If the pair of lines given by \((x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2\right) \sin ^2 \alpha\) are perpendicular to each other, then \(\alpha\) is

\(A\) line \(L\) is passing through points \(A(1,3,2)\) and \(B(2,2,1)\). If mirror image of point \(\mathrm{P}(1,1,-1)\) in the line L is \((\mathrm{x}, \mathrm{y}, \mathrm{z})\) then \(x+\mathrm{y}+\mathrm{z}=\)

\(\frac{\sin A+\sin 7 A+\sin 13 A}{\cos A+\cos 7 A+\cos 13 A}=\)

The differential equation of family of circles, whose centres are on the X -axis and also touch the Y -axis is

A polygon has 44 diagonals. Then the number of sides of the polygon are

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\), then the probability, that he is unable to solve less than two problems, is

The positive root of \(x^{2}-78.8=0\) after first approximation by Newton Raphson method assuming initial approximation to the root is 14, is

What happens to the emf for the cell , \({ }^{\ominus} \mathrm{Zn}_{(\mathrm{s})}\left|\mathrm{Zn}^{+2}(1 \mathrm{M})\right|\left|\mathrm{Ag}^{+1}(1 \mathrm{M})\right| \stackrel{\oplus}{\mathrm{Ag}}_{(\mathrm{s})}\) if concentration of \(\mathrm{Ag}^{+1}\) decreases to 0.1 M ?

The region represented by the inequalities \(x \geq 6, y \geq 3\), \(2 x+y \geq 10, x \geq 0, y \geq 0\) is

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

Two positively charged identical spheres separated by a distance 'd' exert some force (F) on each other when they are kept in air. If both the spheres are immersed in a liquid of dielectric constant 5 , the force experienced by each is (All other parameters are unchanged.)

The velocity time graph of a body moving in a straight line is shown in figure. The ratio of displacement to distance travelled by the body in time 0 to 8 s is

The degree and order of the differential equation ${\left[1+{\left(\frac{dy}{dx}\right)}^3\right]}^{\frac{7}{3}}=7\left(\frac{d^{2}y}{d{x}^2}\right)$ respectively are