For a certain organ pipe, the first three resonance frequencies are in the ratio of \(1:3:5\) respectively. If the frequency of fifth harmonic is \(405\,Hz\) and the speed of sound in air is \(324 \,ms ^{-1}\) the length of the organ pipe is \(..........m.\)

The value of the acceleration due to gravity is \(g _{1}\) at a height \(h =\frac{ R }{2}( R =\) radius of the earth) from the surface of the earth. It is again equal to \(g _{1}\) at a depth \(d\) below the surface of the earth. The ratio \(\left(\frac{ d }{ R }\right)\) equals

A triangular plate is shown. A force \(\overrightarrow{ F }=4 \hat{ i }-3 \hat{ j }\) is applied at point \(P\). The torque at point \(P\) with respect to point \('O'\) and \('Q'\) are

A perfectly dimagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. The whole system is placed in a uniform magnetic field \(\overrightarrow{ B }\). Then the field inside the paramagnetic substance is:

A plane electromagnetic wave propagates along the \(+x\) direction in free space. The components of the electric field, \(\vec{E}\) and magnetic field, \(\vec{B}\) vectors associated with the wave in Cartesian frame are

Given below are two statements : one is labeled as Assertion \(A\) and the other is labeled as Reason \(R\) Assertion \(A:\) Photodiodes are used in forward bias usually for measuring the light intensity. Reason \(R:\) For a \(p-n\) junction diode, at applied voltage \(V\) the current in the forward bias is more than the current in the reverse bias for \(\left|V_z\right| > \pm V \geq\left|V_0\right|\) where \(V_0\) is the threshold voltage and \(V _{ z }\) is the breakdown voltage. In the light of the above statements, choose the correct answer from the options given below

An electric toaster has resistance of \(60\  \Omega\) at room temperature \(\left(27^{\circ} \mathrm{C}\right)\). The toaster is connected to a \(220 \mathrm{~V}\) supply. If the current flowing through it reaches \(2.75 \mathrm{~A}\), the temperature attained by toaster is around _______. (if \(\alpha=2 \times 10^{-4} /{ }^{\circ} \mathrm{C}\) )

Charge is distributed within a sphere of radius \(R\) with a volume charge density \(\rho (r) = \frac{A}{{{r^2}}}{e^{ - 2r/a}}\) where \(A\) and \(a\) are constants. If \(Q\) is the total charge of this charge distribution, the radius \(R\) is.

A square shaped coil of area \(70\,cm ^2\) having \(600\) turns rotates in a magnetic field of \(0.4\,wbm ^{-2}\), about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes \(500\) revolution in a minute, the instantaneous emf when the plane of the coil is inclined at \(60^{\circ}\) with the field, will be \(..........V\). (Take \(\pi=\frac{22}{7}\) )

A bead of mass ' \(m\) ' slides without friction on the wall of a vertical circular hoop of radius ' \(R\) ' as shown in figure. The bead moves under the combined action of gravity and a massless spring ( k ) attached to the bottom of the hoop. The equilibrium length of the spring is ' \(R\) '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' R ', would be (spring constant is ' k ', g is accleration due to gravity)