If the radius of a nucleus with mass number 125 is 1.5 fermi, then radius of a nucleus with mass number 64 is

A solid sphere of mass 4 kg and radius 28 cm is on an inclined plane. If the acceleration of the sphere when it rolls down without sliding is \(3.5 \mathrm{~m} \mathrm{~s}^{-2}\), then the acceleration of the sphere when it slides down without rolling is

Match the following lists.
\begin{array}{|c|c|c|c|}
\hline & List I & & List II \\
\hline A & \begin{array}{l}\text { Zeroth law of } \\
\text { thermodynamics }\end{array} & 1 & \begin{array}{l}\text { Direction of flow of } \\
\text { heat }\end{array} \\
\hline B & \begin{array}{l}\text { First law of } \\
\text { thermodynamics }\end{array} & \| & Work done is zero \\
\hline\underline{\mathrm{C}} & Free expansion of a gas & III & Thermal equilibrium \\
\hline D & \begin{array}{l}\text { Second law of } \\
\text { Thermodynamics }\end{array} & IV & \begin{array}{l}\text { Law of conservation } \\
\text { of energy }\end{array} \\
\hline
\end{array}
The correct answer is
\(\begin{array}{lccccccccc} & \text { A } & \text { B } & \text { C } & \text { D } \end{array}\)

A hydraulic lift is shown in the figure. The movable pistons \(A, B\) and \(C\) are of radius \(10 \mathrm{~cm}, 100 \mathrm{~m}\) and \(5 \mathrm{~cm}\) respectively. If a body. of mass \(2 \mathrm{~kg}\) is placed on piston \(A\), the maximum masses that can be lifted by piston \(B\) and \(C\) are respectively.

The masses of two fixed spheres are \(\mathrm{M}\) and \(2 \mathrm{M}\) and the radius of each sphere is \(\mathrm{R}\). Their centres are \(10 \mathrm{R}\) apart. The minimum speed with which a particle of mass \(\frac{\mathrm{M}}{10}\) be projected from the mid-point of the line joining the centres of the two spheres so that it escapes to infinity is _________ .

A rocket with an initial mass \(m_0\) is going up with a constant acceleration \(a\) by exhausting gases with a velocity \(v\) relative to the rocket motion, then the mass of the rocket at any instant of time is (assume that no other forces act on it)

The length of a wire required to make a solenoid of length \(l\) and self-induction \(L\) is

In a \(\triangle A B C\), if \(\angle A=60^{\circ}\), then \((a+b+c)(b+c-a)=\)

\(\int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x=\)

If \(y=\operatorname{Tan}^{-1}\left(\frac{2 x}{1-x^2}\right),|x| < 1\), then \(\left(\frac{d y}{d x}\right)_{x=\frac{1}{2}}=\)

\(\begin{aligned} & \int(1+x) \log \left(1+x^2\right) d x=\left(x+\frac{x^2}{2}+\frac{1}{2}\right) \\ & \log \left(1+x^2\right)+g(x)+C, \text { then } g(x)=\end{aligned}\)

Two identical condensers are joined as shown in the figure. When the switch \(\mathrm{S}\) is closed, the total energy of the system is \(\mathrm{U}_1\). If the switch is opened and both the condensers are filled with a dielectric of dielectric constant 3 , then the energy of the system becomes \(\mathrm{U}_2\). The value of \(\frac{\mathrm{U}_1}{\mathrm{U}_2}\) is

An inclined plane making an angle \(30^{\circ}\) with the horizontal is placed in a uniform horizontal electric field of \(100 \mathrm{Vm}^{-1}\) as shown in the figure. A small block of mass \(\mathrm{I} \mathrm{kg}\) and charge, 0.01C is allowed to slide down from rest from a height, \(h=1 \mathrm{~m}\). If the coefficient of friction is 0.2 , then the acceleration of the block is nearly,
(Acceleration due to gravity, \(g=10 \mathrm{~ms}^{-2}\) )