Distance between an object and three times magnified real image is 40 cm . The focal length of the mirror used is _________ cm .

In a physical balance working on the principle of moments, when \(5\, mg\) weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

A person driving car at a constant speed of \(15\,m / s\) is approaching a vertical wall. The person notices a change of \(40\,Hz\) in the frequency of his car's horn upon reflection from the wall. The frequency of horn is ............ \(Hz\). (Given : Speed of sound : \(330\,m / s\) )

A faulty thermometer reads \(5^{\circ} C\) in melting ice and \(95^{\circ} C\) in steam. The correct temperature on absolute scale will be. \(.........K\) when the faulty thermometer reads \(41^{\circ} C\).

The Young's modulus of a steel wire of length \(6\,m\) and cross-sectional area \(3\,mm ^2\), is \(2 \times 11^{11}\,N / m ^2\). The wire is suspended from its support on a given planet. A block of mass \(4\,kg\) is attached to the free end of the wire. The acceleration due to gravity on the planet is \(\frac{1}{4}\) of its value on the earth. The elongation of wire is  (Take \(g\) on the earth \(=10\) \(\left.m / s ^2\right):\)

An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension \(0.095 \mathrm{~J} / \mathrm{m}^2\) and density \(10^3 \mathrm{~kg} / \mathrm{m}^3\). The difference between pressure inside the bubble and atmospheric pressure is _____ \(\mathrm{N} / \mathrm{m}^2\). (Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

A metallic rod of length ' \(L\) ' is rotated with an angular speed of ' \(\omega\) ' normal to a uniform magnetic field ' \(B\) ' about an axis passing through one end of rod as shown in figure. The induced emf will be :

In a Young's double slit experiment, the intensities at two points, for the path difference \(\frac{\lambda}{4}\) and \(\frac{\lambda}{3}\) ( \(\lambda\) being the wavelength of light used) are \(I_1\) and \(I_2\) respectively. If \(I_0\) denotes the intensity produced by each one of the individual slits, then \(\frac{ I _1+ I _2}{ I _0}=..............\)

A uniform disc with mass \(M=4\,kg\) and radius \(R=\) \(10\,cm\) is mounted on a fixed horizontal axle as shown in figure. A block with mass \(m =2\,kg\) hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is_______ \(N\) \(\left(\right.\) Take \(\left.g =10\,ms ^{-2}\right)\)

Consider a solid sphere of radius \(\mathrm{R}\) and mass density \(\rho(\mathrm{r})=\rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right),  0<\mathrm{r} \leq \mathrm{R} .\) The minimum density of a liquid in which it will float is

A force is represented by \(\mathrm{F}=a \mathrm{x}^2+\mathrm{bt}^{1 / 2}\). Where \(\mathrm{x}=\) distance and \(\mathrm{t}=\) time. The dimensions of \(\mathrm{b}^2 / \mathrm{a}\) are _______.

A parallel plate capacitor is made of two square plates of side \(a\), separated by a distance \(d\,(d  < < a)\). The lower triangular portion is filled with a dielectric of dielectric constant \(K\), as shown in the figure. Capacitance of this capacitor is