A body cools from \(60^{\circ} C\) to \(40^{\circ} C\) in \(6\) minutes. If, temperature of surroundings is \(10^{\circ} C\). Then, after the next 6 minutes, its temperature will be \(.........{ }^{\circ} C\).

The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength \(500\,nm\) and \(600\, nm\), each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in \(nm\) is

The output \(\mathrm{Y}\) of following circuit for given inputs is _______.

If the refractive index of the material of a prism is \(\cot \left(\frac{A}{2}\right)\), where \(A\) is the angle of prism then the angle of minimum deviation will be

For a certain radioactive process the graph between \(In\, {R}\) and \({t}\,({sec})\) is obtained as shown in the figure. Then the value of half life for the unknown radioactive material is approximately \(....\,{sec}.\)

A particle of charge \(1.6 \mu \mathrm{C}\) and mass \(16 \mu \mathrm{~g}\) is present in a strong magnetic field of 6.28 T. The particle is then fired perpendicular to magnetic field. The time required for the particle to return to original location for the first time is ________ S. \((\pi=3.14)\)

A hanging mass \(M\) is connected to a four times bigger mass by using a string-pulley arrangement. as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by \(2\,Mg\) force. In this situation. tension in the string is \(\frac{x}{5}\) \(Mg\) for \(x =\) Neglect mass of the string and friction of the block (bigger mass) with ice slab. (Given \(g=\) acceleration due to gravity)

Let the solution curve of the differential equation \(x d y=\left(\sqrt{x^{2}+y^{2}}+y\right) d x, x>0\), intersect the line \(x =1\) at \(y =0\) and the line \(x =2\) at \(y =\alpha\). Then the value of \(\alpha\) is.

Let \(z =1+ i\) and \(z _1=\frac{1+ i \overline{ z }}{\overline{ z }(1- z )+\frac{1}{ z }}\). Then \(\frac{12}{\pi}\) \(\arg \left(z_1\right)\) is equal to \(..........\).

Let \(3, a, b, c\) be in \(A.P.\) and \(3, a-1, b+1, c+9\) be in \(G.P.\) Then, the arithmetic mean of \(a, b\) and \(c\) is :

If the common tangent to the parabolas, \(y ^{2}=4 x\) and \(x ^{2}=4 y\) also touches the circle, \(x^{2}+y^{2}=c^{2},\) then \(c\) is equal to

Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum\limits_{n=1}^{100} a_{2 n+1}=200\) and \(\sum\limits_{n=1}^{100} a_{2 n}=100,\) then \(\sum\limits_{n=1}^{200} a_{n}\) is equal to 

Refer the figure given below. \(\mu_1\) and \(\mu_2\) are refractive indices of air and lens material. The height of image will be _______ cm.