Two particles of equal mass \(\mathrm{m}\) have respective initial velocities \(u\hat{i}\) and \(u\left(\frac{\hat{\mathrm{i}}+ \hat{\mathrm{j}}}{2}\right) .\) They collide completely inelastically. The energy lost in the process is

Two identical springs of spring constant \('2k'\) are attached to a block of mass \(m\) and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this sytem is ...... .

In the given figure, the face \(AC\) of the equilateral prism is immersed in a liquid of refractive index ' \(n\) '. For incident angle \(60^{\circ}\) at the side \(AC\), the refracted light beam just grazes along face \(AC\). The refractive index of the liquid \(n =\frac{\sqrt{ x }}{4}\). The value of \(x...\) is (Given refractive index of glass \(=1.5\) )

A body moves on a frictionless plane starting from rest. If \(\mathrm{S}_{\mathrm{n}}\) is distance moved between \(\mathrm{t}=\mathrm{n}-1\) and \(\mathrm{t}\) \(=\mathrm{n}\) and \(\mathrm{S}_{\mathrm{n}-1}\) is distance moved between \(\mathrm{t}=\mathrm{n}-2\) and \(t=n-1\), then the ratio \(\frac{S_{n-1}}{S_n}\) is \(\left(1-\frac{2}{x}\right)\) for \(n\) \(=10\). The value of \(x\) is _______.

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants \(G, h\) and \(c\) . Which of the following correctly gives the Planck length?

The escape velocity of a body on a planet  \('A'\)  is \(12 \,kms ^{-1}\). The escape velocity of the body on another planet  \('B'\) , whose density is four times and radius is half of the planet  \('A'\), is ................... \(kms ^{-1}\)

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

Bag \(I\) contains \(3\) red,\(4\) black and \(3\) white balls and Bag \(II\) contains \(2\) red,\(5\) black and \(2\) white balls. One ball is transferred from Bag \(I\) to Bag \(II\) and then a ball is draw from Bag \(II\). The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red,is.

A closed circular tube of average radius \(15\,cm\), whose inner walls are rough, is kept in vertical plane. A block of mass \(1\,kg\) just fit inside the tube. The speed of block is \(22\,m / s\), when it is introduced at the top of tube. After completing five oscillations, the block stops at the bottom region of tube. The work done by the tube on the block is \(......J\). [Given \(g=10\,m / s ^2\) ]

Let \([ x ]\) be the greatest integer \(\leq x\). Then the number of points in the interval \((-2,1)\), where the function \(f(x)=|[x]|+\sqrt{x-[x]}\) is discontinuous is \(........\).

Let \(A=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) and \(B=\left[\begin{array}{ccc}9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2}\end{array}\right]\), then the value of \(A ^{\prime} BA\) is.

The solution curve of the differential equation \(y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0 \text { passing }\) through the point\((\mathrm{e}, 1)\) is

If a wave gets refracted into a denser medium, then which of the following is true?