$A$ point on the plane determined by the points $A(1,1,-1)$, $B(2,-1,0)$, and $C(-1,0,2)$ among the following is:

The vector equation of the plane passing through the point $A(-2, 7, 5)$ and parallel to the vectors $\vec{b} = 4\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ is:

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2, 1, 1)$,$(1, -\lambda^2, 1)$,and $(1, 1, -\lambda^2)$ also passes through the point $(-1, -1, 1)$. Then $S$ is equal to

The distance of a point $(2, 3, -5)$ from the plane $\vec{r} \cdot (4 \hat{i} - 3 \hat{j} + 2 \hat{k}) = 4$ is

In a three-dimensional $xyz$ space,the equation $x^2 - 5x + 6 = 0$ represents:

If the plane $\frac{x}{2} + \frac{y}{3} + \frac{z}{3} = 1$ intersects the coordinate axes at $A, B, C$,then the area of $\Delta ABC$ is: