The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x² + y² + z² + 4x - 2y - 6z = 155,is

• 26

• $11\frac{4}{13}$

• 13

• 39

The acute angle between the line joining the points (2, 1, - 3), (- 3, 1, 7)and a line parallel to $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y}}{4} = \frac{\mathrm{z} + 3}{5}$, through the point (- 1, 0, 4), is

• ${\cos }^{-1}\left(\frac{7}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{1}{\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{3}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{1}{5\sqrt{10}}\right)$

Two systems ofrectangular axis have the same origin. If a plane cuts them at distances a, b, c and d', b', c' from the origin, then

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} + \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}}$ = 0

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

• $\frac{1}{{\mathrm{a}}^2} - \frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} + \frac{1}{{\mathrm{c}}^2} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} - 1}{0} = \frac{\mathrm{z} - 1}{4}$ then its perpendicular distance from the origin is

• $\frac{3}{4}$

• $\frac{4}{3}$

• $\frac{7}{5}$

• 1

The intersection of the spheres x² + y² + z² + 7x - 2y - z = 13 and x² + y² + z² - 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane

• x - y - z = 1

• x - 2y - z = 1

• x - y - 2z = 1

• 2x - y - z = 1

If $\mathrm{\theta }$ is the angle between the vectors a = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and b = $6\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then

• $\cos \left(\mathrm{\theta }\right) = \frac{4}{21}$

• $\cos \left(\mathrm{\theta }\right) = \frac{3}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{2}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{5}{21}$

The ratio in which the xy-plane divides the join of (a, b, c) and (- a, - c, - b), is

• a : b

• b : c

• c : a

• c : b

Lines $\frac{\mathrm{x} - 2}{1} = \frac{\mathrm{y} - 3}{1} = \frac{\mathrm{z} - 4}{- \mathrm{k}}$ and $\frac{\mathrm{x} - 1}{\mathrm{k}} = \frac{\mathrm{y} - 4}{2} = \frac{\mathrm{z} - 5}{1}$ will be coplanar, if

• k = 0 or - 1

• k = 1 or - 1

• k = 0 or - 3

• k = 3 or - 3

The equation of the plane containing the line

$\frac{\mathrm{x} - {\mathrm{x}}_1}{\mathrm{l}} = \frac{\mathrm{y} - {\mathrm{y}}_1}{\mathrm{m}} = \frac{\mathrm{z} - {\mathrm{z}}_1}{\mathrm{n}}$ is

a(x - x₁) + b(y - y₁) + c(z - z₁) = 0, where

• ax₁ + by₁ + cz₁ = 0

• al + bm + cn = 0

• $\frac{\mathrm{a}}{\mathrm{l}} + \frac{\mathrm{b}}{\mathrm{m}} + \frac{\mathrm{c}}{\mathrm{n}}$

• lx₁ + my₁ + nz₁ = 0

Distance between the is $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} + 2}{- 2} = \frac{\mathrm{z} - 1}{2}$ and the plane 2x + 2y - z = 6, is

• 9 unit

• 1 unit

• 2 unit

• 3 unit