A particle executes simple harmonic motion and is located at \(x = a, b\)  and  \(c\) at times \(t_0, 2t_0\) and \(3t_0\) respectively. The frequency of the oscillation is

A particle of mass \(20\,g\) is released with an initial velocity \(5\,m/s\) along the curve from the point \(A,\) as shown in the figure. The point \(A\) is at height \(h\) from point \(B.\) The particle slides along the frictionless surface. When the particle reaches point \(B,\) its angular momentum about \(O\) will be ......... \(kg - m^2/s\). [Take \(g = 10\,m/s^2\) ]

A slanted object \(A B\) is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:

Substance \(A\) has atomic mass number \(16\) and half life of \(1\) day. Another substance \(B\) has atomic mass number \(32\) and half life of \(\frac{1}{2}\) day. If both \(A\) and \(B\) simultaneously start undergo radio activity at the same time with initial mass \(320\,g\) each, how many total atoms of \(A\) and \(B\) combined would be left after \(2\) days \(.........\times 10^{24}\)

A ball of mass \(200\,g\) rests on a vertical post of height \(20\,m\). A bullet of mass \(10\,g\), travelling in horizontal direction, hits the centre of the ball. After collision both travels independently. The ball hits the ground at a distance \(30\,m\) and the bullet at a distance of \(120\,m\) from the foot of the post. The value of initial velocity of the bullet will be \(............m/s\) (if \(\left.g =10 m / s ^2\right)\)

A body of mass \(5\,kg\) is moving with a momentum of \(10\,kg\,ms ^{-1}\). Now a force of \(2\,N\) acts on the body in the direction of its motion for \(5\,s\). The increase in the Kinetic energy of the body is \(...........J\).

The give graph shown variation (with distance \(r\) from centre) of

There are two identical chambers, completely thermally insulated from surroundings. Both chambers have a partition wall dividing the chambers in two compartments. Compartment \(1\) is filled with an ideal gas and Compartment \(3\) is filled with a real gas. Compartments \(2\) and \(4\) are vacuum . A small hole (orifice) is made in the partition walls and the gases are allowed to expand in vacuum Statement \(-1\) : No change in the temperature of the gas takes place when ideal gas expands in vacuum. However, the temperature of real gas goes down (cooling) when it expands in vacuum Statement \(-2\) : The internal energy of an ideal gas is only kinetic. The internal energy of a real gas is kinetic as well as potential

If \(x^{2}+9 y^{2}-4 x+3=0, x, y \in R\), then \(x\) and \(y\) respectively lie in the intervals:

Let \(f : R \rightarrow R\) be a function defined by \(f ( x )=\) \(\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}\), for some \(m\), such that the range of \(f\) is \([0,2]\). Then the value of \(m\) is \(............\)

Let \(E_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \mathrm{a}\,>\,\mathrm{b} .\) Let \(\mathrm{E}_{2}\) be another ellipse such that it touches the end points of major axis of \(E_{1}\) and the foci \(E_{2}\) are the end points of minor axis of \(E_{1}\). If \(E_{1}\) and \(E_{2}\) have same eccentricities, then its value is :

Let \(a_1, a_2 , a_3,.....\) be an \(A.P\), such that \(\frac{{{a_1} + {a_2} + .... + {a_p}}}{{{a_1} + {a_2} + {a_3} + ..... + {a_q}}} = \frac{{{p^3}}}{{{q^3}}};p \ne q\). Then \(\frac{{{a_6}}}{{{a_{21}}}}\) is equal to

The value of \(\left(2 .{ }^{1} P _{0}-3 .{ }^{2} P _{1}+4 .{ }^{3} P _{2}-\ldots .\right.\) up to \(51\) th term)+\(\left(1 !-2 !+3 !-\ldots . .\right.\) up to \(51^{\text {th }}\) term \()\) is equal to