The temperature of an ideal gas in \(3-\)dimensions is \(300\, {K}\). The corresponding de-Broglie wavelength of the electron approximately at \(300\,{K}\), is \(....\,nm\) \(\left\lfloor{m}_{e}=\text { mass of electron }=9 \times 10^{-31} \,{kg}\right.\) \({h}=\text { Planck constant }=6.6 \times 10^{-34} {Js}\) \(\left.{k}_{{B}}=\text { Boltzmann constant }=1.38 \times 10^{-23}\, {JK}^{-1}\right]\)

Identify the valid statements relevant to the given circuit at the instant when the key is closed.

A. There will be no current through resistor \(R\).
B. There will be maximum current in the connecting wires.
C. Potential difference between the capacitor plates A and B is minimum.
D. Charge on the capacitor plates is minimum.
Choose the correct answer from the options given below:

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:

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is _______.

For extrinsic semiconductors; when doping level is increased;

At \(300\,K\), the rms speed of oxygen molecules is \(\sqrt{\frac{\alpha+5}{\alpha}}\) times to that of its average speed in the gas. Then, the value of \(\alpha\) will be (used \(\pi=\frac{22}{7}\) )

A sub-atomic particle of mass \(10^{-30} \mathrm{~kg}\) is moving with a velocity \(2.21 \times 10^6 \mathrm{~m} / \mathrm{s}\). Under the matter wave consideration, the particle will behave closely like _______. \(\left(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s}\right)\)

For plan electromagnetic waves propagating in the \(z-\) direction, which one of the following combination gives the correct possible direction for \(\vec E\) and \(\vec B\) field respectively?

The domain of the definition of the function \(f\left( x \right) = \frac{1}{{4 - {x^2}}} + \log \,\left( {{x^3} - x} \right)\) is

Let \(P=\{z \in \mathbb{C}:|z+2-3 i| \leq 1\}\) and \(Q=\{z \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}\). Let in \(\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|\) be maximum and minimum at \(z_1\) and \(z_2\) respectively. If \(\left|z_1\right|^2+2|z|^2=\alpha+\beta \sqrt{2}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) equals ...........

Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}x+a, & x<0 \\ |x-1|, & x \geq 0\end{array}\right.\) and \(g(x)=\left\{\begin{array}{cc}x+1, & x<0 \\ (x-1)^{2}+b, & x \geq 0\end{array}\right.\) where \(a , b\) are non-negative real numbers. If \((gof)\,( x )\) is continuous for all \(x \in R\), then \(a + b\) is equal to ...... .

On a linear temperature scale \(Y\), water freezes at \(- 160^o\, Y\) and boils at \(- 50^o\, Y\). On this \(Y\) scale , a temperature of \(340\, K\) would be read as........ \(^oY\)  ( water freezes at \(273\, K\) and boils at \(373\, K\))

The percentage error in the calculated volume of a sphere, if there is \(2\%\) error in its diameter measurement, is __________.