On the \(x-\)axis and a dsitance \(x\) from the origin, the gravitational field due to a mass distribution is given by \(\frac{A x}{\left(x^{2}+a^{2}\right)^{3 / 2}}\) in the \(x-\)direction. The magnitude of gravitational potential on the \(x-\)axis at a distance \(x,\) taking its value to be zero at infinity, is

The maximum height reached by a projectile is \(64 \mathrm{~m}\). If the initial velocity is halved, the new maximum height of the projectile is _______ \(\mathrm{m}\).

Two coherent monochromatic light beams of intensities \(I\) and \(4I\) are superimposed. The difference between maximum and minimum possible intensities in the resulting beam is \(x\)I. The value of \(x\) is _______.

Using Bohr's model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the \(2^{\text{nd}}\) and \(4^{\text{th}}\) orbits of hydrogen atom _______.

The increase in pressure required to decrease the volume of a water sample by \(0.2 \%\) is \(\mathrm{P} \times 10^5 \mathrm{Nm}^{-2}\). Bulk modulus of water is \(2.15 \times 10^9 \mathrm{Nm}^{-2}\). The value of P is ________.

A person is swimming with a speed of \(10\, m /s\) s at an angle of \(120^{\circ}\) with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is \('x'\) \(m / s\). The value of \('x'\) to the nearest integer is ..............

Four closed surfaces and corresponding charge distributions are shown below Let the respective electric fluxes through the surfaces be \({\phi _1},{\phi _2},{\phi _3}\) and \({\phi _4}\) . Then

\(2.{}^{20}{C_0} + 5.{}^{20}{C_1} + 8.{}^{20}{C_2} + 11.{}^{20}{C_3} + ......62.{}^{20}{C_{20}}\) is equal to

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

Let \([x]\) denote the greatest integer \(\leq x\), where \(x \in R\). If the domain of the real valued function \(\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}\) is \((-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}\), then the value of \(\mathrm{a}+\mathrm{b}+\mathrm{c}\) is:

Let \(O\) be the origin and \(OP\) and \(OQ\) be the tangents to the circle \(x^2+y^2-6 x+4 y+8=0\) at the point \(P\) and \(Q\) on it. If the circumcircle of the triangle OPQ passes through the point \(\left(\alpha, \frac{1}{2}\right)\), then a value of \(\alpha\) is

The above \(P-V\) diagram represents the thermodynamic cycle of an engine, operating with an ideal monatomic gas. The amount of heat, extracted from the source in a single cycle is

Match the List\(-I\) with List\(-II\).

List \(I\)	List \(II\)
\(A\) \(AC\) generator	\(I\) Presence of both \(L\) and \(C\)
\(B\) Transformer	\(II\) Electromagnetic Induction
\(C\)Resonance phenomenon to occur	\(III\) Quality factor
\(D\)Sharpness of resonance	\(IV\) Mutual Inductance

Choose the correct answer from the options given below: