Let \(A=\left[\begin{array}{rrr}-1 & -2 & -3 \\ 3 & 4 & 5 \\ 4 & 5 & 6\end{array}\right], B=\left[\begin{array}{cr}1 & -2 \\ -1 & 2\end{array}\right] \quad\) and \(C=\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]\), if \(a, b\) and \(c\) respectively, denote the ranks of \(A, B\) and \(C\), then the correct order of these number is

\(\int_0^{\mathrm{x}} \frac{\mathrm{t}^2}{\sqrt{\mathrm{a}^2+\mathrm{t}^2}} \mathrm{dt}=\)

If \(\alpha\) and \(\beta\) are the roots of the equation
\(2 x^2-4 x+3=0\), then \(\frac{2\left(\alpha^4+\beta^4\right)+3\left(\alpha^2+\beta^2\right)}{\alpha+\beta}=\)

'The product of perpendiculars from the two foci of the ellipse \(\frac{x^2}{9}+\frac{y^2}{25}=1\) on the tangent at any point on the ellipse is

\(\sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin \frac{5 \pi}{16} \sin \frac{7 \pi}{16}\) is equal to

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

The angle between the pair of tangents drawn from $(1,1)$ to the circle $x^2+y^2+4x+4y-$$1=0$ is

Observe the following reaction
\(\mathrm{Cl}_2(\mathrm{~g})+2 \mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{ClO}^{-}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})+\mathrm{H}_2 \mathrm{O}(\mathrm{l})\)
Identify the correct statements about this reaction
I. \(\mathrm{Cl}_2(\mathrm{~g})\) is oxidized to \(\mathrm{ClO}^{-}\)(aq)
II. \(\mathrm{Cl}_2(\mathrm{~g})\) is oxidized to \(\mathrm{Cl}^{-}(\mathrm{aq})\)
III. \(\mathrm{Cl}_2(\mathrm{~g})\) is reduced to \(\mathrm{ClO}^{-}\)(aq)
IV. \(\mathrm{Cl}_2(\mathrm{~g})\) is reduced to \(\mathrm{Cl}^{-}(\mathrm{aq})\)
The correct answer is

Find the transformed equation of \(x \cos \theta+y \sin \theta=p\), when the axes are rotated through an angle \(\theta\).

The volume of one mole of the gas is changed from \(V\) to \(2 V\) at constant pressure \(p\). If \(\gamma\) is the ratio of specific heats of the gas, change in internal energy of the gas is

\(K_p\) for the conversion of oxygen to ozone at \(400 \mathrm{~K}\) is \(1.0 \times 10^{-30}\), its standard Gibbs energy change in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is approximately

The equation of any in the complex plane is of the form \(z \bar{z}+b \bar{z}+\bar{b} z+c=0\) where \((b \in \mathbf{C}, c \in \mathbf{R})\)

The ore which is concentrated by leaching