An antenna is placed in a dielectric medium of dielectric constant \(6.25\). If the maximum size of that antenna is \(5.0\, mm\). it can radiate a signal of minimum frequency of \(GHz .\) (Given \(\mu_{ r }=1\) for dielectric medium)

The magnetic field of a plane electromagnetic wave is given by \(\vec B\, = {B_0}\hat i\,[\cos \,(kz - \omega t)]\, + \,{B_1}\hat j\,\cos \,(kz - \omega t)\) where \({B_0} = 3 \times {10^{-5}}\,T\) and \({B_1} = 2 \times {10^{-6}}\,T\). The rms value of the force experienced by a stationary charge \(Q = 10^{-4} \,C\) at \(z = 0\) is closet to

If a copper wire is stretched to increase its length by \(20 \% \). The percentage increase in resistance of the wire is \(..........\%\)

A straight rod of length \(L\) extends from \(x = a\) to \(x = L + a\). The gravitational force it exerts on a point mass \(‘m’\) at \(x = 0\), if the mass per unit length of the rod is \(A + Bx^2\), is given by

The displacement of simple harmonic oscillator after \(3\) seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is \(\dots \; s\)

In the given figure, there is a circuit of potentiometer of length \(A B=10 \,{m}\). The resistance per unit length is \(0.1 \,\Omega\) per \({cm}\). Across \({AB}\), a battery of emf \({E}\) and internal resistance ' \({r}^{\prime}\) is connected. The maximum value of emf measured by this potentiometer is : (In \(V\))

Young's modulus is determined by the equation given by \(\mathrm{Y}=49000 \frac{\mathrm{m}}{\ell} \frac{\text { dyne }}{\mathrm{cm}^2}\) where \(\mathrm{M}\) is the mass and \(\ell\) is the extension of wre used in the experiment. Now error in Young modules \((\mathrm{Y})\) is estimated by taking data from \(M-\ell\) plot in graph paper. The smallest scale divisions are \(5 \mathrm{~g}\) and \(0.02\) \(\mathrm{cm}\) along load axis and extension axis respectively. If the value of \(M\) and \(\ell\) are \(500 \mathrm{~g}\) and \(2 \mathrm{~cm}\) respectively then percentage error of \(\mathrm{Y}\) is _______.

The value of the integral \(\int \limits_{0}^{1} \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}\) is:

Let \(\quad S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.\) is a real number \(\}\). Then which of the following is NOT correct?

A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by 30° from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is ________ mm. (Take \( \pi=22/7 \))

Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by 15 cm length of wire Q is ___________.

\(\left(\mu_0=4 \pi \times 10^{-7} T . m / A \right)\)

The number of real roots of the equation \(e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0\) is:

Given below are two statements: Statement \(I :\) Pressure in a reservoir of water is same at all points at the same level of water. Statement \(II :\) The pressure applied to enclosed water is transmitted in all directions equally. In the light of the above statements, choose the correct answer from the options given below: