The equation of the line passing through the point $\left(-3, 1\right)$ and bisecting the angle between coordinate axes is

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

If \(f(x)=b \cdot e^{a x}+a \cdot e^{b x}\), then \(f^{\prime \prime}(0)=\)

\(\int_0^{\frac{\pi}{2}} \frac{\mathrm{~d} x}{1+(\cot x)^{101}}=\)

If \(x=\phi(t), y=\psi(t)\), then \(\frac{d^{2} y}{d x^{2}}\) is equal to

If the lines \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) are tangent to circle, then the radius of the circle is

A line with positive direction cosines passes through the point \(\mathrm{P}(2,-1,2)\) and makes equal angles with co-ordinate axes. The line meets the plane \(2 x+y+z=9\) at point Q . Then the length of the line segment PQ equals

A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m . The period of oscillation of the sphere is (acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) )

In compound microscope, the focal length and aperture of the objective used is respectively

Which among the following elements is obtained in pure form by liquation process of refining?

What is the boiling point of heavy water?

Buna-S is obtained from

Following combination of gates is equivalent to