A compound microscope is designed with two symmetric biconvex lenses. The objective lens is cut vertically, creating two identical plano-convex lenses. One of them is used in place of original objective lens. To retain same magnification keeping the object distance unchanged, the tube length has to be

If minimum possible work is done by a refrigerator in converting \(100\; grams\) of water at \(0^{\circ} C\) to ice, how much heat (in calories) is released to the surrounding at temperature \(27^{\circ} C\) (Latent heat of ice \(=80 Cal / gram\) ) to the nearest integer?

A particle of charge \(-q\) and mass \(m\) moves in a circle of radius \(r\) around an infinitely long line charge of linear density \(+\lambda\). Then time period will be given as _______. (Consider \(k\) as Coulomb's constant)

In photoelectric effect, the stopping potential \(\left(\mathrm{V}_0\right) \mathrm{v} / \mathrm{s}\) frequency \((\nu)\) curve is plotted.
( h is the Planck's constant and \(\phi_0\) is work function of metal )
(A) \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) is linear.
(B) The slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) curve \(=\frac{\phi_0}{\mathrm{~h}}\)
(C) h constant is related to the slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) line.
(D) The value of electric charge of electron is not required to determine \(h\) using the \(V_0 \mathrm{v} / \mathrm{s} \nu\) curve.
(E) The work function can be estimated without knowing the value of \(h\).
Choose the correct answer from the options given below :

In a collinear collision, a particle with an initial speed \(v_0\) strikes a stationary particle of the same mass. If the final total kinetic energy is \(50\%\) greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is:

A train moves towards a stationary observer with speed \(34\, m/s\). The train sounds a whistle and its frequency registered by the observer is \(f_1\). If the speed of the train is reduced to \(17\, m/s\), the frequency registered is \(f_2\). If speed of sound is \(340\, m/s\), then the ratio \(f_1/f_2\) is

\(c_P\) and \(c_V\) are specific heats at constant pressure and constant volume respectively. It is observed that \(c_P - c_V = a\) for hydrogen gas \(c_P - c_V = b\) for nitrogen gas The correct relation between \(a\) and \(b\) is

If the differential equation representing the family of all circles touching \(x-\) axis at the origin is \(\left( {{x^2} - {y^2}} \right)\frac{{dy}}{{dx}} = g\left( x \right)y\) , then \(g(x)\) equals

Let \(x=x(y)\) be the solution of the differential equation \(2 y \,e^{x / y^{2}} d x+\left(y^{2}-4 x e^{x / y^{2}}\right) d y=0\) such that \(x(1)=0\). Then, \(x(e)\) is equal to

The induced \(emf\) can be produced in a coil by \(A.\) moving the coil with uniform speed inside magnetic field \(B.\) moving the coil with non-uniform speed inside uniform magnetic field \(C.\) rotating the coil inside the uniform magnetic field \(D.\) changing the area of the coil inside the uniform magnetic field Choose the correct answer from the options given below:

Let a line \(L_1\) pass through the origin and be perpendicular to the lines
\(L_2: \vec{r} = (3+t)\hat{i} + (2t-1)\hat{j} + (2t+4)\hat{k}\) and
\(L_3: \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}\), \(t, s \in \mathbb{R}\).
If \((a, b, c)\), \(a \in \mathbb{Z}\), is the point on \(L_3\) at a distance of \(\sqrt{17}\) from the point of intersection of \(L_1\) and \(L_2\), then \((a+b+c)^2\) is equal to ________.

A survey shows that \(73 \%\) of the persons working in an office like coffee, whereas \(65 \%\) like tea. If \(x\) denotes the percentage of them, who like both coffee and tea, then \(x\) cannot be

Let \(y=y(x)\) be the solution of the differential equations \(\frac{d y}{d x}+\frac{5}{x\left(x^5+1\right)} y=\frac{\left(x^5+1\right)^2}{x^7}, x > 0\). If \(y(1)=2\), then \(y(2)\) is equal to