\(LCR\) circuit is at resonance for a capacitor \(C\), inductance \(L\) and resistance \(R\). Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now what?

A radioactive nucleus \(A\) with a half life \(T\), decays into a nucleus \(B.\) At \(t = 0\), there is no nucleus \(B\). At sometime \(t\), the ratio of the number of \(B\) to that of \(A\) is \(0.3\). Then, \(t\) is given by

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 .........\(cm\)

A tennis ball (treated as hollow spherical shell) starting from \(O\) rolls down a hill. At point \(A\) the ball becomes air borne leaving at an angle of \(30^o\) with the horizontal. The ball strikes the ground at \(B\). What is the value of the distance \(AB\) ? (Moment of inertia of a spherical shell of mass \(m\) and radius \(R\) about its diameter .......... \(m\). \(= \frac {2}{3}\,mR^2\))

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 scooter accelerates from rest for time \(t_{1}\) at constant rate \(a _{1}\) and then retards at constant rate \(a _{2}\) for time \(t _{2}\) and comes to rest. The correct value of \(\frac{t_{1}}{t_{2}}\) will be ..... .

In the given circuit, an ideal voltmeter connected across the \(10\,\Omega \) resistance reads \(2\, V\). The internal resistance \(r\), of each cell is ................... \(\Omega\)

A diverging lens with magnitude of focal length \(25\ cm\) is placed at a distance of \(15\ cm\) from a converging lens of magnitude of focal length \(20\ cm\). A beam of parallel light falls on the diverging lens. The final image formed is

One mole of an ideal diatomic gas expands from volume V to 2 V isothermally at a temperature \(27^{\circ} C\) and does W joule of work. If the gas undergoes same magnitude of expansion adiabatically from \(27^{\circ} C\) doing the same amount of work W , then its final temperature will be (close to) ___________ \({ }^{\circ} C\).

The de Broglie wavelength associated with an electron accelerated through a potential difference V is \(\lambda_e\) and the de Broglie wavelength associated with a proton accelerated through the same potential difference is \(\lambda_p\). If their corresponding masses are \(m_e\) and \(m_p\), respectively, then the ratio of their de Broglie wavelengths \(\left(\dfrac{\lambda_e}{\lambda_p}\right)\) is ______.

A physical quantity C is related to four other quantities \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) and s as follows \(\mathrm{C}=\frac{\mathrm{pq}^2}{\mathrm{r}^3 \sqrt{\mathrm{~s}}}\)
The percentage errors in the measurement of \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) and s are \(1 \%, 2 \% 3 \%\) and \(2 \%\) respectively. The percentage error in the measurement of C will be _____\(\%\).