Mathematics : Three Dimensional Geometry

The angle between the straight line $\mathrm{r} = \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and the plane $\mathrm{r} . \left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 4 is

• ${\sin }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{6}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

If a straight line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ with the coordinate axes, then $\frac{1 - {\tan }^2\left(\mathrm{\alpha }\right)}{1 - {\tan }^2\left(\mathrm{\alpha }\right)} + \frac{1}{\sec \left(2\mathrm{\beta }\right)} - 2{\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• - 1

• 1

• - 2

• 2

The equation of the plane which bisects the· line segment joining the points (3, 2, 6) and (5, 4, 8) and is perpendicular to the same line segment, is

• x + y + z = 16

• x + y + z = 10

• x + y + z = 12

• x + y + z = 14

The foot of the perpendicular from the point (1, 6, 3) to the line $\frac{\mathrm{x}}{1} = \frac{\mathrm{y} - 1}{2} = \frac{\mathrm{z} - 2}{3}$ is

• (1, 3, 5)

• (- 1, - 1, - 1)

• (2, 5, 8)

• (- 2, - 3, - 4)

The plane x + 3y + 13 = 0 passes through the line of intersection of the planes 2x - 8y + 4z = p and 3x - 5y + 4z + 10 = 0. If the plane is perpendicular to the plane 3x - y - 2z - 4 = 0, then the value of p is

• 2

• 5

• 9

• 3

If a straight line makes the angles 60°, 45° and $\mathrm{\alpha }$ with X, Y and Z-axes respectively, then ${\sin }^2\left(\mathrm{\alpha }\right)$ is equal to

• $\frac{3}{4}$

• $\frac{3}{2}$

• $\frac{1}{2}$

• 1

The angle between a and b is $\frac{5\mathrm{\pi }}{6}$ and  and the projection of a on b is $\frac{- 9}{\sqrt{3}}$, then $\left|\mathrm{a}\right|$ is equal to

• 12

• 8

• 10

• 6

The direction cosines of the straight line given by the planes x = 0 and z = 0 are

• 1, 0, 0

• 0, 0, 1

• 1, 1, 0

• 0, 1, 0

If a . b = 0 and a + b makes an angle of 60° with b, then $\left|\mathrm{a}\right|$ is equal to

• 0

• $\frac{1}{\sqrt{3}}\left|\mathrm{b}\right|$

• $\frac{1}{\left|\mathrm{b}\right|}$

• $\sqrt{3}\left|\mathrm{b}\right|$

The straight line r = $(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}) + \mathrm{a}(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}})$ meets the XY - plane at the point

• (2, - 1, 0)

• (3, 4, 0)

• $\left(\frac{1}{2}, \frac{3}{4}, 0\right)$

• $\left(\frac{1}{2}, \frac{5}{4}, 0\right)$