The temperature \(\theta\) at the junction of two insulating sheets, having thermal resistances \(R _{1}\) and \(R _{2}\) as well as top and bottom temperatures \(\theta_{1}\) and \(\theta_{2}\) (as shown in figure) is given by

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 :

A block of mass \(5 \mathrm{~kg}\) is placed on a rough inclined surface as shown in the figure.If \(\vec{F}_1\) is the force required to just move the block up the inclined plane and \(\vec{F}_2\) is the force required to just prevent the block from sliding down, then the value of \(\left|\vec{F}_1\right|-\left|\vec{F}_2\right|\) is _______. [Use \(g=10 \mathrm{~m} / \mathrm{s}^2\) ]

Distance of the centre of mass of a solid uniform cone from its vertex is \(z_0\) . If the radius of its base is \(R\) and its height is \(h\) then \(z_0\) is equal to

A block of mass \(m\) slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is : Given \({m}=8 \,{kg}, {M}=16\, {kg}\) Assume all the surfaces shown in the figure to be frictionless.

The cut-off voltage of the diodes (shown in figure) in forward bias is \(0.6 \,V\). The current through the resister of \(40 \,\Omega\) is ........... \(mA\).

At a given point of time the value of displacement of a simple harmonic oscillator is given as \(y = A \cos \left(30^{\circ}\right)\). If amplitude is \(40\,cm\) and kinetic energy at that time is \(200\, J\), the value of force constant is \(1.0 \times 10^{ x }\,Nm ^{-1}\). The value of \(x\) is ......

What equal length of an iron wire and a copper-nickel alloy wire, each of \(2 \; {mm}\) diameter connected parallel to give an equivalent resistance of \(3 \Omega ?\) (Given resistivities of iron and copper-nickel alloy wire are \(12 \;\mu \Omega {cm}\) and \(51\; \mu \Omega {cm}\) respectively) (in \({m}\))

An electron in the ground state of the hydrogen atom has the orbital radius of \(5.3 \times 10^{-11} \mathrm{~m}\) while that for the electron in third excited state is \(8.48 \times 10^{-10} \mathrm{~m}\). The ratio of the de Broglie wavelengths of electron in the excited state to that in the ground state is _______.

Match the LIST-I with LIST-II

List-I	List-II
A. Planck's constant	I. \(ML^2T^{-2}\)
B. Stopping potential	II. \(T^{-1}\)
C. Work function	III. \(ML^2T^{-1}\)
D. Threshold frequency	IV. \(ML^2T^{-3}A^{-1}\)

Choose the correct answer from the options given below: