A rod of length \(50\,cm\) is pivoted at one end. It is raised such that if makes an angle of \(30^o\) fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in \(rad\,s^{-1}\) ) will be \((g = 10\,ms^{-2})\)

A particle of mass \(100\,g\) is projected at time \(t =0\) with a speed \(20\,ms ^{-1}\) at an angle \(45^{\circ}\) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time \(t=2\,s\) is found to be \(\sqrt{ K }\,kg\,m ^2 / s\). The value of \(K\) is \(............\) \(\left(\right.\) Take \(\left.g =10\,ms ^{-2}\right)\)

When a gas filled in a closed vessel is heated by raising the temperature by \(1^{\circ} C\), its pressure increase by \(0.4 \%\). The initial temperature of the gas is ..........\(K\)

The equivalent resistance between points \(A\) and \(B\) in the given network is ............ \(\Omega\)

A coil in the shape of an equilateral triangle of side \(10\, {cm}\) lies in a vertical plane between the pole pieces of permanent magnet producing a horizontal magnetic field \(20\, {mT}\). The torque acting on the coil when a current of \(0.2\, {A}\) is passed through it and its plane becomes parallel to the magnetic field will be \(\sqrt{{x}} \times 10^{-5} \,{Nm}\). The value of \({x}\) is ..... .

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modules of the material of the wire will :

In a meter bridge experiment null point is obtained at \(40\, cm\) from one end of the wire when resistance \(X\) is balanced against another resistance \(Y\). If \(X < Y\), then the new position of the null point from the same end, if one decides to balance a resistance of \(3X\) against \(Y\), will be close to .............. \(cm\)

Let \( C_{r} \) denote the coefficient of \( x^{r} \) in the binomial expansion of \( (1+x)^{n} \),\(n \in N , 0 \leq r \leq n\). If \( P_{n}=C_{0}-C_{1}+\frac{2^{2}}{3}C_{2}-\frac{2^{3}}{4}C_{3}+....+\frac{(-2)^{n}}{n+1}C_{n}, \) then the value of \( \sum_{n=1}^{25}\frac{1}{P_{2n}} \) equals.

Escape velocity of a body from earth is \(11.2 \mathrm{~km} / \mathrm{s}\). If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the plate is _______.

A bag contains \(8\) balls, whose colours are either white or black. \(4\) balls are drawn at random without replacement and it was found that \(2\) balls are white and other \(2\) balls are black. The probability that the bag contains equal number of white and black balls is :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.
Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor \(\left(R / \sqrt{R^2+\omega^2 L^2}\right)\), where \(\omega\) is frequency of the supply across resistor \(R\) and inductor L . If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.
In the light of the above statements, choose the most appropriate answer from the options given below :

Let \(L_1\) be the length of the common chord of the curves \(x^2 + y^2\, = 9\) and \(y^2\, = 8x\), and \(L_2\) be the length of the latus rectum of \(y^2\, = 8x\), then

\(\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{{\rm{sin}}\left( {\pi {{\cos }^2}x} \right)}}{{{x^2}}} = \)