A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is$\frac{nq}{6{\epsilon }_0}$  (in $\mathrm{SI}$ units). The value of $n$ is _________.

In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are $2.7\times {10}^{-8}\mathrm{\Omega }\mathrm{}\mathrm{m}$ and $1.0\times {10}^{-7}\mathrm{\Omega }\mathrm{}\mathrm{m}$ , respectively. The electrical resistance between the two faces P and Q of the composite bar is

A cylindrical cavity of diameter a exists inside a cylinder of diameter \(2 a\) as shown in the figure. Both the cylinder and the cavity are infinity long. A uniform current density \(J\) flows along the length. If the magnitude of the magnetic field at the point \(P\) is given by \(\frac{N}{12} \mu_{0} a J\), then the value of \(N\) is

A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.

When this pulse reaches the other end of the pipe,

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When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.Question :
After the drop detaches, its surface energy is

A rigid uniform bar $AB$ of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\mathrm{\theta }$. Which of the following statements about its motion is correct?

A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $\theta ={30}^o$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are:

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

Consider a titration of potassium dichromate solution with acidified Mohr's salt solution using diphenylamine as indicator. The number of moles of Mohr's salt required per mole of dichromate is

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

Question:
The speed of the block at point \(C\), immediately before it leaves the second incline is

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

The maximum value of the expression \(\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta}\) is

Scheme $1$ and $2$ describe the conversion of $\mathrm{P}$ to $\mathrm{Q}$ and $\mathrm{R}$ to $\mathrm{S},$ respectively. Scheme $3$ describes the synthesis of $\mathrm{T}$ from $\mathrm{Q}$ and $\mathrm{S}.$ The total number of $\mathrm{B}\mathrm{r}$ atoms in a molecule of $\mathrm{T}$ is
Scheme $1$:

Scheme $2$:

Scheme$3$: