A radioactive nucleus \(A\) with a half life \(T\), decays into a nucleus \(B.\) At \(t = 0\), there is no nucleus \(B\). At sometime \(t\), the ratio of the number of \(B\) to that of \(A\) is \(0.3\). Then, \(t\) is given by

If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is _________ m.
(Atomic number of gold \(=79\) and \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9\)in SI units)

An electron rotates in a circle around a nucleus having positive charge \(\mathrm{Ze}\). Correct relation between total energy ( \(E\) ) of electron to its potential energy \((U)\) is _______.

A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure. Distance of the image of an object ' O ' is :
( C is the center of curvature of the spherical surface and \(R\) is the radius of curvature)

Consider a moving coil galvanomenter (MCG):
A. The torsional constant in moving coil galvanometer has dimentions \(\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]\)
B. Increasing the current sensitivity may not necessarily increase the voltage sensitivity.
C. If we increase number of turns \((\mathrm{N})\) to its double \((2 \mathrm{~N})\), then the voltage sensitivity doubles.
D. MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer.
E. Current sensitivity of MCG depends inversely on number of turns of coil.
Choose the correct answer from the options given below:

A plane electromagnetic wave of frequency \(20\,MHz\) propagates in free space along \(x\)-direction. At a particular space and time, \(\overrightarrow{ E }=6.6 \hat{ j } V / m\). What is \(\overrightarrow{ B }\) at this point?

Given below are two statements: One is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\): If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude. Reason \(R \): At equator, the direction of acceleration due to the gravity is towards the center of earth. In the light of above statements, choose the correct answer from the options given below

A chord is drawn through the focus of the parabola \(y^2\, = 6x\) such that its distance from the vertex of this parabola is \(\frac{{\sqrt 5 }}{2}\) , then its slope can be:

Two masses \(M _{1}\) and \(M _{2}\) are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass \(M _{2}\) is twice that of \(M_{1}\). the acceleration of the system is \(a_{1}\). When the mass \(M_{2}\) is thrice that of \(M_{1}\). The acceleration of The system is \(a_{2}\). The ratio \(\frac{a_{1}}{a_{2}}\) will be.

Match List\(-I\) with List\(-II\)

List\(-I\)	List\(-II\)
\((a)\) \(h\) (Planck's constant)	\((i)\) \(\left[ M L T ^{-1}\right]\)
\((b)\) \(E\) (kinetic energy)	\((ii)\) \(\left[ M L ^{2} T ^{-1}\right]\)
\((c)\) \(V\) (electric potential)	\((iii)\) \(\left[ M L ^{2} T ^{-2}\right]\)
\((d)\) \(P\) (linear momentum)	\(( iv )\left[ M L ^{2} I ^{-1} T ^{-3}\right]\)

Choose the correct answer from the options given below

The area of the region described by \(A=\{(x,y):x^2 + y^2 \le 1\,and\,y^2 \le 1-x \}\) is

A stationary particle breaks into two parts of masses \(m_A\) and \(m_B\) which move with velocities \(v_A\) and \(V_B\) respectively. The ratio of their kinetic energies \(\left(\mathrm{K}_B: \mathrm{K}_{\mathrm{A}}\right)\) is _______.

\(\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}\) is equal to