The potential energy of an object is $U\left(x\right)=\left(5x^2-4x^3\right) \mathrm{J}$, where $x$ is the position in meter. The position at which the force becomes zero is

The surface of a hill inclined at an angle 30° to the horizontal. A stone is thrown from the summit of the hill (point A) at an initial speed 10 m/s at angle 60° to the vertical. If the stone strikes the hill at point B as shown in the figure, the distance between A and B is (Take g = 10 m/s2)

A long wire carries a current of \(18 \mathrm{~A} \mathrm{kept}\) along the axis of a long solenoid of radius \(1 \mathrm{~cm}\). The field due to the solenoid is \(8.0 \times 10^{-3} \mathrm{~T}\). The magnitude of the resultant field at a point \(0.6 \mathrm{~mm}\) from the solenoid axis is (Assume, \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\) )

An object of mass \(2 m\) is projected with a speed of \(100 \mathrm{~ms}^{-1}\) at an angle \(\theta=\sin ^{-1}\left(\frac{3}{5}\right)\) to the horizontal. At the highest point, the object breaks into two pieces of same mass \(m\) and the first one comes to rest. The distance between the point of projection and the point of landing of the bigger piece (in metre) is (Given, \(g=10 \mathrm{~m} / \mathrm{s}^2\) )

When a sound wave of wavelength \(\lambda\) is propagating in a medium, the maximum velocity of the particle is equal to the wave velocity. The amplitude of wave is

The angle between force \(\vec{F}=3 \hat{i}+4 \hat{j}-5 \hat{k}\) and displacement \(\vec{d}=5 \hat{i}+4 \hat{j}+3 \hat{k}\) is

Two metal wires \(A\) and \(B\) have length \(L\) and \(3 L\) respectively. The radius of cross-sectional circular area of wire \(A\) and \(B\) are \(R\) and \(2 R\), respectively. These wires are joined end to end along their axis. When one end of the combined system is fixed and other end is pulled with a constant force \(F\), the elongation in both the wires is equal. If \(Y_A\) and \(Y_B\) are Young's modulus of wire \(A\) and \(B\), then the \(Y_B / Y_A\) is

Match the following table. \begin{array}{|c|c|c|c|} \hline & List-I & & List-II \ \hline A. & \begin{array}{l}\text { Michelson-Morley } \ \text { experiment }\end{array} & I. & \begin{array}{l}\text { The existence of } \ \text { anti-matter }\end{array} \ \hline B. & \begin{array}{l}\text { Stern-Gerlach } \ \text { experiment }\end{array} & II. & \begin{array}{l}\text { The existence of } \ \text { de-Broglie matter waves }\end{array} \ \hline C. & \begin{array}{l}\text { Davisson-Germer } \ \text { experiment }\end{array} & III. & Electrons have spins \ \hline D. & \begin{array}{l}\text { Anderson discovery } \ \text { of positron }\end{array} & \mathrm{IV}. & \begin{array}{l}\text { The non-existence of } \ \text { ether }\end{array} \ \hline \end{array} \(\begin{array}{llll}A & B & C & D\end{array}\)

Two towns \(\mathrm{X}\) and \(\mathrm{Y}\) are connected by a regular bus service. A bus leaves in either direction at every \(t=T\) minutes. A man moving with some speed in the direction \(\mathrm{X}\) to \(\mathrm{Y}\) finds that a bus goes past him every \(t=t_1\) minutes in the direction of his motion, and every \(t=t_2\) minutes in the opposite direction. Then \(T\) is given by

In Sun, the important source of energy is

Two identical wires, carrying equal currents are bent into circular coils \(A\) and \(B\) with 2 and 3 turns respectively. The ratio of the magnetic fields at the centres of the coils \(A\) and B is

A body of mass $1 \mathrm{kg}$ is executing simple harmonic motion (SHM). Its displacement $y$ (in $\mathrm{cm}$) at time t given by $y=\left[6\sin \left(100\mathrm{t}+\frac{\pi }{4}\right)\right]\mathrm{cm}.$ Its maximum kinetic energy is

A potentiometer balances at \(44 \mathrm{~cm}\) when a cell of internal resistance \(1 \Omega\) is in the secondary circuit. To obtain the balancing point at \(40 \mathrm{~cm}\), the resistance to be connected parallel to cell is