The line \(x+y=1\) meets the lines represented by the equation \(y^3-6 x y^2+11 x^2 y-6 x^3=0\) at the points \(P, Q, R\). If \(O\) is the origin, then \((O P)^2+(O Q)^2+(O R)^2=\)

If \(z_1=10+6 i, z_2=4+6 i\) an \(z\) is any complex number such that the argument of \(\frac{\left(z-z_1\right)}{\left(z-z_2\right)}\) is \(\frac{\pi}{4}\), then

The interval in which \(y=\ln (\ln (x)), x>1\) is decreasing is

If the distance between the points \((a \cos \theta, a \sin \theta)\) and \((a \cos \phi, a \sin \phi)\) is \(2 a\), then \(\theta\) is equal to

If \(X \sim B(5, p)\) is a binomial variate such that \(p(X=3)=\) \(p(X=4)\), then \(p(|X-3| \lt 2)=\)

If \(1, \omega, \omega^2, \ldots \omega^{10}\) are 11 th roots of unity, then product of these roots is 1

If \((1+\tan \alpha)(1+\tan 4 \alpha)=2, \alpha \in\left(0, \frac{\pi}{16}\right)\), then \(\alpha\) is equal to

The correct equation for one mole of a real gas is (a, b are constants)

The standard reduction potentials of \(\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Ag}\) are respectively \(-0.76,0.34\) and \(0.8 \mathrm{~V}\). The following cells were constructed :
(1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}\)
(2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}\)
(3) \(\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}\)
What is the correct order of \(E_{\text {cell }}^{\circ}\) of these cells?

Let the length of the latus rectum of an ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) be equal to the length of its semi-major axis. If the radius of its director circle is \(\sqrt{3}\) and \(\mathrm{e}\) is its eccentricity, then the length of its latus rectum is

Match the following
\begin{array}{lll}
& \begin{array}{l}
List I \\
(Complex)
\end{array} & \begin{array}{l}
List II \\
(Color)
\end{array} \\
A. \left[\mathrm{Ni}(\mathrm{en})_3\right]^{2+} & I. & Green \\
B. \left[\mathrm{Ni}\left(\mathrm{H}_2 \mathrm{O}\right)_4(\mathrm{en})\right]^{2+} & II. Blue \\
C. \left[\mathrm{Ni}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+} & III. Pale blue \\
D. \left[\mathrm{Ni}\left(\mathrm{H}_2 \mathrm{O}\right)_6(\mathrm{en})_2\right]^{2+} & IV. Violet
\end{array}

If the reflection of a point \(\mathrm{A}(2,3)\) in X -axis is B ; reflection of B in the line \(x+y=0\) is C and the reflection of C in \(x-y=0\) is D then the point of intersection of the lines \(\mathrm{CD}, \mathrm{AB}\) is

The circle \(x^2+y^2-4 x-8 y+16=0\) rolls up along the tangent drawn to it at \((2+\sqrt{3}, 3)\) by 2 units. The equation of the circle in the new position is