If \(\cos 2 \theta=\sin \propto, \quad\) then \(\theta=\)

The area of the region bounded by the curve \(\mathrm{y}=|x-2|\) between \(x=1, x=3\) and X -axis is ......

The third element in the second row of adjoint of a matrix \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{3 \times 3}\left(\right.\) where \(\left.\mathrm{a}_{\mathrm{ij}}=2 \mathrm{i}+\mathrm{j}\right)\) is

The differential equation \(x \frac{\mathrm{dy}}{\mathrm{d} x}=2 \mathrm{y}\) represents ....

Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be three non-coplanar vectors and \(\overline{\mathrm{p}}, \overline{\mathrm{q}}, \overline{\mathrm{r}}\) defined by the relations
\(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}\)
then the value of the expression \((\bar{a}+\bar{b}) \cdot \bar{p}+(\bar{b}+\bar{c}) \cdot \bar{q}+(\bar{c}+\bar{a}) \cdot \bar{r}\) is equal to

The principal value of \(\sin ^{-1}\left(\sin \frac{3 \pi}{4}\right)\) is

If \(\int\left(\frac{4 \mathrm{e}^x-25}{2 \mathrm{e}^x-5}\right) \mathrm{d} x=\mathrm{A} x+\operatorname{Blog}\left(2 \mathrm{e}^x-5\right)+\mathrm{c} \quad\) (where c is a constant of integration) then

What is the number of moles of water removed from one mole of cane sugar when charred completely using conc. \(\mathrm{H}_2 \mathrm{SO}_4\) ?

A uniform rope of length ' \(L\) ' and mass ' \(\mathrm{m}_1\) ' hangs vertically from a rigid support. A block of mass ' \(\mathrm{m}_2\) ' is attached to the free end of the rope. A transverse wave of wavelength ' \(\lambda_1\) ' is produced at the lower end of the rope. The wavelength of the wave when it reaches the top of the rope is ' \(\lambda_2\) '. The ratio \(\frac{\lambda_1}{\lambda_2}\) is

Three isolated metal spheres A, B, C have radius R, 2R, 3R respectively, and the same charge, \(Q . U_A, U_B\) and \(U_C\) be the energy density just outside the surface of the spheres. The relation between \(\mathrm{U}_{\mathrm{A},} \mathrm{U}_{\mathrm{B}}\) and \(\mathrm{U}_{\mathrm{C}}\) is

If \(\int e^{x^2} \cdot x^3 d x=e^{x^2} f(x)+C\)
(where \(C\) is a constant of integration)
and \(f(1)=0\), then value of \(f(2)\) will be

Two surfaces \(A\) and \(B\) are enclosing the charges as shown below. The total normal electric induction (T.N.E.I) through the surfaces A and B are respectively.

For a ray of light, the critical angle is minimum, when it travels from