Five persons \(P_1, P_2, P_3, P_4\) and \(P_5\) recorded object distance (u) and image distance (v) using same convex lens having power +5 D as \((25,96),(30,62)\), \((35,37),(45,35)\) and \((50,32)\) respectively. Identify correct statement

If two vectors \(\vec{P}=\hat{i}+2 m \hat{j}+m \hat{k}\) and \(\vec{Q}=4 \hat{i}-2 \hat{j}+ mk\) are perpendicular to each other. Then, the value of \(m\) will be :

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length \(L ( R < L )\) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M ) :

A light wave described by \(E =60 \sin \left(3 \times 10^{15}\right) t + \sin \left(12 \times 10^{15}\right)\) t] (in SI units) falls on a metal surface of work function 2.8 eV . The maximum kinetic energy of ejected photoelectron is (approximately) _________ eV .

A galaxy is moving away from the earth at a speed of \(286\, kms ^{-1}\). The shift in the wavelength of a red line at \(630\, nm\) is \(x \times 10^{-10}\, m\). The value of \(x ,\) to the nearest integer, is........ [Take the value of speed of light \(c,\) as \(3 \times 10^{8}\) \(\left. ms ^{-1}\right]\)

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 square loop of side \(2.0\,cm\) is placed inside a long solenoid that has \(50\) turns per centimetre and carries a sinusoidally varying current of amplitude \(2.5\,A\) and angular frequency \(700\,rad\,s ^{-1}\). The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is \(x \times 10^{-4} V\). The value of \(x\) is \(.........\)(Take, \(\pi=\frac{22}{7}\))

Find the magnetic field at the point \(P\) in figure. The curved portion is a semicircle connected to two long straight wires.

In a meter bridge experiment to determine the value of unknown resistance, first the resistances \(2 \Omega\) and \(3 \Omega\) are connected in the left and right gaps of the bridge and the null point is obtained at a distance \(l cm\) from the left. Now when an unknown resistance \(x \Omega\) is connected in parallel to \(3 \Omega\) resistance, the null point is shifted by 10 cm to the right of wire. The value of unknown resistance \(x\) is _________ \(\Omega\).

If three moles of monoatomic gas \(\left(\gamma=\frac{5}{3}\right)\) is mixed with two moles of a diatomic gas \(\left(\gamma=\frac{7}{5}\right)\), the value of adiabatic exponent \(\gamma\) for the mixture is _______.

Consider a frame that is made up of two thin massless rods \(AB\) and \(AC\) as shown in the figure. \(A\) vertical force \(\overrightarrow{ P }\) of magnitude \(100 \;N\) is applied at point \(A\) of the frame. Suppose the force is \(\overrightarrow{ P }\) resolved parallel to the arms \(AB\) and \(AC\) of the frame. The magnitude of the resolved component along the arm \(AC\) is \(xN\). The value of \(x\), to the nearest integer, is ............ [Given : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)