For a series \(LCR\) circuit, \(I\) vs \(\omega\) curve is shown \((A)\) To the left of \(\omega_{r}\), the circuit is mainly capacitive. \((B)\) To the left of \(\omega_{r}\), the circuit is mainly inductive. \((C)\) At \(\omega_{r}\), impedance of the circuit is equal to the resistance of the circuit. \((D)\) At \(\omega_{t}\), impedance of the circuit is \(0\). Choose the most appropriate answer from the options given below

Two cars are travelling towards each other at speed of \(20 \mathrm{~m} \mathrm{~s}^{-1}\) each. When the cars are \(300 \mathrm{~m}\) apart, both the drivers apply brakes and the cars retard at the rate of \(2 \mathrm{~m} \mathrm{~s}^{-2}\). The distance between them when they come to rest is _______.

The value of acceleration due to gravity at Earth’s surface is \(9.8\, m\,s^{-2}\). The altitude above its surface at which the acceleration due to gravity decreases to \(4.9\, m\,s^{-2}\), is close to: (Radius of earth \(= 6.4\times10^6\, m\))

A soap bubble is blown to a diameter of \(7 \mathrm{~cm}\). \(36960 \mathrm{erg}\) of work is done in blowing it further. If surface tension of soap solution is \(40 \mathrm{dyne} / \mathrm{cm}\) then the new radius is ______ \(\mathrm{cm}\). Take \(:\left(\pi=\frac{22}{7}\right)\).

A ceiling fan having 3 blades of length \(80 \mathrm{~cm}\) each is rotating with an angular velocity of \(1200\  \mathrm{rpm}\). The magnetic field of earth in that region is \(0.5 \ \mathrm{G}\) and angle of dip is \(30^{\circ}\). The emf induced across the blades is \(N \pi \times 10^{-5} \mathrm{~V}\). The value of \(\mathrm{N}\) is _______.

A particle of mass \(1\, {kg}\) is hanging from a spring of force constant \(100\, {Nm}^{-1 .}\) The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period \({T}\). The time when the kinetic energy and potential energy of the system will become equal, is \(\frac{{T}}{{x}}\). The value of \({x}\) is ..... .

A thin biconvex lens is prepared from the glass (\(\mu=1.5\)) both curved surfaces of which have equal radii of \(20\) cm each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ________ cm.

In the reported figure, heat energy absorbed by a system in going through a cyclic process is \(......\,\pi J\)

Two ships \(A\) and \(B\) are sailing straight away from a fixed point \(O\) along routes such that \(\angle AOB\) is always \(120^o\) . At a certain instance, \(OA\, = 8\, km\), \(OB\, = 6\, km\) and the ship \(A\) is sailing at the rate of \(20\, km/hr\) while the ship \(B\) sailing at the rate of \(30\, km/hr\). Then the distance between \(A\) and \(B\) is changing at the rate (in \(km/hr\))

Read the following statements: \(A.\) When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of the liquid becomes twice. \(B.\) Two bodies \(P\) and \(Q\) having equal surface areas are maintained at temperature \(10^{\circ}\,C\) and \(20^{\circ}\,C\). The thermal radiation emitted in a given time by \(P\) and \(Q\) are in the ratio \(1: 1.15\) \(C.\) A carnot Engine working between \(100\,K\) and \(400\,K\) has an efficiency of \(75 \%\) \(D.\) When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice. Choose the correct answer from the options given below :

Let the shortest distance between the lines \(L : \frac{ x -5}{-2}=\frac{ y -\lambda}{0}=\frac{ z +\lambda}{1}, \lambda \geq 0\) and \(L _1: x +1= y -\) \(1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\), then which of the following is NOT possible?

The coefficient of \(x^{4}\) in the expansion of \(\left(1+x+x^{2}+x^{3}\right)^{6}\) in powers of \(x,\) is

A wire of resistance \(R\) is bent to form a square \(ABCD\) as shown in the figure. The effective resistance between \(E\) and \(C\) is ( \(E\) is mid-point of arm \(CD\) )