A planet \((P_1)\) is moving around the star of mass \(2M\) in the orbit of radius \(R\). Another planet \((P_2)\) is moving around another star of mass \(4M\) in a orbit of radius \(2R\). Ratio of time periods of revolution of \(P_2\) and \(P_1\) is _______.

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights \(h_{sph}\) and \(h_{cyl}\) on the incline. The radio \(\frac{{{h_{sph}}}}{{{h_{cyl}}}}\) is given by

Two identical symmetric double convex lenses of focal length \(f\) are cut into two equal parts \(L_1, L_2\) by \(A B\) plane and \(L_3, L_4\) by \(X Y\) plane as shown in figure respectively. The ratio of focal lengths of lenses \(L_1\) and \(L_3\) is ________.

Unpolarised light of intensity \(32\,Wm ^{-2}\) passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is \(3\,Wm ^{-2}\), then the angle between pass axis of first two polaroids is \(............\,^{\circ}\)

In an experiment, a small steel ball falls through a Iiquid at a constant speed of \(10\, cm/s\). If the steel ball is pulled upward with a force equal to twice its effective weight, how fast will it move upward ? ......... \(cm/s\)

Given below are two statements : \(Statement\) \((I)\) : Planck's constant and angular momentum have same dimensions. \(Statement\) \((II)\) : Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below :

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

Match List \(- I\) (Electromagnetic wave type) with List \(- II\) (Its association/application) and select the correct option from the choices given below the lists

લિસ્ટ\(-I\)	લિસ્ટ\(-II\)
\(a\). Infrared waves	\(i\). To treat muscular strain
\(b\). Radio waves	\(ii\). For broadcasting
\(c\). \(X-\)rays	\(iii\). To detect fracture of bones
\(d\). Ultraviolet rays	\(iv\). Absorbed by the ozone layer of the atmosphere

\(a\)         \(b\)         \(c\)         \(d\)

If the coefficient of \(x ^{15}\) in the expansion of \(\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}\), where \(a\) and \(b\) are positive real numbers, then for each such ordered pair \((a, b) :\)

A coil is placed in magnetic field such that plane of coil is perpendicular to the direction of magnetic field. The magnetic flux through a coil can be changed: \(A.\) By changing the magnitude of the magnetic field within the coil. \(B.\) By changing the area of coil within the magnetic field. \(C.\) By changing the angle between the direction of magnetic field and the plane of the coil. \(D.\) By reversing the magnetic field direction abruptly without changing its magnitude. Choose the most appropriate answer from the options given below

A train is moving with a speed of \(12 \mathrm{~m} / \mathrm{s}\) on rails which are \(1.5 \mathrm{~m}\) apart. To negotiate a curve radius \(400 \mathrm{~m}\), the height by which the outer rail should be raised with respect to the inner rail is (Given, \(g=\) \(10 \mathrm{~m} / \mathrm{s}^2\) ) :

Two blocks (\(P\) and \(Q\)) with respectively masses \(2\) kg and \(1.5\) kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube (\(S\)), as shown in the figure below. Block \(P\) is positioned on the top surface which has no friction and block \(Q\) is in contact with side-surface, having coefficient friction \(\mu\). The cube (\(S\)) moves towards the right with acceleration of \(\dfrac{g}{2}\), where \(g\) is gravitational acceleration. During this movement the block \(P\) and \(Q\) remain stationary. The value of \(\mu\) is _______. (take \(g = 10\) m/s\(^2\))

If two distinct point \(Q , R\) lie on the line of intersection of the planes \(- x +2 y - z =0\) and \(3 x-5 y+2 z=0\) and \(P Q=P R=\sqrt{18}\) where the point \(P\) is \((1,-2,3)\), then the area of the triangle \(PQR\) is equal to