Let the mirror image of the point $P(1, 3, a)$ with respect to the plane $\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) - b = 0$ be $Q(-3, 5, 2)$. Then the value of $|a + b|$ is equal to ......

If a plane meets the axes $X, Y, Z$ in $A, B, C$ respectively such that the centroid of $\triangle ABC$ is $(1, 2, 3)$,then the equation of the plane is

The equation of the plane passing through the point $(2, -1, -3)$ and parallel to the lines $\frac{x - 1}{3} = \frac{y + 2}{2} = \frac{z}{-4}$ and $\frac{x}{2} = \frac{y - 1}{-3} = \frac{z - 2}{2}$ is

Assertion: The points $(2, 1, 5)$ and $(3, 4, 3)$ lie on opposite sides of the plane $2x + 2y - 2z - 1 = 0$.
Reason: The algebraic perpendicular distances from the given points to the plane have opposite signs.

The equation of the plane passing through the point $(1, 2, 3)$ and parallel to the plane $x + 2y + 5z = 0$ is

Let $A=(-3,-2,7)$ and $B=(3,1,-2)$. Let a plane perpendicular to the line segment $AB$ divide $AB$ in the ratio $2:1$. Then the intercept made by the plane on the $y$-axis is