The angle between the two planes $x + y - z = 3$ and $3x + 2y + z = 5$ is :

The correct order of steps from A to E, for finding value of p, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angle is:

A. $p = \frac{70}{11}$

B. $(-3) \times \left(\frac{-3p}{7}\right) + 1 \times \left(\frac{2p}{7}\right) + 2 \times (-5) = 0$

C. $\frac{x-1}{-3} = \frac{y-2}{2\frac{p}{7}} = \frac{z-3}{2}$, $\frac{x-1}{-\frac{3p}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$

D. $\frac{9p}{7} + \frac{2p}{7} - 10 = 0$

E. $\frac{11p}{7} = 10$

Choose the correct answer from the options given below:

If lines $\frac{x+5}{5\lambda+2} = \frac{4-2y}{10} = \frac{1-3z}{-3}$ and $\frac{x-2}{1} = \frac{1+2y}{4\lambda} = \frac{2+z}{3}$ are perpendicular, than value of '$\lambda$' is

A line passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction of the vector $\hat{i} + \hat{j} - 2\hat{k}$. The equation of the line in Cartesian form is:

Let $L_1$ and $L_2$ be any two lines given by

$L_1: 3\hat{i}+2\hat{j}-4\hat{k} + \lambda(\hat{i}+2\hat{j}+2\hat{k})$

$L_2: 5\hat{i}-2\hat{j} + \mu(3\hat{i}+2\hat{j}+6\hat{k})$ Where $\lambda$ and $\mu$ are parameters

then which of the following statements are TRUE?

A. Equation of a line passing through the point $(1,2,3)$ which is parallel to line $L_1$ is $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{2}$

B. Direction cosines of line $L_2$ are $\dfrac{3}{7}, \dfrac{2}{7}, \dfrac{6}{7}$

C. Line $L_1$ is parallel to the line $L_2$

D. The angle between the two lines $L_1$ and $L_2$ is $\cos^{-1}\left(\dfrac{19}{21}\right)$

Choose the correct answer from the options given below:

The vector equation of line passing through (2, -1, 3) and perpendicular to the lines

$\frac{x-2}{3} = \frac{y-1}{1} = \frac{z+2}{2}$ and $\frac{x+3}{-4} = \frac{y-5}{-3} = \frac{z+1}{2}$ is

(Here $\lambda$ is a parameter)

The sum of all the values of $\lambda$ for which the distance of the point P (2, 3, $\lambda$) from the plane $x + 2y - 2z = 9$ is 3 units, is