A line makes angles \(\propto, \beta, \gamma\) with the co-ordinate axes and \(\propto+\beta=90^{\circ}\), then \(\gamma=\)

\( \int \frac{x^2+1}{x\left(x^2-1\right)} \mathrm{d} x= \)

Find \(\frac{d y}{d x}\), if \(x=2 \cos \theta-\cos 2 \theta\) and
\(y=2 \sin \theta-\sin 2 \theta .\)

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point \(X\) on the circumference of the circle, then 2 r equals

The mean and variance of 7 observations are 8 and 16 respectively. If first five observations are \(2,4,10,12,14\), then absolute difference of remaining two observations is

Considering only the principal values of the inverse trigonometric function, the value of \(\tan \left(\cos ^{-1} \frac{1}{5 \sqrt{2}}-\sin ^{-1} \frac{4}{\sqrt{17}}\right)\) is

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

The shaded region in the following figure is the solution set of the inequations

If ${\log }_{10}\left(\frac{x^2-y^2}{x^2+y^2}\right)=2,$ then $\frac{dy}{dx}=$ _______

The inductive reactance of a coil is ' R ' \(\Omega\). If inductance of a coil is tripled and frequency of a.c supply is also tripled, then new inductive reactance will be

When both source and listener are approaching each other the observed frequency of sound is given by \(\left(V_L\right.\) and \(V_S\) is the velocity of listener and source respectively, \(\mathrm{n}_0=\) radiated frequency)

The approximate value of $f\left(x \right)=x^3+5x^2-7x+9 \mathrm{a}\mathrm{t} x=1.1$ is

Identify correct statement for the reaction.
\(\mathrm{I}_2+\mathrm{KClO}_3 \xrightarrow{\Delta} \mathrm{ICl}+\mathrm{KIO}_3\)