A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

For x \in \left( {0,{{5\pi } \over 2}} \right), define $$f\left(x\right)=\underset{0}{\overset{x}{\int }}\sqrt{t}\sin tdt.$$ Then $$f$$ has

The shortest distance between line $$y-x=1$$ and curve $$x=y^2$$ is

The area of the region enclosed by the curves y = x,x = e,y = {1 \over x} and the positive $$x$$-axis is :

Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement, “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as :

A screw gauge gives the following reading when used to measure the diameter of a wire. Main scale reading : 0 mm Circular scale reading : 52 divisions Given that 1 mm on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above date is:

Let $$x$$-$$z$$ plane be the boundary between two transparent media. Medium $$1$$ in $$z\ge 0$$ has a refractive index of $$\sqrt{2}$$ and medium $$2$$ with $$z<0$$ has a refractive index of $$\sqrt{3}.$$ A ray of light in medium $$1$$ given by the vector $$\overrightarrow{A}=6\sqrt{3}\hat{i}+8\sqrt{3}\hat{j}-10\hat{k}$$ is incident on the plane of separation. The angle of refraction in medium $$2$$ is:

A car is fitted with a convex side-view mirror of focal length $$20$$ $$cm$$. A second car $$2.8m$$ behind the first car is overtaking the first car at a relative speed of $$15$$ $$m/s$$. The speed of the image of the second car as seen in the mirror of the first one is :

If a wire is stretched to make it $$0.1\%$$ longer, its resistance will:

Water is flowing continuously from a tap having an internal diameter $$8\times 10^{-3}m.$$ The water velocity as it leaves the tap is $$0.4m{s}^{-1}$$ . The diameter of the water stream at a distance $$2\times 10^{-1}m$$ below the tap is close to :

Work done in increasing the size of a soap bubble from a radius of $$3$$ $$cm$$ to $$5$$ $$cm$$ is nearly (Surface tension of soap solution $$=0.03N{m}^{-1},$$

Two particles are executing simple harmonic motion of the same amplitude $$A$$ and frequency $$\omega$$ along the $$x$$-axis. Their mean position is separated by distance $$X_0\left(X_0>A\right)$$. If the maximum separation between them is $$\left(X_0+A\right),$$ the phase difference between their motion is:

A mass $$M,$$ attached to a horizontal spring, executes $$S.H.M.$$ with amplitude $$A_1.$$ When the mass $$M$$ passes through its mean position then a smaller mass $$m$$ is placed over it and both of them move together with amplitude $$A_2.$$ The ratio of \left( {{{{A_1}} \over {{A_2}}}} \right) is :

A fully charged capacitor $$C$$ with initial charge $$q_0$$ is connected to a coil of self inductance $$L$$ at $$t=0.$$ The time at which the energy is stored equally between the electric and the magnetic fields is :

A resistor $${}^{\prime }{R}^{\prime }$$ and $$2\mu F$$ capacitor in series is connected through a switch to $$200$$ $$V$$ direct supply. Across the capacitor is a neon bulb that lights up at $$120$$ $$V.$$ Calculate the value of $$R$$ to make the bulb light up $$5$$ $$s$$ after the switch has been closed. $$\left({\log }_{10}2.5=0.4\right)$$

This question has Statement - $$1$$ and Statement - $$2$$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement -$$1$$ : Sky wave signals are used for long distance radio communication. These signals are in general, less stable then ground wave signals. Statement -$$2$$ : The state of ionosphere varies from hour to hour, day to day and season to season.

The half life of a radioactive substance is $$20$$ minutes. The approximate time interval $$\left(t_2-t_1\right)$$ between the time $$t_2$$ when {2 \over 3} of it had decayed and time $$t_1$$ when {1 \over 3} of it had decayed is :

Energy required for the electron excitation in $$L{i}^{++}$$ from the first to the third Bohr orbit is :

$$100g$$ of water is heated from $$30^{\circ }C$$ to $$50^{\circ }C$$. Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is $$4184$$ $$J/kg/K$$):

Three perfect gases at absolute temperatures $$T_1,T_2$$ and $$T_3$$ are mixed. The masses of molecules are $$m_1,m_2$$ and $$m_3$$ and the number of molecules are $$n_1,$$ $$n_2$$ and $$n_3$$ respectively. Assuming no loss of energy, the final temperature of the mixture is: