Two charges, each equal to \(q\), are kept at \(x = -a\) and \(x = a\) on the \(x-\)axis. A particle of mass \(m\) and charge \(q_0=\frac{q}{2}\) is placed at the origin. If charge \(q_0\) is given a small displacement \((y < < a)\) along the \(y-\)axis, the net force acting on the particle is proportional to

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 : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

When a car is approaching the observer, the frequency of horn is \(100 Hz\). After passing the observer, it is \(50\,Hz\). If the observer moves with the car, the frequency will be \(\frac{ x }{3} Hz\) where \(x =.....\)

A small circular loop of conducting wire has radius \(a\) and carries current \(I\). It is placed in a uniform magnetic field \(\mathrm{B}\) perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period \(T\). If the mass of the loop is \(m\) then 

A deuteron and an alpha particle having equal kinetic energy enter perpendicular into a magnetic field. Let \(r_{d}\) and \(r_{\alpha}\) be their respective radii of circular path. The value of \(\frac{r_{d}}{r_{\alpha}}\) is equal to

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modules of the material of the wire will :

In a meter bridge experiment the balance point in obtained if the gaps are closed by \(2\,\Omega\) and \(3\,\Omega\). A shunt of \(x\,\Omega\) is added to \(3\,\Omega\) resistor to shift the balancing point by \(22.5\,cm\). The value of \(X\) is \(................\)

A closed vessel contains \(0.1\) mole of a monoatomic ideal gas at \(200\, K\). If \(0.05\) mole of the same gas at \(400\, K\) is added to it, the final  equilibrium temperature (in \(K\) ) of the gas in the vessel will be closed to

Let \(A = \{1, 4, 7\}\) and \(B = \{2, 3, 8\}\). Then the number of elements, in the relation \(R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + b_2 \text{ divides } a_2 + b_1\}\) is _______.

If  \(2\int_0^1 {{{\tan }^{ - 1}}}\,xdx = \int_0^1 {{{\cot }^{ - 1}}}\,(1 - x + {x^2})dx,\) then \(\int_0^1 {{{\tan }^{ - 1}}}\, (1 - x + {x^2})dx\)  is equal to 

In the experiment to determine the galvanometer resistance by half-deflection method, the plot of \(\frac{1}{\theta}\) vs the resistance \((R)\) of the resistance box is shown in the figure. The figure of merit of the galvanometer is _______ \(\times 10^{-1} \mathrm{~A} /\) division. [The source has emf 2V]

The equation of a transverse wave travelling along a string is \(y(x, t)=4.0 \sin \left[20 \times 10^{-3} x+600 t\right] \mathrm{mm}\), where \(x\) is in mm and \(t\) is in second. The velocity of the wave is :

lf a line \(L\) is perpendicular to the line \(5x - y\,= 1\) , and the area of the triangle formed by the line \(L\) and the coordinate axes is \(5\), then the distance of line \(L\) from the line \(x + 5y\, = 0\) is