The solid cylinder of length \(80\, {cm}\) and mass \({M}\) has a radius of \(20\, {cm}\). Calculate the density of the material used if the moment of inertia of the cylinder about an axis \(CD\) parallel to \({AB}\) as shown in figure is \(2.7\, {kg} {m}^{2}\)

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

A convex lens of focal length 30 cm is placed in contact with a concave lens of focal length 20 cm. An object is placed at 20 cm to the left of this lens system. The distance of the image from the lens in cm is ________

A proton and a deutron ( \(\mathrm{q}=+\mathrm{e}, m=2.0 \mathrm{u})\) having same kinetic energies enter a region of uniform magnetic field \(\vec{B}\), moving perpendicular to \(\vec{B}\). The ratio of the radius \(r_d\) of deutron path to the radius \(r_p\) of the proton path is _______.

Diameter of a steel ball is measured using a Vernier callipers which has divisions of \(0. 1\,cm\) on its main scale \((MS)\) and \(10\) divisions of its vernier scale \((VS)\) match \(9\) divisions on the main scale. Three such measurements for a ball are given as

S.No.	\(MS\;(cm)\)	\(VS\) divisions
\((1)\)	\(0.5\)	\(8\)
\((2)\)	\(0.5\)	\(4\)
\((3)\)	\(0.5\)	\(6\)

If the zero error is \(- 0.03\,cm,\) then mean corrected diameter is  ........... \(cm\)

A slanted object \(A B\) is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities \(d_1\) and \(d_2\) are filled in the tube. Each liquid subtends \(90^o\) angle at centre. Radius joining their interface makes an angle \(\alpha\) with vertical. Ratio \(\frac{{{d_1}}}{{{d_2}}}\) is

A ball is dropped from a height of \(20\,m\). If the coefficient of restitution for the collision between ball and floor is \(0.5\), after hitting the floor, the ball rebounds to a height of \(.............m\).

The image of an illuminated square is obtained on a screen with the help of a converging lens. The distance of the square from the lens is \(40\,cm.\) The area of the image is \(9\,times\) that of the square. The focal length of the lens is........\(cm\)

A diatomic gas with rigid molecules does \(10\, J\) of work when expanded at constant pressure. What would be the heat energy absorbed by the gas, in this process ..... \(J\).

If the area of the bounded region \(R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) is, \(\alpha\left(\log _{e} 2\right)^{-1}+\beta\left(\log _{e} 2\right)+\gamma\), then the value of \((\alpha+\beta-2 \gamma)^{2}\) is equal to:

Let \(f( x)\) be a polynomial of degree four having extreme values at \( x=1 \) and \( x=2\) . If \(\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f\left( x \right)}}{{{x^2}}}} \right] = 3\),then \(f\left( 2 \right)\) is equal to :

If \(x\, = a\), \(y\, = b\), \(z\, = c\) is a solution of the system of linear equations \(x+8y+ 7z\,= 0\) ; \(9x+ 2y+ 3z\, = 0\) ; \(x+y+z\, = 0\) such that the point \((a, b, c)\) lies on the plane \(x + 2y + z\, = 6\), then \(2a + b + c\) equals