The driver of a bus approaching a big wall notices that the frequency of his bus's horn changes from \(420\, Hz\) to \(490\, Hz ,\) when he hears it after it gets reflected from the wall. Find the speed of the bus (in \(kmh^{-1}\)) if speed of the sound is \(330\, ms ^{-1}\).

In a hypothetical fission reaction \({ }_{92} \mathrm{X}^{236} \rightarrow{ }_{56} \mathrm{Y}^{141}+{ }_{36} \mathrm{Z}^{92}+3 \mathrm{R}\) The identity of emitted particles \((R)\) is _______.

Two light beams fall on a transparent material block at point 1 and 2 with angle \(\theta_1\) and \(\theta_{2^{\prime}}\) respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2, \(\mathrm{d}=4 \sqrt{3} \mathrm{~cm}\) and \(\theta_1=\theta_2=\cos ^{-1}\left(\frac{\mathrm{n}_2}{2 \mathrm{n}_1}\right)\), where refractive index of the block \(\mathrm{n}_2\gt\) refractive index of the outside medium \(\mathrm{n}_1\), then the thickness of the block is ________ cm.

Two radioactive materials \(A\) and \(B\) have decay constants \(10\,\lambda \) and \(\lambda \), respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of a to that of \(B\) will be \(1/e\) after a time

A thin rod of mass \(0.9\, kg\) and length \(1 \,m\) is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move \(0.1\, kg\) moving in a straight line with velocity \(80\,m / s\) hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in \(rad/s\)) of the rod immediately after the collision will be

In the cube of side \(a\) shown in the figure, the vector from the central point of the face \(ABOD\) to the central point of the face \(BEFO\) will be

Three parallel plate capacitors each with area A and separation d are filled with two dielectric ( \(k _1\) and \(k _2\) ) in the following fashion. Which of the following is true? \(\left( k _1> k _2\right)\)

Match List\(-I\) with List\(-II\).

List\(-I\)	List\(-II\)
\(A\). Gauss's Law in Electrostatics	\(I\). \(\oint \vec{E} \cdot d \vec{l}=-\frac{d \phi_B}{d t}\)
\(B\). Faraday's Law	\(II\). \(\oint \overrightarrow{ B } \cdot d \overrightarrow{ A }=0\)
\(C\). Gauss's Law in Magnetism	\(III\). \(\oint \vec{B} \cdot d \vec{l}=\mu_0 i_C+\mu_0 \in_0 \frac{d \phi_E}{d t}\)
\(D\). Ampere-Maxwell Law	\(IV\). \(\oint \overrightarrow{ E } \cdot d \overrightarrow{ s }=\frac{ q }{\epsilon_0}\)

Choose the correct answer from the options given below:

A force \(F=\left(5+3 y^2\right)\) acts on a particle in the \(y\) direction, where \(F\) is newton and \(y\) is in meter. The work done by the force during a displacement from \(y =2\,m\) to \(y =5\,m\) is \(.............\,j\).

The period of oscillation of a simple pendulum is \(T =2 \pi \sqrt{\frac{ L }{ g }} .\) Measured value of \( L \) is \(1.0\, m\) from meter scale having a minimum division of \(1 \,mm\) and time of one complete oscillation is \(1.95\, s\) measured from stopwatch of \(0.01 \,s\) resolution. The percentage error in the determination of \(g\) will be ..... \(\%.\)

A rod of length \(60 \mathrm{~cm}\) rotates with a uniform angular velocity \(20 \mathrm{rad} \mathrm{s}^{-1}\) about its perpendicular bisector, in a uniform magnetic field \(0.5 \mathrm{~T}\). The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is________\(V\).