As shown in the figure, two parallel plate capacitors having equal plate area of \(200\,cm ^2\) are joined in such a way that \(a \neq b\). The equivalent capacitance of the combination is \(x \varepsilon_0 F\). The value of \(x\) is \(..........\).

A cricket ball of mass \(0.15\, kg\) is thrown vertically up by a bowling machine so that it rises to a maximum height of \(20 \;m\) after leaving the machine. If the part pushing the ball applies a constant force \(F\) on the ball and moves horizontally a distance of \(0.2\, m\) while launching the ball, the value of \(F(\) in \(N)\) is  \(\left(g=10\, m s^{-2}\right)\)

A travelling microscope is used to determine the refractive index of a glass slab. If \(40\) divisions are there in \(1 \; cm\) on main scale and \(50\) Vernier scale divisions are equal to \(49\) main scale divisions, then least count of the travelling microscope is \(\dots \; \times 10^{-6} \; m\)

A small ball of mass \(m\) is thrown upward with velocity \(u\) from the ground. The ball experiences a resistive force \(mkv ^{2}\) where \(v\) is its speed. The maximum height attained by the ball is 

A body of mass \(1\) kg moves along a straight line with a velocity \(v = 2x^2\). The work done by the body during displacement from \(x = 0\) to \(5\) m is __________ J.

If the percentage errors in measuring the length and the diameter of a wire are \(0.1 \%\) each. The percentage error in measuring its resistance will be _______.

A concave-convex lens of refractive index 1.5 and the radii of curvature of its surfaces are 30 cm and 20 cm , respectively. The concave surface is upwards and is filled with a liquid of refractive index 1.3. The focal length of the liquid-glass combination will be _______.

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is \(0.04\) , the acceleration of the system in \(\mathrm{ms}^{-2}\) is _______. (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless)

A point object, \(O\) is placed in front of two thin symmetrical coaxial convex lenses \(L _1\) and \(L _2\) with focal length \(24\,cm\) and \(9\,cm\) respectively. The distance between two lenses is \(10\,cm\) and the object is placed \(6\,cm\) away from lens \(L _1\) as shown in the figure. The distance between the object and the image formed by the system of two lenses is .........\(cm\)

In a closed organ pipe, the frequency of fundamental note is \(30 \mathrm{~Hz}\). A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to \(110 \mathrm{~Hz}\). If the organ pipe has a cross-sectional area of \(2 \mathrm{~cm}^2\), the amount of water poured in the organ tube is _______ \(g.\) (Take speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\) )

The acceleration of an electron in the first orbit of the hydrogen atom \((z\, = 1)\) is