An electromagnetic wave of frequency \(1\times10^{14}\, hertz\) is propagating along \(z-\) axis. The amplitude of electric field is \(4\, V/m\) . lf \({\varepsilon_0}=\, 8.8\times10^{-12}\, C^2/Nm^2\) , then average energy density of electric field will be:

Distance between an object and three times magnified real image is 40 cm . The focal length of the mirror used is _________ cm .

The resistance \(R =\frac{ V }{ I },\) where \(V =(50 \pm 2) \;V\) and \(I=(20 \pm 0.2)\;A.\) The percentage error in \(R\) is \(x\%\). The value of \(x\) to the nearest integer is .........

A ball is dropped from the top of a \(100\; m\) high tower on a planet. In the last \(\frac{1}{2}\;s \) before hitting the ground, it covers a distance of \(19\; \mathrm{m}\). Acceleration due to gravity (in \(\mathrm{ms}^{-2}\) ) near the surface on that planet is

A \(100 \,\Omega\) resistance, a \(0.1\, \mu {F}\) capacitor and an inductor are connected in series across a \(250 \,{V}\) supply at variable frequeny. Calculate the value of inductance of inductor at which resonance will occur. Given that the resonant frequency is \(60\, {Hz}\). (In \({H}\))

Two electric dipoles of dipole moments \(1.2 \times 10^{-30}\,cm\) and \(2.4 \times 10^{-30}\,cm\) are placed in two difference uniform electric fields of strengths \(5 \times 10^{4}\,NC ^{-1}\) and \(15 \times 10^{4}\,NC ^{-1}\) respectively. The ratio of maximum torque experienced by the electric dipoles will be \(\frac{1}{ x }\). The value of \(x\) is \(.....\)

A tightly wound long solenoid carries a current of 1.5 A . An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns . The number of turns per metre in the solenoid is ________.
[Take mass of electron \(\mathrm{m}_{\mathrm{e}}=9 \times 10^{-31} \mathrm{~kg}\), charge of electron \(\left|\mathrm{q}_{\mathrm{e}}\right|=1.6 \times 10^{-19} \mathrm{C}\),
\(\left.\mu_0=4 \pi \times 10^{-7} \frac{\mathrm{~N}}{\mathrm{~A}^2}, 1 \mathrm{~ns}=10^{-9} \mathrm{~s}\right]\)

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 :

The distance of the point \((1,-2,3)\) from the plane \(x-y+z=5\) measured parallel to the line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\) is

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of \(81 \mathrm{~cm}^3 / \mathrm{min}\) and the thickness of the ice-cream layer decreases at the rate of \(\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}\). The surface area (in \(\mathrm{cm}^2\) ) of the chocolate ball (without the ice-cream layer) is :

The energy released when \(\dfrac{7}{17.13}\) kg of \(^{7}_{3}\text{Li}\) is converted into \(^{4}_{2}\text{He}\) by proton bombardment is \(\alpha \times 10^{32}\) eV. The value of \(\alpha\) is _______. (Nearest integer) (Mass of \(^{7}_{3}\text{Li} = 7.0183\) u, mass of \(^{4}_{2}\text{He} = 4.004\) u, mass of proton \(= 1.008\) u and \(1\) u \(= 931\) MeV/c\(^2\) and Avogadro number \(= 6.0 \times 10^{23}\))

Let \(f(x) = \begin{cases} x^3 + 8 ; & x < 0 \\ x^2 - 4 ; & x \geq 0 \end{cases}\) and \(g(x) = \begin{cases} (x-8)^{1/3} ; & x < 0 \\ (x+4)^{1/2} ; & x \geq 0 \end{cases}\).
Then the number of points, where the function \(g \circ f\) is discontinuous, is __________.

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed \(v,\) he sees that rain drops are coming at an angle \(60^{\circ}\) from the horizontal. On further increasing the speed of the car to \((1+\beta) v ,\) this angle changes to \(45^{\circ} .\) The value of \(\beta\) is close to\(......\)