The area bounded by the parabolas \(\mathrm{y}=9 x^2, \mathrm{y}=\frac{x^2}{16}\) and the line \(\mathrm{y}=1\) is

The general solution of the differential equation \(\left(1-x^{2}\right) \frac{d y}{d x}+2 x y=x\left(1-x^{2}\right)^{\frac{1}{2}}\) is

The money invested in a company is compounded continuously. If ₹ \(400\) invested today becomes ₹800 in 6 years, then at the end of 30 years, it will become (in ₹)

If the radius of a circle \(x^{2}+y^{2}-4 x+6 y-k=0\) is 5, then \(\mathrm{k}=\)

\(\int \sin \sqrt{x} \mathrm{~d} x=\ldots+C\)
(where \(C\) is a constant of integration.)

A line drawn from a point \(\mathrm{A}(-2,-2,3)\) and parallel to the line \(\frac{\mathrm{x}}{-2}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}}{-1}\) meets the \(\mathrm{YOZ}-\) plane in point \(\mathrm{P}\), then the co-ordinates of the point \(\mathrm{P}\) are

The distance of a point \((2,5)\) from the line \(3 x+y+4=0\) measured along the line \(\mathrm{L}_1\) and \(\mathrm{L}_2\) are same. If slope of line \(L_1\) is \(\frac{3}{4}\), then slope of the line \(\mathrm{L}_2\) is

Photons of energy \(10 \mathrm{eV}\) are incident on a photosensitive surface of threshold frequency \(2 \times 10^{15} \mathrm{~Hz}\). The kinetic energy in \(\mathrm{eV}\) of the photoelectrons emitted is [Planck's constant \(\mathrm{h}=6.63 \times 10^{34}\) \(\mathrm{Js}]\)

In photoelectric effect, the kinetic energy of electron emitted from the metal surface depends upon

Water is flowing through a horizontal pipe of non-uniform cross-section. In the region of narrowest part inside the pipe, the water will have

A moving body is covering distances which are proportional to square of the time. Then the acceleration of the body is

What will be the resistance of the shunt when \(5 \%\) of the main current is passed through a galvanometer of resistance G?

Let \(\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection at \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is equal to