The pole of the line $\frac{x}{a}+\frac{y}{b}=1$ with respect of the circle $x^2+y^2=c^2$ is

If the curves \(2 x^2+k y^2=30\) and \(3 y^2=28 x\) cut each other orthogonally, then \(k=\)

If the coefficients of \(r\) th and \((r+1)\) th terms in the expansion of \((1+x)^{24}\) are in the ratio \(12: 13\), then \(r\) is the root of the quadratic equation

If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets $$ the $$ straight $$ line $5x-2y+6=0$ at a point $Q$ on the $Y$ -axis, then the length of $PQ$ is

If one of the lines given by the pair of lines \(3 x^2-2 y^2+a x y=0\) is making an angle \(60^{\circ}\) with x -axis then \(\mathrm{a}=\)

\(\sum_{n=1}^5 n\left(n^2+n+1\right)=\)

\(\sin \frac{\pi}{12} \sin \frac{2 \pi}{12} \sin \frac{3 \pi}{12} \sin \frac{4 \pi}{12} \sin \frac{5 \pi}{12} \sin \frac{6 \pi}{12}=\)

If the equation to the locus of points equidistant from the points \((-2,3),(6,-5)\) is \(a x+b y+c=0\), where \(a>0\), then the ascending order of \(a, b, c\) is

If the least positive integer n satisfying the equation \(\left(\frac{\sqrt{3}+\mathrm{i}}{\sqrt{3}-\mathrm{i}}\right)^{\mathrm{n}}=-1\) is p and the least positive integer \(m\) satisfying the equation \(\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^m=\operatorname{cis} \frac{2 \pi}{3}\) is \(q\), then \(\sqrt{p^2+q^2}=\)

A neutral ammonia \(\left(\mathrm{NH}_3\right)\) in its vapour state has an electric dipole moment of magnitude \(5 \times 10^{-30} \mathrm{C}\)-m. How far apart are the molecules centres of positive and negative charge.

Identify the reactions in which dichromate acts as an oxidising reagent.
I. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+6 \mathrm{Fe}^{2+}+14 \mathrm{H}^{+} \longrightarrow 2 \mathrm{Cr}^{3+}+6 \mathrm{Fe}^{3+}+7 \mathrm{H}_2 \mathrm{O}\)
II. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+2 \mathrm{OH}^{-} \longrightarrow 2 \mathrm{CrO}_4^{2-}+\mathrm{H}_2 \mathrm{O}\)
III. \(\mathrm{Cr}_2 \mathrm{O}_7^{2-}+6 \mathrm{I}^{-}+14 \mathrm{H}^{+} \longrightarrow 2 \mathrm{Cr}^{3+}+3 \mathrm{I}_2+7 \mathrm{H}_2 \mathrm{O}\)
IV. \(\mathrm{Na}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{KCl} \longrightarrow \mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{NaCl}\)

To find the coefficient of \(x^4\) in the expansion of \(\frac{3 x}{(x-2)(x-1)}\), the interval in which the expansion isvalid, is