A simple pendulum of string length 30 cm performs 20 oscillations in 10s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]

A \(9\, V\) battery with internal resistance of \(0.5\,\Omega \) is connected across an infinite network as shown in the figure. All ammeters \(A_1 , A_2, A_3\) and voltmeter \(V\) are ideal. Choose correct statement

A \(30\) cm long solenoid has \(10\) turns per cm and area of \(5\) cm\(^2\). The current through the solenoid coil varies from \(2\) A to \(4\) A in \(3.14\) s. The e.m.f. induced in the coil is \(\alpha\times 10^{-5}\) V. The value \(\alpha\) is ________.

The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength \(500\,nm\) and \(600\, nm\), each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in \(nm\) is

Consider two uniform discs of the same thickness and different radii \(R _{1}= R\) and \(R _{2}=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_{1}\) and \(I_{2}\), respectively, about their axes is \(I _{1}: I _{2}=1: 16\) then the value of \(\alpha\) is

A body of mass \(8\,kg\) and another of mass \(2\, kg\) are moving with equal kinetic energy. The ratio of their respective momenta will be.

A solenoid of \(1000\) turns per metre has a core with relative permeability \(500\). Insulated windings of the solenoid carry an electric current of \(5\, A\). The magnetic flux density produced by the solenoid is (permeability of free space \(\left.=4 \pi \times 10^{-7} H / m \right)\)

Let \(a_{1}, a_{2}, \ldots, a_{10}\) be an \(AP\) with common difference \(-3\) and \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}\) be a \(GP\) with common ratio \(2.\) Let \(c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .\) If \(c_{2}=12\) and \(\mathrm{c}_{3}=13\), then \(\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}\) is equal to ..... .

If the quadratic equation \((\lambda+2)x^2-3\lambda x+4\lambda=0\), \(\lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:

A circle with centre \((2,3)\) and radius \(4\) intersects the line \(x + y =3\) at the points \(P\) and \(Q\). If the tangents at \(P\) and \(Q\) intersect at the point \(S(\alpha, \beta)\), then \(4 \alpha-7 \beta\) is equal to \(........\).

Let \(A\) be a set of all \(4 -\)digit natural numbers whose exactly one digit is \(7 .\) Then the probability that a randomly chosen element of  \(A\) leaves remainder \(2\) when divided by \(5\) is ..... .

The radii of two planets \(A\) and \(B\) are \(R\) and \(4 R\) and their densities are \(\rho\) and \(\rho / 3\) respectively. The ratio of acceleration due to gravity at their surfaces \(\left(g_A: g_B\right)\) will be

The plane containing the line \(\frac{{x - 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{3}\) and parallel to the line \(\frac{x}{1} = \frac{y}{1} = \frac{z}{4}\) passes through the point