The temperature dependence of resistances of \(Cu\) and undoped \(Si\) in the temperature range \(300-400\;K\), is best described by

Radius of a certain orbit of hydrogen atom is \(8.48\mathring A \). If energy of electron in this orbit is \(\mathrm{E} / \mathrm{x}\), then \(\mathrm{x}=\) _______. (Given \(a_0=0.529 \mathring A, E=\) energy of electron in ground state)

Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media \(A\) and \(B\) are \(2.4 \times 10^8 m / s\) and \(2.7 \times 10^8 m / s\) respectively, then the value of critical angle is :

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

From the photoelectric effect experiment, following observations are made. Identify which of these are correct \(A.\) The stopping potential depends only on the work function of the metal. \(B.\) The saturation current increases as the intensity of incident light increases. \(C.\) The maximum kinetic energy of a photo electron depends on the intensity of the incident light. \(D.\) Photoelectric effect can be explained using wave theory of light. Choose the correct answer from the options given below:

A sphere of capacitance \(100\) pF is charged to a potential of \(100\) V. Another identical uncharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is \(\alpha \times 10^{-7}\) J. The value of \(\alpha\) is __________. (combined capacitance of spheres is \(200\) pF)

In thermodynamics, heat and work are

Let the foci and length of the latus rectum of an ellipse \(\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) be \(( \pm 5,0)\) and \(\sqrt{50}\), respectively. Then, the square of the eccentricity of the hyperbola \(\frac{\mathrm{x}^2}{\mathrm{~b}^2}-\frac{\mathrm{y}^2}{\mathrm{a}^2 \mathrm{~b}^2}=1\) equals

If \(\alpha ,\beta \ne 0\) and \(f\left( n \right) = {\alpha ^n} + {\beta ^n}\) and \(\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 - \alpha } \right)^2}\) \({\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}\) ,then \(K=\) . . . . . .

If \(y=y(x)\) is the solution curve of the differential equation \(\left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0\), \(x>2, y(4)=\frac{3}{2}\) and the slope of the curve is never zero, then the value of \(y(10)\) equals :

Let \(R\) be a relation on \(N \times N\) defined by \((a, b) R\) (c, d) if and only if \(a d(b-c)=b c(a-d)\). Then \(R\) is

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R.\) Assertion \(A:\) An electric fan continues to rotate for some time after the current is switched off. Reason \(R:\) Fan continuous to rotate due to inertia of motion. In the light of above statements, choose the most appropriate answer from the options given below.

The equation \(y = \sin \,x\,\sin \,\left( {x + 2} \right) - {\sin ^2}\,\left( {x + 1} \right)\) represents a straight line lying in