Using monochromatic light of wavelength \(\lambda \), an experimentalist sets up the Young 's double slit experiment in three ways as shown. If she observes that \(y = \beta '\), the wavelength of light used is.....\(nm\)

The rain drop of mass \(1\) g, starts with zero velocity from a height of \(1\) km. It hits the ground with a speed of \(5\) m/s. The work done by the unknown resistive force is _______ J. (take \(g = 10\) m/s\(^2\))

If momentum \((P),\) area \((A)\) and time \((T)\) are taken to be the fundamental quantities then the dimensional formula for energy is :

The temperature of a gas having \(2.0 \times 10^{25}\) molecules per cubic meter at \(1.38 \mathrm{~atm}\) (Given, \(\mathrm{k}=\) \(\left.1.38 \times 10^{-23} \mathrm{JK}^{-1}\right)\) is _______.

Identify the pair of physical quantities which have different dimensions 

Given below are two statements : one is labeled as Assertion \(A\) and the other is labeled as Reason \(R\) Assertion \(A:\) Photodiodes are used in forward bias usually for measuring the light intensity. Reason \(R:\) For a \(p-n\) junction diode, at applied voltage \(V\) the current in the forward bias is more than the current in the reverse bias for \(\left|V_z\right| > \pm V \geq\left|V_0\right|\) where \(V_0\) is the threshold voltage and \(V _{ z }\) is the breakdown voltage. In the light of the above statements, choose the correct answer from the options given below

A real gas within a closed chamber at \(27^{\circ} \mathrm{C}\) undergoes the cyclic process as shown in figure. The gas obeys \(P V^3=\mathrm{RT}\) equation for the path \(A\) to \(B\). The net work done in the complete cycle is _______.(assuming \(R=8 \mathrm{~J} / \mathrm{molK}\) )

If the constant term in the expansion of \(\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0\), is \(\alpha \times 2^8 \times \sqrt[5]{3}\), then \(25 \alpha\) is equal to :

A boy’s catapult is made of rubber cord which is \(42\, cm\) long, with \(6\, mm\) diameter of cross -section and of negligible mass. The boy keeps a stone weighing \(0.02\, kg\) on it and stretches the cord by \(20\, cm\) by applying a constant force. When released, the stone flies off with a velocity of \(20\, ms^{-1}\). Neglect the change in the area of cross section of the cord while stretched. The Young’s modulus of rubber is closest to

A poly-atomic molecule ( \(C_V=3 R, C_P=4 R\), where \(R\) is gas constant) goes from phase space point \(\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^5 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5 \times 10^4 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^4\right.\) \(\left.\mathrm{Pa}, \mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^3\right)\). A to B is an adiabatic path and \(B\) to \(C\) is an isothermal path. The net heat absorbed per unit mole by the system is :

Consider \(\mathrm{I}_1\) and \(\mathrm{I}_2\) are the currents flowing simultaneously in two nearby coils 1 & 2, respectively. If \(\mathrm{L}_1=\) self inductance of coil \(1, \mathrm{M}_{12}=\) mutual inductance of coil 1 with respect to coil 2, then the value of induced emf in coil 1 will be ________.

If \(f(x)\) is a non-zero polynomial of degree four, having local extreme points at \(x = -1, 0, 1;\) then the set \(S = \{x \in R; f(x) = f(0)\}\) contains exactly

A gas balloon is going up with a constant velocity of \(10\) m/s. When this balloon reached a height of \(75\) m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ________ m. (Take \(g=10\) m/s\(^2\))