Two radioactive substances \(A\) and \(B\) have decay constants \(5\lambda \) and \(\lambda \) respectively. At \(t = 0\), a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become \((\frac {1}{e})^2\) will be

Dimension of \(\frac{1}{\mu_0 \varepsilon_0}\) should be equal to

In a series \(L R\) circuit \(X_{L}=R\) and power factor of the circuit is \(P _{1}\). When capacitor with capacitance \(C\) such that \(X _{ L }= X _{ C }\) is put in series, the power factor becomes \(P_{2}\). The ratio \(\frac{ P _{1}}{ P _{2}}\) is

A diatomic gas \((\gamma=1.4)\) does \(400 J\) of work when it is expanded isobarically. The heat given to the gas in the process is ............ \(J\)

The centre of a wheel rolling on a plane surface moves with a speed \(v_{0} .\) A particle on the rim of the wheel at the same level as the centre will be moving at a speed \(\sqrt{x}\, v_{0}\). Then the value of \(x\) is ...... .

In a potentiometer arrangement, a cell gives a balancing point at \(75\, cm\) length of wire. This cell is now replaced by another cell of unknown emf. If the ratio of the emf's of two cells respectively is \(3: 2\), the difference in the balancing length of the potentiometer wire in above two cases will be.........\(cm .\)

A brass wire of length 2 m and radius 1 mm at \(27^{\circ} C\) is held taut between two rigid supports. Initially it was cooled to a temperature of \(-43^{\circ} C\) creating a tension T in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to 1.4 T , is _________ \({ }^{\circ} C\)

If \((a, b)\) be the orthocentre of the triangle whose vertices are \((1,2),(2,3)\) and \((3,1)\), and \(I_1=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}, \mathrm{I}_2=\int_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}\) , then \(36 \frac{\mathrm{I}_1}{\mathrm{I}_2}\) is equal to :

A ball is released from a height \(h\). If \(t_{1}\) and \(t_{2}\) be the time required to complete first half and second half of the distance respectively. Then, choose the correct relation between \(t_{1}\) and \(t_{2}\).

For the given cyclic process \(CAB\) as shown for a gas, the work done is ..... \(J\)

The number of distinct real roots of \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) 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 \(x =\sin \left(2 \tan ^{-1} \alpha\right)\) and \(y =\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)\). If \(S =\left\{\alpha \in R : y ^{2}=1- x \right\}\), then \(\sum_{\alpha \in S } 16 \alpha^{3}\) is equal to \(...........\)