A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of \(10\,cm\). The separation between the two lenses is \(2\,cm\). The focal lengths of the component lenses

A cylinder of mass \(M_c\) and sphere of mass \(M_s\) are placed at points \(A\) and \(B\) of two inclines, respectively (See Figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio \(\frac{{\sin \,{\theta _c}}}{{\sin \,{\theta _s}}}\) is

When the temperature of a metal wire is increased from \(0^{\circ} \,C\) to \(10^{\circ}\, C\), its length increases by \(0.02 \% .\) The percentage change in its mass density will be closest to:

In \(1^{\text {st }}\) case, Carnot engine operates between temperatures \(300\,K\) and \(100\,K\). In \(2^{\text {nd }}\) case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in \(2^{\text {ad }}\) case) will be.

If earth has a mass nine times and radius twice to the of a planet \(P\). Then \(\frac{v_e}{3} \sqrt{x}\; ms ^{-1}\) will be the minimum velocity required by a rocket to pull out of gravitational force of \(P\), where \(v_e\) is escape velocity on earth. The value of \(x\) is

A capacitor of capacitance \(900\,\mu F\) is charged by a \(100\,V\) battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capacitor connected to negative plate of the charged capacitor. The loss of energy in this process is measured as \(x \times 10^{-2}\,J\). The value of \(x\) is \(..............\)

In the figure given below. a block of mass \(M =490\,g\) placed on a frictionless table is connected with two springs having same spring constant \(\left( K =2 N m ^{-1}\right)\). If the block is horizontally displaced through ' \(X\) 'm then the number of complete oscillations it will make in \(14 \pi\) seconds will be \(.........\)

The planet Mars has two moons, if one of them has a period \(7\, hours,\, 30\, minutes\) and an orbital radius of \(9.0 \times 10^{3}\, {km} .\) Find the mass of Mars. \(\left\{\operatorname{Given} \frac{4 \pi^{2}}{G}=6 \times 10^{11} {N}^{-1} {m}^{-2} {kg}^{2}\right\}\)

The energy of an electromagnetic wave contained in a small volume oscillates with

A particle is moving in a circular path of radius \(a,\) with a constant velocity \(v\) as shown in the figure.The centre of circle is marked by \('C'\). The angular momentum from the origin \(O\) can be written as

The electron in a hydrogen atom first jumps from the third excited state to the second excited state and subsequently to the first excited state. The ratio of the respective wavelengths, \(\lambda _1/\lambda _2\), of the photons emitted in this process is

Two polarisers \(P_1\) and \(P_2\) are placed in such a way that the intensity of the transmitted light will be zero. A third polariser \(P_3\) is inserted in between \(P_1\) and \(\mathrm{P}_2\), at the particular angle between \(\mathrm{P}_2\) and \(\mathrm{P}_3\). The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers \(\mathrm{P}_2\) and \(\mathrm{P}_3\) is :