If \(\log (x+y)=\log (x y)+a\), where a is constant, then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=2\) and \(y=4\) is

If \(A=\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1\end{array}\right]\), then \(A^{4} A^{-1}=\)

The direction cosines of a normal to the plane passing throgh \((4,2,3)\), \((-1,4,2)\) and \((3,2,1)\) are .....

If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are three non coplanar vectors and \(\overrightarrow{\mathbf{p}}, \overrightarrow{\mathbf{q}}, \overrightarrow{\mathbf{r}}\) are defined by the relations
\(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}, \quad \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]}\)
then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{p}}+\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{q}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{r}}\) is equal to

The area inside the parabola \(y^2=4 a x\), between the lines \(x=a\) and \(x=4 a\) is equal to

\(\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cdot \cos x} d x\) is equal to

If the vectors \(\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}+3 \overrightarrow{\mathbf{c}},-2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}-4 \overrightarrow{\mathbf{c}}\)
and \(\overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}+5 \overrightarrow{\mathbf{c}}\) are coplanar, then the value of \(\lambda\) is

The resultant logic gate from the combination of following gates is

Mole fraction of solute in it's 2 molal aqueous solution is

Which one of the following conversions does not involve either oxidation or reduction?

If \(\theta\) is a parameter, then the parametric equations of the circle
\(x^{2}+y^{2}-6 x+4 y-3=0\) are given by

A rectifier is used to

Plants usually absorb Nitrogen in the form of nitrates because Nitrogen is ________.