Two blocks of mass \(2 \mathrm{~kg}\) and \(4 \mathrm{~kg}\) are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is \(4.0 \times 10^{-5}\) \(\mathrm{m}\) and Young's modulus of the metal is \(2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^2\). The longitudinal strain developed in the wire is \(\frac{1}{\alpha \pi}\). The value of \(\alpha\) is _______. [Use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

Given below are two statements: Statement \(I :\) An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero but the electric field is not zero anywhere in the sphere. Statement \(II :\) If \(R\) is the radius of a solid metallic sphere and \(Q\) be the total charge on it. The electric field at any point on the spherical surface of radius \(r ( < R )\) is zero but the electric flux passing through this closed spherical surface of radius \(r\) is not zero. In the light of the above statements, choose the correct answer from the options given below:

As shown in figure, a \(70\,kg\) garden roller is pushed with a force of \(\overrightarrow{ F }=200\,N\) at an angle of \(30^{\circ}\) with horizontal. The normal reaction on the roller is \(.......\,N\) \(\left(\right.\) Given \(\left.g =10\,m s ^{-2}\right)\)

A mirror is used to produce an image with magnification of \(\frac{1}{4}\). If the distance between object and its image is 40 cm , then the focal length of the mirror is ________

A magnetic dipole experiences a torque of \(80 \sqrt{3} \mathrm{~N} \mathrm{~m}\) when placed in uniform magnetic field in such a way that dipole moment makes angle of \(60^{\circ}\) with magnetic field. The potential energy of the dipole is :

From the circuit given below, the capacitance between terminals \(A\) and \(B\) shown in the circuit is ______ \(\mu\)F.
(take \(C_1=C_2=C_3=1\) \(\mu\)F and \(C_4=2\) \(\mu\)F.)

Two transparent media having refractive indices \(1.0\) and \(1.5\) are separated by a spherical refracting surface of radius of curvature \(30\,cm\). The centre of curvature of surface is towards denser medium and a point object is placed on the principle axis in rarer medium at a distance of \(15\,cm\) from the pole of the surface. The distance of image from the pole of the surface is .......\(cm\)

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

Figure below shows a liquid being pushed out of the tube by a piston having area of cross section \(2.0\,cm ^2\). The area of cross section at the outlet is \(10\,mm ^2\). If the piston is pushed at a speed of \(4\,cm s ^{-1}\), the speed of outgoing fluidis \(.........\,cm s ^{-1}\).

Let the mean and the variance of 6 observation \(a, b\), \(68,44,48,60\) be \(55\) and \(194 \), respectively if \(a>b\), then \(a+3 b\) is

The equivalent capacitance between \(A\) and \(B\) in the circuit given below is.....\(μF\)

A function \(f(x)\) is given by \(f(x)=\frac{5^{x}}{5^{x}+5}\), then the sum of the series \(f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)\) is equal to ....... .

The largest value of \(a,\) for which the perpendicular distance of the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})\) from the point \((2,1,4)\) is \(\sqrt{3}\), is\(...\)