Assertion (a) : A particle of mass \(m\) dropped into a hole made along the diameter of the earth from one end to the other and possess simple harmonic motion.Reason (R) : Gravitational force between any two particles is inversely proportional to the square of the distance between them.

In a hydrogen atom, an electron of mass $9.1\times {10}^{-31} \mathrm{kg}$ revolves about a proton in circular orbit of radius $0.53 \mathrm{A}.$ The radial acceleration and angular velocity of electron are respectively

A gas is expanded from an initial state to a final state along a path that consists of (a) an isothermal expansion doing \(40 \mathrm{~J}\) work, (b) an adiabatic expansion doing \(\mathrm{W}\) work, (c) an isothermal expansion doing \(30 \mathrm{~J}\) work. If the total change in the internal energy of the gas is \(-20 \mathrm{~J}\), the work done by the gas during the adiabatic expansion \(\mathrm{W}=\)

The maximum magnetic field produced by a current of 12 A passing through a copper wire of diameter 1.2 mm is

The class of materials having the largest band gap in the following is

Potential difference between the points \(P\) and \(Q\) in the circuit shown is

A hydraulic lift is shown in the figure. The radii of the movable pistons \(P_1\) and \(P_2\) are \(2 \mathrm{~m}\) and \(5 \mathrm{~m}\) respectively. If a block of mass \(x\) is placed on \(P_2\) then the minimum mass that should be kept on \(P_1\) to lift the block on \(P_2\) is

If the probability distribution of a random variable \(X\) is as follows, then \(P(x \leq 2)=\)

\(x_i\)	0	1	2	3	4
\(P\left(X = x_i\right)\)	3K	5K	\(3k^2\)	\(4k^2 + k\)	\(3k^2\)

The equation of the hyperbola with focus \((1,2), e=\sqrt{3}\) and directrix \(2 x+y=1\) is given by

For \(f(x)=\frac{\sin \pi[x]}{1+[x]}+\frac{x}{2+3 x}\), where \([x]\) denotes the greatest integer function, the domain and range in \(R\) are respectively

Using the standard electrode potentials given below identify the correct statements from the following.
\[
\begin{aligned}
\mathrm{Fe}^{2+}+2 e^{-} \longrightarrow \mathrm{Fe} ; E^{\circ} & =-0.44 \mathrm{~V} \\
\mathrm{Cu}^{2+}+2 e^{-} \longrightarrow \mathrm{Cu} ; E^{\circ} & =+0.34 \mathrm{~V} \\
\mathrm{Ag}^{+}+e^{-} \longrightarrow \mathrm{Ag} ; E^{\circ} & =+0.80 \mathrm{~V}
\end{aligned}
\]
(i) Copper can displace iron from \(\mathrm{FeSO}_4\) solution.
(ii) Iron can displace copper from \(\mathrm{CuSO}_4\) solution.
(iii) Silver can displace copper from \(\mathrm{CuSO}_4\) solution.
(iv) Iron can displace silver from \(\mathrm{AgNO}_3\) solution.

The differential equation representing the family of circles having their centres on Y-axis is \(\left(y_1=\frac{d y}{d x}\right.\) and \(\left.y_2=\frac{d^2 y}{d x^2}\right)\)

If \(\alpha, \beta, \gamma\) are the roots of \(2 x^3-2 x-1=0\), then \((\Sigma \alpha \beta)^2\) is equal to