In the given circuit, \(C _1=2\,\mu F , C _2=0.2\,\mu F\), \(C _3=2\,\mu F , C _4=4\,\mu F,\) \(C _5=2\,\mu F , C _6=2\,\mu F\), the charge stored on capacitor \(C _4\) is \(.....\mu C\).

A planoconvex lens becomes an optical system of \(28\, cm\) focal length when its plane surface is silvered and illuminated from left to right as shown in Fig \(-A\). If the same lens is instead silvered on the curved surface and illuminated from other side as in Fig. \(B\), it acts like an optical system of focal length \(10 \,cm\). The refractive index of the material of lens is

Two blocks of masses \(m\) and \(M,(M \gt m)\), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
(\(\mu=\) coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is \(\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}\)
(B) The acceleration of the blocks is \(\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}\) (\(\mathrm{x}=\) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \(\frac{m \mu|x|}{M+m}\)
(D) The maximum amplitude of the upper block, if it does not slip, is \(\frac{\mu(M+m) g}{k}\)
(E) Maximum frictional force can be \(\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}\).
Choose the correct answer from the options given below:

In a single slit experiment, a parallel beam of green light of wavelength \(550 \mathrm{~nm}\) passes through a slit of width \(0.20 \mathrm{~mm}\). The transmitted light is collected on a screen \(100 \mathrm{~cm}\) away. The distance of first order minima from the central maximum will be \(\mathrm{x} \times 10^{-5} \mathrm{~m}\). The value of \(\mathrm{x}\) is _______.

 A small speaker delivers \(2\, W\) of audio output. At what distance from the speaker will one detect \(120\, dB\) intensity sound ... \(cm\) ? [Given reference intensity of sound as \(10^{-12}\,W/m^2\)]

Energy released when two deuterons \(\left({ }_1 \mathrm{H}^2\right)\) fuse to form a helium nucleus \(\left({ }_2 \mathrm{He}^4\right)\) is :
(Given : Binding energy per nucleon of \({ }_1 \mathrm{H}^2=1.1 \mathrm{MeV}\) and binding energy per nucleon of \({ }_2 \mathrm{He}^4=7.0 \mathrm{MeV}\))

When a particle executes simple Harmonic motion, the nature of graph of velocity as function of displacement will be.

Monochromatic light of frequency \(6 \times 10^{14} \mathrm{~Hz}\) is produced by a laser. The power emitted is \(2 \times 10^{-3} \mathrm{~W}\). How many photons per second on an average, are emitted by the source? (Given \(\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}\) )

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then the domain of \(f(x)=\sec ^{-1}(2[x]+1)\) is :

If \({\cos ^{ - 1}}\left( {\frac{2}{{3x}}} \right) + {\cos ^{ - 1}}\,\left( {\frac{3}{{4x}}} \right) = \frac{\pi }{2},\,x > \frac{3}{4}\) then \(x\) is equal to

An electron, moving along the \(x-\) axis with an initial energy of \(100\, eV\), enters a region of magnetic field \(\vec B = (1.5\times10^{-3}T)\hat k\) at \(S\) (See figure). The field extends between \(x = 0\) and \(x = 2\, cm\). The electron is detected at the point \(Q\) on a screen placed \(8\, cm\) away from the point \(S\). The distance \(d\) between \(P\) and \(Q\) (on the screen) is :......\(cm\) (electron's charge \(= 1.6\times10^{-19}\, C\), mass of electron \(= 9.1\times10^{-31}\, kg\))

If the area of the bounded region \(R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) is, \(\alpha\left(\log _{e} 2\right)^{-1}+\beta\left(\log _{e} 2\right)+\gamma\), then the value of \((\alpha+\beta-2 \gamma)^{2}\) is equal to:

The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is