In a double slit experiment, at a certain point on the screen the path difference between the two interfering waves is \(\frac{1}{8} ^{th}\) of a wavelength. The ratio of the intensity of light at that point to that at the centre of a bright fringe is

A block moving horizontally on a smooth surface with a speed of \(40\, {ms}^{-1}\) splits into two equal parts. If one of the parts moves at \(60\, {ms}^{-1}\) in the same direction, then the fractional change in the kinetic energy will be \(x: 4\) where \(x=..... .\)

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass \(4\;kg\). (The coordinates of the same are shown in figure) are

The current flowing through a conductor connected across a source is \(2\,A\) and \(1.2\,A\) at \(0^{\circ}\,C\) and \(100^{\circ}\,C\) respectively. The current flowing through the conductor at \(50^{\circ}\,C\) will be \(......\times 10^2\,mA\).

A uniform wire of length \(l\) of weight \(w\) is suspended from the roof with a weight of \(W\) at the other end. The stress in the wire at \(\dfrac{l}{3}\) distance from the top is \(\left(\dfrac{W}{A} + \dfrac{2}{\gamma}\dfrac{w}{A}\right)\), where, \(A\) is the cross sectional area of the wire. The value of \(\gamma\) is _______.

A mirror is used to produce an image with magnification of \(\frac{1}{4}\). If the distance between object and its image is 40 cm , then the focal length of the mirror is ________

In a resonance tube experiment when the tube is filled with water up to height of \(17.0\, cm\) from bottom, it resonates with a given tuning fork. When the water level is raised the next resonance with the same tuning fork occurs at a height of \(24.5\, cm\). If the velocity of sound in air is \(330\, m / s ,\) the tuning fork frequency is\(......Hz\)

A Spherical ball of radius \(1 mm\) and density \(10.5 g / cc\) is dropped in glycerine of coefficient of viscosity \(9.8\) poise and density \(1.5 g / cc\). Viscous force on the ball when it attains constant velocity is \(3696 \times 10^{-x} N\). The value of \(x\) is \(\text { (Given, } g =9.8 m / s ^2 \text { and } \pi=\frac{22}{7} \text { ) }\)

Let \(x=x(t)\) and \(y=y(t)\) be solutions of the differential equations \(\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0\) and \(\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0\) respectively, \(\mathrm{a}, \mathrm{b} \in \mathrm{R}\). Given that \(x(0)=2 ; y(0)=1\) and \(3 y(1)=2 x(1)\), the value of \(t\), for which \(x(t)=y(t)\), is :

Identify the correct statements :
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
Choose the correct answer from the options given below.

A radioactive sample is undergoing \(\alpha\) decay. At any time \(t_{1}\), its activity is \(A\) and another time \(t _{2}\), the activity is \(\frac{ A }{5}\). What is the average life time for the sample?

If \(A\) is the area in the first quadrant enclosed by the curve \(C: 2 x^2-y+1=0\), the tangent to \(C\) at the point \((1,3)\) and the line \(x+y=1\), then the value of \(60 A\) is

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{ dy }{ dx }+\frac{1}{ x ^{2}-1} y =\left(\frac{ x -1}{ x +1}\right)^{\frac{1}{2}}\), \(x>1\) passing through the point \(\left(2, \sqrt{\frac{1}{3}}\right)\). Then \(\sqrt{7} y (8)\) is equal to.