In an experment ot verify Newton's law of cooling, a graph is plotted between, the temperature difference \((\Delta T )\) of the water and surroundings and time as shown in figure. The initial temperature of water is taken as \(80^{\circ} \,C\). The value of \(t _{2}\) as mentioned in the graph will be...........

A conducting loop of radius \(\frac{10}{\sqrt{\pi}}\,cm\) is placed perpendicular to a uniform magnetic field of \(0.5\,T\). The magnetic field is decreased to \(zero\) in \(0.5\,s\) at a steady rate. The induced emf in the circular loop at \(0.25\,s\) is:

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

When one moves from a point \(16\) km below the earth's surface to a point \(16\) km above the earth's surface. The change in \(g\) is approximately \(\alpha\) %. The value of \(\alpha\) is _____.
(Take radius of the earth \(= 6400\) km.)

Magnitude of magnetic field (in \(SI\) units) at the centre of a hexagonal shape coil of side \(10\, cm\), \(50\) turns and carrying current \(I\) (Ampere) in units of \(\frac{\mu_{0} I}{\pi}\) is

A hollow spherical shell at outer radius \(R\) floats just submerged under the water surface. The inner radius of the shell is \(r\). If the specific gravity of the shell material is \(\frac{27}{8}\) \(w.r.t.\) water, the value of \(r\) is\(......R\)

Which one of the following is not a measurable quantity ?

If for the complex numbers \(z\) satisfying \(|z-2-2 i| \leq 1\), the maximum value of \(|3 i z+6|\) is attained at \(\mathrm{a}+i \mathrm{~b}\), then \(\mathrm{a}+\mathrm{b}\) is equal to .... .

For the two circles \(x^2 + y^2 = 16\) and \(x^2 + y^2 -2y = 0,\) there is/are

An observer moves towards a stationary source of sound with a velocity equal to one-fifth of the velocity of sound. The percentage change in the frequency will be \(\dots \;\)%

The relation \(R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}\) is:

If the mean of the following probability distribution of a random variable \(X\);

\(X\)	\(0\)	\(2\)	\(4\)	\(6\)	\(8\)
\(P(X)\)	\(a\)	\(2a\)	\(a+b\)	\(2b\)	\(3b\)

is \(\frac{46}{9}\) , then the variance of the distribution is

The activity of a radioactive material is \(6.4 \times 10^{-4}\) curie. Its half life is \(5\; days\). The activity will become \(5 \times 10^{-6}\) curie after \(.......day\)