The longest wavelength associated with Paschen series is _______. (Given \(R_H=1.097 \times 10^7 \mathrm{SI}\) unit)

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:

A particle moving with kinetic energy \(E\) has de Broglie wavelength \(\lambda\). If energy \(\Delta \mathrm{E}\) is added to its energy, the wavelength become \(\frac {\lambda}{2} .\) Value of \(\Delta \mathrm{E},\) is 

Velocity \((v)\) and acceleration \((a)\) in two systems of units \(1\) and \(2\) are related as \(V _{2}=\frac{ n }{ m ^{2}} v _{1}\) and \(a_{2}=\frac{a_{1}}{m n}\) respectively. Here \(m\) and \(n\) are constants. The relations for distance and time in two systems respectively are

A bulb and a capacitor are connected in series across an AC supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb:

In an experiment with a closed organ pipe, it is filled with water by \(\left(\frac{1}{5}\right)\) th of its volume. The frequency of the fundamental note will change by

Three masses \(m_1 = 4\) kg, \(m_2 = 4\) kg and \(m_3 = 6\) kg are suspended from a fixed smooth frictionless pulley as shown in the figure below. The value of \(T_1/T_2\) is _____.
(take \(g = 10\) m/s\(^2\))

A thin rod having a length of \(1\; m\) and area of cross-section \(3 \times 10^{-6}\,m ^2\) is suspended vertically from one end. The rod is cooled from \(210^{\circ}\,C\) to \(160^{\circ}\,C\). After cooling, a mass \(M\) is attached at the lower end of the rod such that the length of rod again becomes \(1\,m\). Young's modulus and coefficient of linear expansion of the rod are \(2 \times 10^{11} Nm ^{-2}\) and \(2 \times 10^{-5} K ^{-1}\), respectively. The value of \(M\) is \(.......kg .\left(\right.\) Take \(\left.g=10\,m s ^{-2}\right)\)

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at \(A\) and \(B\) are \(6 cm^2\) and \(3 cm^2\) respectively.
The rate of flow will be ___________ \(cm ^3 / s\). (take \(g =10 m / s ^2\) )

Two light beams of intensities \(4\,I\) and \(9\,I\) interfere on a screen. The phase difference between these beams on the screen at point \(A\) is \(zero\) and at point \(B\) is \(\pi\). The difference of resultant intensities, at the point \(A\) and \(B\), will be \(....I\).