If \(\overrightarrow{ A }=(2 \hat{ i }+3 \hat{ j }-\hat{ k }) \;m\) and \(\overrightarrow{ B }=(\hat{ i }+2 \hat{ j }+2 \hat{ k })\; m\). The magnitude of component of vector \(\overrightarrow{ A }\) along vector \(\vec{B}\) will be \(......m\).

Two wires A and B are made of same material having ratio of lengths \(\frac{L_A}{L_B}=\frac{1}{3}\) and their diameters ratio \(\frac{d_A}{d_B}=2\). If both the wires are stretched using same force, what would be the ratio of their respective elongations?

Match List-I with List-II.

List-I (Relation)		List-II (Law)
A.	\(\oint \overrightarrow{ E } \cdot \overrightarrow{ d } l =-\frac{ d }{ dt } \oint \overrightarrow{ B } \cdot \overrightarrow{ d a}\)	I.	Ampere's circuital law.
B.	\(\oint\) B. \(\vec{d} l=\mu_0\left(1+\epsilon_0 \frac{d \phi_E}{d t}\right)\)	II.	Faraday's laws of electromagnetic induction.
C.	\(\oint \overrightarrow{ E } . \overrightarrow{ d a}=\frac{1}{\epsilon_0} \int \rho dv\)	III.	Ampere-Maxwell law
D.	\(\oint \overrightarrow{ B } \cdot \overrightarrow{ d } l =\mu_0 I\)	IV.	Gauss's law of electrostatics

Choose the correct answer from the options given below :

For a train engine moving with speed of \(20 \;ms ^{-1}\). the driver must apply brakes at a distance of \(500 \;m\) before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \(\sqrt{ x }\; ms ^{-1}\). The value of \(x\) is \(..............\) (Assuming same retardation is produced by brakes)

Given below are two statements : one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : When a body is projected at an angle \(45^{\circ}\), it's range is maximum. Reason \(R\) : For maximum range, the value of \(\sin 2 \theta\) should be equal to one. In the light of the above statements, choose the correct answer from the options given below :

A vertical electric field of magnitude \(4.9 \times 10^{5} N / C\) just prevents a water droplet of a mass \(0.1\, g\) from falling. The value of charge on the droplet will be  ........ \(\times 10^{-9} \;C\) \(\left(\right.\) Given \(\left.g =9.8 m / s ^{2}\right)\)

A 20 m long uniform copper wire held horizontally is allowed to fall under the gravity ( \(g =10 m / s ^2\) ) through a uniform horizontal magnetic field of 0.5 Gauss perpendicular to the length of the wire. The induced EMF across the wire it travells a vertical distance of 200 m is ___________ mV .

Moment of inertia of a dise of mass \(M\) and radius ' \(R\) ' about any of its diameter is \(\frac{ MR ^2}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \(\frac{ x }{2} MR ^2\). The value of \(x\) is \(..........\).

Light from a point source in air falls on a spherical glass surface (refractive index, \(\mu=1.5\) and radius of curvature \(=50 \mathrm{~cm}\)). The image is formed at a distance of 200 cm from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is ________ m.