A thin convex lens of focal length \('f'\) is put on a plane mirror as shown in the figure. When an object is kept at a distance \('a'\) from the lens - mirror combination, its image is formed at a distance \(\frac{a}{3}\) in front of the combination. The value of \('a'\) is

A resistance of \(40 \,\Omega\) is connected to a source of alternating current rated \(220\, V , 50 Hz\). Find the time taken by the current to change from its maximum value to \(ms\) value

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of \(24\,W\) The radius of curvature of hemisphere is \(10\,cm\) and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is \(..........\times 10^{-8}\,N\).

In the given circuit, the current in resistance \(\mathrm{R}_3 \) is _______.

A particle is subjected two simple harmonic motions as :
\(\mathrm{x}_1=\sqrt{7} \sin 5 \mathrm{tcm}\)
and \(x_2=2 \sqrt{7} \sin \left(5 t+\frac{\pi}{3}\right) \mathrm{cm}\)
where x is displacement and \(t\) is time in seconds. The maximum acceleration of the particle is \(\mathrm{x} \times 10^{-2} \mathrm{~ms}^{-2}\). The value of x is :

A body is in simple harmonic motion with time period half second \((T\, = 0.5\, s)\) and amplitude one \(cm\, (A\,= 1\, cm)\). Find the average velocity in the interval in which it moves form equilibrium position to half of its amplitude .... \(cm/s\)

The energy \(E\) and momentum \(p\) of a moving body of mass \(m\) are related by some equation. Given that c represents the speed of light, identify the correct equation.

If \(\lim _{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{c x e^{-c x}}{2}}{1-\cos (2 x)}=17\), then \(5 a ^2+ b ^2\) is equal to

If a curve \(y=y(x)\) passes through the point \(\left(1, \frac{\pi}{2}\right)\) and satisfies the differential equation \(\left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5, x \geq 1\), then at \(x=2\), the value of cosy is:

If the distance between object and its two times magnified virtual image produced by a curved mirror is \(15 \mathrm{~cm}\), the focal length of the mirror must be _______.

If \(Y, K\) and \(\eta\) are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

The mean and variance of \(10\) observations were calculated as \(15\) and \(15\) respectively by a student who took by mistake \(25\) instead of \(15\) for one observation. Then, the correct standard deviation is\(.....\)

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______