431.

The equation of the plane passing through the origin and containing the line

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

• x + 5y - 3z = 0

• x - 5y + 3z = 0

• x - 5y - 3z = 0

• 3x - 10y + 5z = 0

A flagpole stands on a building of height 450 ft and an observer on a level ground is 300 ft from the base of the building. The angle of elevation of the bottom of the flagpole is 30° and the height of the flagpole is SO ft. If 8 is the angle of elevation of the top of the flagpole, then tan$\left(\mathrm{\theta }\right)$ is equal to

• $\frac{4}{3\sqrt{3}}$

• $\frac{\sqrt{3}}{2}$

• $\frac{9}{2}$

• $\frac{\sqrt{3}}{5}$

If A (0, 0), B (12, 0), C (12, 2), D (6, 7) and E (0, 5) are the vertices of the pentagon ABCDE, then its area in square units, is

• 58

• 60

• 61

• 63

The equation of the plane perpendicular to the line $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y} - 2}{- 1} = \frac{\mathrm{z} + 1}{2}$  afd passing through the point (2, 3, 1) is

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) = 1$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) = 1$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) = 7$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right) = 10$

If the planes $\overrightarrow{\mathrm{r}} . \left(2\hat{\mathrm{i}} - \mathrm{\lambda }\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) = 0 \mathrm{and} \overrightarrow{\mathrm{r}} . \left(\mathrm{\lambda }\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) = 5$  are perpendicular to each other, then the value of ${\mathrm{\lambda }}^2 + \mathrm{\lambda }$ is

• 0

• 2

• 1

• 3

The cartesian form of the plane $\overrightarrow{\mathrm{r}} = \left(\mathrm{s} - 2\mathrm{t}\right)\hat{\mathrm{i}} + \left(3 - \mathrm{t}\right)\hat{\mathrm{j}} + \left(2\mathrm{s} + \mathrm{t}\right)\hat{\mathrm{k}}$ is

• 2x - 5y -  z - 15 = 0

• 2x - 5y +  z - 15 = 0

• 2x - 5y -  z + 15 = 0

• 2x + 5y -  z + 15 = 0

Let P(- 7, 1, - 5) be a point on a plane and let O be the origin. If OP is normal to the plane, then the equation of the plane is

• 7x - y + 5z + 75 = 0

• 7x + y - 5z + 73 = 0

• 7x + y + 5z + 73 = 0

• 7x - y - 5z + 75 = 0

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 point in the xy-plane which is equidistant from the point (2, 0, 3), (0, 3, 2) and (0, 0, 1) is

• (1, 2, 3)

• (- 3, 2, 0)

• (3, - 2, 0)

• (3, 2, 0)

The angle between the line $\frac{3\mathrm{x} - 1}{3} = \frac{\mathrm{y} +\hspace{0.17em}3}{- 1} = \frac{5 - 2\mathrm{z}}{4}$ and the plane 3x - 3y - 6z = 10 is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$