The distance of closest approach of an alpha particle to a nucleus when the alpha particle moves towards the nucleus with linear momentum \(\mathrm{P}\) is \(\mathrm{d}\). The distance of closest approach of alpha particle to nucleus, if the linear momentum of the alpha particle is \(1.5 \mathrm{P}\)

A charge \(\mathrm{Q}\) is to be divided between two objects. The values of the charges on the objects so that the electrostatic force between them will be maximum is

A block of metal 4 kg is in rest on a frictionless surface. It was targeted by a jet releasing water of \(2 \mathrm{~kg} \mathrm{~s}^{-1}\) at a speed of \(10 \mathrm{~ms}^{-1}\). The acceleration of the block is

In a metal, the charge carrier density is \(9.1 \times 10^{28} \mathrm{~m}^{-3}\) and its electrical conductivity is \(6.4 \times 10^7 \mathrm{~S} \mathrm{~m}^{-1}\). When an electric field of \(10 \mathrm{~N} \mathrm{C}^{-1}\) is applied to the metal, then the average time between two successive collisions of electrons in the metal is
\(\left(\right.\) Mass of electron \(=9.1 \times 10^{-31} \mathrm{~kg}\); charge of electron \(\left.=1.6 \times 10^{-19} \mathrm{C}\right)\)

If the earth stops rotating in its orbit about the sun, there will be variation in the weight of our bodies at

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.
\(\begin{array}{llll} \hline & \text { List- I } & & \text { List- II } \\ \hline \text {(A) } & X_L & \text { (i) } & 200 \Omega \\ \hline \text {(B) } & X_C & \text { (ii) } & 300 \Omega \\ \hline \text {(C) } & R & \text { (iii) } & 500 \Omega \\ \hline \text {(D) } & Z & \text { (iv) } & 600 \Omega \\ \hline \end{array}\)

The cold junction of a thermocouple is at \(0^{\circ} \mathrm{C}\). The thermo e.m.f produced in the thermocouple is given by an equation \(E=16 T-0.04 T^2\), where \(T\) is the temperature of the hot junction. The temperature of inversion and neutral temperature of the thermocouple are

The arithmetic mean of five natural numbers is 40 . The largest exceeds the smallest number by 10 . If \(\alpha\) is the maximum possible value for the largest of these 5 numbers, then the number of positive integral divisors of \(\alpha\) is

A transmitting antenna of height \(20 \mathrm{~m}\) and the receiving antenna of height \(h\) are separated by a distance of \(40 \mathrm{~km}\) for satisfactory communication in line of sight (Los) mode. Then the value of \(h\) is (Give, radius of earth is \(6400 \mathrm{~km}\).)

In the face centered unit cell, the lattice points are present at

For any real number \(\lambda \neq 1\), the centre of the circle that passes through \(\mathrm{A}(1, \lambda), \mathrm{B}(\lambda, 1)\) and \((\lambda, \lambda)\) is

\(\int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x=\)

The points whose position vectors are \(2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, 3 \mathbf{i}+4 \mathbf{j}+2 \mathbf{k}\) and \(4 \mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\) are the vertices of