An electric dipole with dipole moment $5\times {10}^{-7} \mathrm{C} \mathrm{m}$ is in the electric field of $2\times {10}^4 \mathrm{N} {\mathrm{C}}^{-1}$ at an angle of $60°$ with the direction of the electric field. The torque acting on the dipole is

In the equation \(\left(\mathrm{P}+\frac{a}{\mathrm{~V}^2}\right)(\mathrm{V}-b)=\mathrm{RT}\), where P is pressure, V is volume, T is temperature, R is universal gas constant, \(a\) and \(b\) are constants. The dimensions of \(a\) are

When an inductor of inductance \(\frac{6}{\pi} \mathrm{H}\), a capacitor of capacitance \(\frac{50}{\pi} \mu \mathrm{F}\) and resistor of resistance \(R\) are connected in series with an AC supply of rms voltage \(220 \mathrm{~V}\) and frequency \(50 \mathrm{~Hz}\), the rms current through the circuit is \(440 \mathrm{~mA}\). Match the inductive reactance, \(X_L\) the capacitive reactance, \(X_C\) the resistance \(R\) and the impedance \(Z\) of the circuit given in List-I with the corresponding values given in List-II.
\(\begin{array}{llll} \hline & \text { List- I } & & \text { List- II } \\ \hline \text {(A) } & X_L & \text { (i) } & 200 \Omega \\ \hline \text {(B) } & X_C & \text { (ii) } & 300 \Omega \\ \hline \text {(C) } & R & \text { (iii) } & 500 \Omega \\ \hline \text {(D) } & Z & \text { (iv) } & 600 \Omega \\ \hline \end{array}\)

A truck moving with a constant velocity \(12 \mathrm{~ms}^{-1}\) crosscs a car moving from rest with uniform acceleration \(2 \mathrm{~ms}^{-2}\). The distance the car has to travel from the starting point to cross the truck again is

A carrier wave of peak voltage \(60 \mathrm{~V}\) is used to transmit a message signal. Then the peak voltage of the modulating signal in order to have a modulation index of \(90 \%\) is

A toy car of mass \(100 \mathrm{~g}\) is moving with velocity of \((\hat{i}+2 \hat{j}-3 \hat{k}) m\), then the kinetic energy of the car is

A wire of resistance \(100 \Omega\) is stretched so that its length increases by \(20 \%\). The stretched wire is then bent in the form of a rectangle whose length and breadth are in the ratio \(3: 2\). The effective resistance between the ends of any diagonal of the rectangle is

If the sum of any two roots of the equation $x^3+p{x}^2+qx+r=0$ is zero, then

An integrating factor of the differential equation \(\left(x^2+1\right) \frac{d y}{d x}+x y=x^3\) is

In a Binomial distribution, if \(p=q\) and \(n \geq 4\), then \(2^n P(X=5)=\)

The volume of sphere is increasing at the rate of 1200 \(\mathrm{cu} \mathrm{cm} / \mathrm{s}\). The rate of increase in its surface area when the radius is \(10 \mathrm{~cm}\) is

\(\mathbf{a}=\mathbf{i}+\mathbf{j}-2 \mathbf{k} \Rightarrow \sum\{(\mathbf{a} \times \mathbf{i}) \times \mathbf{j}\}^2\) is equal to

The midpoint of the diagonal of a rectangle formed by \(x^2+5 x-6=0\) and \(y^2-8 y-20=0\) is