Following statements related to radioactivity are given below \((A)\) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions. \((B)\) The number of un-decayed nuclei in the radioactive sample decays exponentially with time. \((C)\) Slope of the graph of \(\log _{e}\) (no. of undecayed nuclei) \(Vs\). time represents the reciprocal of mean life time \((\tau)\). \((D)\) Product of decay constant ( \(\lambda\) ) and half-life time \(\left(T_{1 / 2}\right)\) is not constant. Choose the most appropriate answer from the options given below

An unpolarized light of intensity \(I_o\) passes through polarizer and then through a certain optically active solution and finally it goes to analyser. If the angle between analyser and polariser is \(0°\) and intensity of light emerged from analyser is \(\dfrac{3}{8}I_o\), the angle of rotation of the light by the solution with respect to analyser is _______ degrees.

For a certain metal, when monochromatic light of wavelength \(\lambda\) is incident, the stopping potential for photoelectrons is \(3V_0\). When the same metal is illuminated by light of wavelength \(2\lambda\), then the stopping potential becomes \(V_0\). The threshold wavelength for photoelectric emission for the given metal is \(\alpha\lambda\). The value of \(\alpha\) is _______.

Water flows in a horizontal tube (see figure). The pressure of water changes by \(700\; \mathrm{Nm}^{-2}\) between \(\mathrm{A}\) and \(\mathrm{B}\) where the area of cross section are \(40\; \mathrm{cm}^{2}\) and \(20\; \mathrm{cm}^{2},\) respectively. Find the rate of flow of water through the tube. ........ \(\mathrm{cm}^{3} / \mathrm{s}\) (density of water \(=1000\; \mathrm{kgm}^{-3}\) )

Two materials having coefficients of thermal conductivity \(3K\) and \(K\) and thickness \(d\) and \(3d\), respectively, are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are  \(\theta_2\) and \(\theta_1\) respectively  \(\left( {\theta _2} > {\theta _1} \right)\) . The temperature at the interface is

At the interface between two materials having refractive indices \(n_1\) and \(n_2\), the critical angle for reflection of an em wave is \(\theta_{1 C}\). The \(n_2\) material is replaced by another material having refractive index \(n_3\) such that the critical angle at the interface between \(n_1\) and \(n_3\) materials is \(\theta_{2 \mathrm{C}}\). If \(\mathrm{n}_3\gt\mathrm{n}_2\gt\mathrm{n}_1 ; \frac{\mathrm{n}_2}{\mathrm{n}_3}=\frac{2}{5}\) and \(\sin \theta_{2 \mathrm{C}}-\sin \theta_{1 \mathrm{C}}=\frac{1}{2}\), then \(\theta_{1 \mathrm{C}}\) is :

An engine is attached to a wagon through a shock absorber of length \(1.5\, {m}\). The system with a total mass of \(40,000\, {kg}\) is moving with a speed of \(72\, {kmh}^{-1}\) when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by \(1.0\, {m}\). If \(90\, \%\) of energy of the wagon is lost due to friction, the spring constant is \(....\, \times 10^{5}\, {N} / {m}\)

A parallel plate capacitor of capacitance \(12.5 \mathrm{pF}\) is charged by a battery connected between its plates to potential difference of \(12.0 \mathrm{~V}\). The battery is now disconnected and a dielectric slab \(\left(\epsilon_{\mathrm{r}}=6\right)\) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is _______.\(\times 10^{-12} \mathrm{~J}\).

Let \(\alpha\) and \(\beta\) be the roots of \(x^{2}-3 x+p=0\) and \(\gamma\) and \(\delta\) be the roots of \(x^{2}-6 x+q=0 .\) If \(\alpha\) \(\beta, \gamma, \delta\) form a geometric progression. Then ratio \((2 q+p):(2 q-p)\) is 

With reference to the observations in photo-electric effect, identify the correct statements from below: \(A.\) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light. \(B.\) The value of saturation current increases on moving the source of light away from the metal surface. \(C.\) The maximum kinetic energy of photo-electrons decreases on decreasing the power of LED (light emitting diode) source of light. \(D.\) The immediate emission of photo-electrons out of metal surface can not be explained by particle nature of light/electromagnetic waves. \(E.\) Existence of threshold wavelength can not be explained by wave nature of light/electromagnetic waves. Choose the correct answer from the options given below:

If \(\left[ {\begin{array}{*{20}{c}}
1&1\\
0&1
\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}
1&2\\
0&1
\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}
1&3\\
0&1
\end{array}} \right]\,........\left[ {\begin{array}{*{20}{c}}
1&{n - 1}\\
0&1
\end{array}} \right]\, = \,\left[ {\begin{array}{*{20}{c}}
1&{78}\\
0&1
\end{array}} \right]\) then the inverse of \(\left[ {\begin{array}{*{20}{c}}
1&n\\
0&1
\end{array}} \right]\) is

Two identical cells, when connected either in parallel or in series gives same current in an external resistance \(5\,\Omega\). The internal resistance of each cell will be \(.............\,\Omega\).

Three urns \(A\), \(B\) and \(C\) contain \(7\) red, \(5\) black; \(5\) red, \(7\) black and \(6\) red, \(6\) black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn \(\mathrm{A}\) is :