At any point for the curve \(3 y^2=(x+5)^3\), if ST represents the length of the subtangent and SN represents the length of the subnormal then \(9(\mathrm{ST})^2=\)

The values of \(\lambda\) and \(\mu\) for which the system of equations \(x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=\mu\) has infinitely many solutions are

The pair of lines joining the origin to the points of intersection of the line \(x-y=2\) with the curve \(5 x^2+12 x y-8 y^2+8 x-4 y+12=0\) are equally inclined to the pair of lines

If the slope of one line of the pair of lines \(2 x^2+h x y+6 y^2\) \(=0\) is thrice the slone of the other line, then \(\mathrm{h}=\)

Two particles $P$ and $Q$ located at the points $P\left(t,t^3-16t-3\right),Q\left(t+1,t^3-6t-6\right)$ are moving in a plane, the minimum distance between the points in their motion is

The equation of the circle which passes through the origin and cuts orthogonally each of the circles \(x^2+y^2-6 x+8=0\) and \(x^2+y^2-2 x-2 y=7\) is

The radius of the circle represented by $(1+i)(1+3 i)(1+7 i)=x+iy$ is $(i=\sqrt{-1})$

If a message signal of frequency \(10 \mathrm{kHz}\) and peak voltage \(12 \mathrm{~V}\) is used to modulate a carrier wave of frequency \(1 \mathrm{MHz}\), the modulation index is 0.6 . To make the modulation index 0.75 , the carrier peak voltage should be

P and Q are the ends of a diameter of the circle \(\mathrm{x}^2+\mathrm{y}^2=\mathrm{a}^2\left(\mathrm{a}>\frac{1}{\sqrt{2}}\right) \cdot s\) and \(t\) are the lengths of the perpendiculars drawn from \(P\) and \(Q\) onto the line \(x+y=1\) respectively. When the product \(s t\) is maximum, the greater value among \(s, t\) is

If \(f(x)=5 \cos ^3 x-3 \sin ^2 x\) and \(g(x)=4 \sin ^3 x+\cos ^2 x\), then the derivative of \(f(x)\) with respect to \(g(x)\) is

A gaseous mixture consists of \(4 \mathrm{~g}\) oxygen and \(4 \mathrm{~g}\) of helium. The ratio \(\frac{C_p}{C_V}\) of mixture is \(\left(C_p\right.\) and \(C_V\) are molar specific heats of the mixture at constant pressure and at constant volume respectively).

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. \(\mathrm{Na}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{KCl} \longrightarrow \mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7+2 \mathrm{NaCl}\)

If \(C_r={ }^n C_r\), then \(C_0+C_4+C_8+C_{12}+\ldots=\)