(V=a r^3+b\), where \(a\) and b are constants. The total charge in the sphere of unit radius is \(\alpha \times \pi a \in_0\). The value of \(\alpha\) is _________ .

If \(50\) Vernier divisions are equal to \(49\) main scale divisions of a travelling microscope and one smallest reading of main scale is \(0.5 \mathrm{~mm}\), the Vernier constant of travelling microscope is _______.

Given below are two statements : one is labelled as Assertion \(A\) and the other is labelled as Reason \(R.\) Assertion \(A :\) Moment of inertia of a circular disc of mass \('M'\) and radius \('R'\) about \(X, Y\) axes (passing through its plane) and \(Z-\)axis which is perpendicular to its plane were found to be \(I_{x}, I_{y}\) and \({I}_{z}\) respectively. The respective radii of gyration about all the three axes will be the same. Reason \(R\) : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below :

In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line. One may conclude that:

Refer to the circuit diagram given in the figure. which of the following observations are correct?
A. Total resistance of circuit is \(6 \Omega\)
B. Current in Ammeter is 1 A
C. Potential across \(A B\) is 4 Volts.
D. Potential across CD is 4 Volts
E. Total resistance of the circuit is \(8 \Omega\).

Choose the correct answer from the options given below:

A common emitter amplifier circuit, built using an \(npn\) transistor, is shown in the figure. Its \(dc\) current gain is \(250\), \(R_C = 1\,k\, \Omega \) and \(V_{CC} = 10\,V\). What is the minimum base current for \(V_{CE}\) to reach saturation?.........\(\mu A\)

A transistor is used in an amplifier circuit in common emitter mode. If the base current changes by \(100 \,\mu A\), it brings a change of \(10 \,mA\) in collector current. If the load resistance is \(2 \,k \Omega\) and input resistance is \(1 \,k \Omega\), the value of power gain is \(x \times 10^{4}\). The value of \(x\) is_________

Four closed surfaces and corresponding charge distributions are shown below Let the respective electric fluxes through the surfaces be \({\phi _1},{\phi _2},{\phi _3}\) and \({\phi _4}\) . Then

If the line \(x = y = z\) intersects the line \(x \sin A+ y\) \(\sin B + z \sin C -18=0= x \sin 2 A + y \sin 2 B + z\) \(\sin 2 C -9\), where \(A , B , C\) are the angles of a triangle \(ABC\), then \(80\left(\sin \frac{ A }{2} \sin \frac{ B }{2} \sin \frac{ C }{2}\right)\) is equal to \(..........\).

Particle \(A\) of mass \(m _{1}\) moving with velocity \((\sqrt{3} \hat{i}+\hat{j})\, ms ^{-1}\) collides with another particle \(B\) of mass \(m _{2}\) which is at rest initially. Let \(\overrightarrow{ V }_{1}\) and \(\overrightarrow{ V }_{2}\) be the velocities of particles \(A\) and \(B\) after collision respectively. If \(m _{1}=2\, m _{2}\) and after collision \(\overrightarrow{ V }_{1}=(\hat{ i }+\sqrt{3} \hat{ j })\, ms ^{-1},\) the angle between \(\overrightarrow{ V }_{1}\) and \(\overrightarrow{ V }_{2}\) is\(......^o\)

\(\mathop {\lim }\limits_{x \to 0} \frac{{x + 2\,\sin \,x}}{{\sqrt {{x^2} + 2\sin \,x + 1}  - \sqrt {{{\sin }^2}\,x - x + 1} }}\) is

The value of \(\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}\) is

A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of crosssectional area \('a'\) is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is \((a\,<\,<\,A)\)