A bullet of mass \(5\, g\), travelling with a speed of \(210\) \(m / s ,\) strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the bullet while the other half is converted into heat in the wood. The rise of temperature of the bullet if the specific heat of its material is \(0.030\, cal /\left( g -{ }^{\circ} C \right)\) \(\left(1\, cal =4.2 \times 10^{7}\, ergs \right)\) close to\(.......^oC\)

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

A charged particle going around in a circle can be considered to be a current loop. A particle of mass \(m\) carrying charge \(q\) is moving in a plane with speed \(v\) under the influence of magnetic field \(\overrightarrow{ B }\). The magnetic moment of this moving particle

A monkey of mass \(50\,kg\) climbs on a rope which can withstand the tension \((T)\) of \(350\,N\). If monkey initially climbs down with an acceleration of \(4\,m / s ^{2}\) and then climbs up with an acceleration of \(5\,m / s ^{2}\). Choose the correct option \(\left( g =10\,m / s ^{2}\right)\)

A particle is moving \(5\) times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is \(1.878 \times 10^{-4}\). The mass of the particle is close to

In a Young's double slit experiment, a laser light of \(560\,nm\) produces an interference pattern with consecutive bright fringes' separation of \(7.2\,mm\). Now another light is used to produce an interference pattern with consecutive bright fringes' separation of \(8.1\,mm\). The wavelength of second light is \(......nm\)

An a.c. source of angular frequency \(\omega\) is connected across a resistor \(R\) and a capacitor \(C\) in series. The current is observed as \(I\). Now the frequency of the source is changed to \(\omega/4\), (keeping the voltage unchanged) the current is found to be \(I/3\). The ratio of resistance to reactance at frequency \(\omega\) is

The total charge enclosed in an incremental volume of \(2 \times 10^{-9} \,{m}^{3}\) located at the origin is ...... \(nC,\) if electric flux density of its field is found as \(D=e^{-x} \sin y \hat{i}-e^{-x} \cos y \hat{j}+2 z \hat{k}\, C / m^{2}\)

The area enclosed between the curves \(y=x|x|\) and \(\mathrm{y}=\mathrm{x}-|\mathrm{x}|\) is :

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 ________.

Let the line \(\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}\) intersect the plane containing the lines \(\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}\) and \(4 a x-y+5 z-7 a=0=2 x -5 y - z -3, a \in R\) at the point \(P(\alpha, \beta, \gamma)\). Then the value of \(\alpha+\beta+\gamma\) equals\(...\)

Let \(S\) be the sum of all solutions (in radians) of the equation \(\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0\) in \([0,4 \pi]\) Then \(\frac{8 \mathrm{~S}}{\pi}\) is equal to ...... .

If two tangents drawn from a point \((\alpha, \beta)\) lying on the ellipse \(25 x^{2}+4 y^{2}=1\) to the parabola \(y^{2}=4 x\) are such that the slope of one tangent is four times the other, then the value of \((10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}\) equals