A particular hydrogen - like ion emits the radiation of frequency \(3 \times 10^{15} \mathrm{~Hz}\) when it makes transition from \(n=2\) to \(n=1\). The frequency of radiation emitted in transition from \(n=3\) to \(n=1\) is \(\frac{x}{9} \times 10^{15} \mathrm{~Hz}\), when \(x=\) = _____.

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:

In a Young's double slit experiment, the source is white light. One of the slits is covered by red filter and another by a green filter. In this case...

Match List-I with List-II.

List-I	List-II
(A) Mass density	(I) [ML2T-3]
(B) Impulse	(II) [MLT-1]
(C) Power	(III) [ML2T0]
(D) Moment of inertia	(IV) [ML-3T0]

Choose the correct answer from the options given below :

Match List \(I\) with List \(II\)

List \(I\)	List \(II\)
\(A\) Troposphere	\(I\) Approximate \(\quad 65-75\, km\) over Earth's surface
\(B\) E-Part of Stratosphere	\(II\) Approximate \(300\,km\) over Earth's surface
\(C\) \(F _2\)-Part of Thermosphere	\(III\)  Approximate \(10\,km\) over Earth's surface
\(D\) \(F _2\)-Part of Thermosphere	\(IV\) Approximate \(100\,km\) over Earth's surface

Choose the correct answer from the options given below:

The resistance per centimeter of a meter bridge wire is \(\mathrm{r}\), with \(\mathrm{X}\  \Omega\) resistance in left gap. Balancing length from left end is at \(40 \mathrm{~cm}\) with \(25\  \Omega\) resistance in right gap. Now the wire is replaced by another wire of \(2 \mathrm{r}\) resistance per centimeter. The new balancing length for same settings will be at _______.

The least count of the main scale of a vernier callipers is \(1\, mm\). Its vernier scale is divided into \(10\) divisions and coincide with \(9\) divisions of the main scale. When jaws are touching each other, the \(7\) th division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between \(3.1\, cm\) and \(3.2\, cm\) and \(4^{th}\) \(VSD\) coincides with a main scale division.The length of the cylinder is \(.....cm\) (\(VSD\) is vernier scale division)

Three distinct numbers are selected randomly from the set \(\{1,2,3, \ldots \ldots, 40\}\). If the probability, that the selected numbers are in an increasing G.P. is \(\frac{m}{n}\), \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to _____.

Distance of the centre of mass of a solid uniform cone from its vertex is \(z_0\) . If the radius of its base is \(R\) and its height is \(h\) then \(z_0\) is equal to

The mean of a data set consisting of \(20\) observations is \(40\). If one observation \(53\) was wrongly recorded as \(33\), then the correct mean will be

The value of \(\int\limits_{ - \pi /2}^{\pi /2} {\frac{{dx}}{{\left[ x \right] + \left[ {\sin \,x} \right] + 4}}} \) where \([t]\) denotes the greatest integer less than or equal to \(t\), is

Let \(x=2\) be a root of the equation \(x^2+p x+q=0\) and \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos \left(x^2-4 p x+q^2+8 q+16\right)}{(x-2 p)^4}, & x \neq 2 p \\ 0, & x=2 p\end{array}\right.\) Then \(\lim _{x \rightarrow 22^{+}}[f(x)]\) where [. ] denotes greatest integer function, is \(........\)

Let \(a , b , c\) be three distinct positive real numbers such that \((2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}\) and \(b^{\log _e 2}=a^{\log _e c}\). Then \(6 a+5 b c\) is equal to \(........\).