If the line \(3 x+4 y-24=0\) intersects \(X\) and \(Y\) axes in points \(A\) and \(B\) respectively then incenter of the triangle \(O A B\) where \(O\) is origin is

The derivative of \(f(\tan x)\) w.r.t. \(g(\sec x)\) at \(x=\frac{\pi}{4}\), where \(f^{\prime}(1)=2\) and \(g^{\prime}(\sqrt{2})=4\) is

The minimum value of $z=10x+25y$ subject to $0\le x\le 3,0\le y\le 3,x+y\ge 5$ is ________

\(\int \frac{1}{\cos x \sqrt{\cos 2 x}} d x=\)
(where C is a constant of integration.)

If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are two unit vectors such that \(5 \bar{a}+4 \bar{b}\) and \(\bar{a}-2 \bar{b}\) are perpendicular to each other, then the between \(\bar{a}\) and \(\bar{b}\) is

If \(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4\), then \(\frac{d y}{d x}=\)

If $y=e^{m{\sin }^{-1}x}\mathrm{ }\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(1-x^2\right){\left(\frac{dy}{dx}\right)}^2=A{y}^2,$ then $A=$ ________

Environmental heterophylly is observed in the leaves of ________.

Two moles of an ideal gas expand freely and isothermally from \(5 \mathrm{dm}^3\) to \(50 \mathrm{dm}^3\). What is the value of \(\Delta \mathrm{H}\) ?

For a reaction \(r=k[A][B]^2\), if concentration of \(A\) is doubled the rate of reaction

A current passing through a circular coil of two turns produces a magnetic field \(B\) as its centre. The coil is then rewound so as to have four turns and the same current is passed through it. The magnetic field at its centre now 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

The magnetic susceptibility of the material of a rod is 599. The absolute permeability of the material of the rod will be \(\left[\mu_0=4 \pi \times 10^{-7}\right.\) SI unit \(]\).