Mathematics : Three Dimensional Geometry

If the pair of lines given by $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right){\cos }^2\left(\mathrm{\theta }\right) = {\left(\mathrm{xcos}\left(\mathrm{\theta }\right) + \mathrm{ysin}\left(\mathrm{\theta }\right)\right)}^2$ are perpendicular to each other, then $\mathrm{\theta }$ is equal to

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{3\mathrm{\pi }}{4}$

If the foot of the perpendicular from (0, 0, 0) to a plane is (1, 2, 3), then the equation of the plane is

• 2x + y + 3z = 14

• x + 2y + 3z = 14

• x + 2y + 3z + 14 = 0

• x + 2y - 3z = 14

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, {\mathrm{r}}_1{\mathrm{r}}_2 + {\mathrm{r}}_2{\mathrm{r}}_3 + {\mathrm{r}}_3{\mathrm{r}}_1 = ?$

• $\frac{{∆}^2}{{\mathrm{r}}^2}$

• $\frac{∆}{\mathrm{r}}$

• $\frac{2∆}{\mathrm{r}}$

• ${∆}^2$

$$\begin{gathered} \mathrm{If} \mathrm{in} \mathrm{a} ∆\mathrm{ABC}, \frac{1}{\mathrm{a} +\hspace{0.17em}\mathrm{c}} + \frac{1}{\mathrm{b} + \mathrm{c}} = \frac{3}{\mathrm{a} + \mathrm{b} + \mathrm{c}}, \mathrm{then} \\ \angle \mathrm{C} = ? \end{gathered}$$

• $30°$

• $45°$

• $60°$

• $90°$

A person observes the top of a tower from a point A on the ground. The elevation of the tower from this point is 60°. He moves 60 min the direction perpendicular to the line joining A and base of the tower. The angle of elevation of the tower from this point is 45°.Then, the height of the tower (in metres) is

• $60\sqrt{\frac{3}{2}}$

• $60\sqrt{2}$

• $60\sqrt{\frac{2}{3}}$

• $60\sqrt{3}$

The direction ratios of the two lines AB and AC are 1, - 1, - 1 and 2, - 1, 1. The direction ratios of the normal to the plane ABC are

• 2, 3, -1

• 2, 2, 1

• 3, 2, - 1

• - 1, 2, 3

A plane passing through(- 1, 2, 3) and whose normal makes equal angles with the coordinate axes is

• x + y + z + 4 = 0

• x - y + z + 4 = 0

• x + y + z - 4 = 0

• x + y + z = 0

A variable plane passes through a fixed point (1, 2, 3). Then, the foot of the perpendicular from the origin to the plane lies on

• a circle

• a sphere

• an ellipse

• a parabola

The angle between the lines $\hat{\mathrm{r}} = \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ and $\hat{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) + \mathrm{\mu }\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ is

• $\frac{\mathrm{\pi }}{2}$

• ${\cos }^{-1}\left(\frac{9}{\sqrt{91}}\right)$

• ${\cos }^{-1}\left(\frac{7}{\sqrt{84}}\right)$

• $\frac{\mathrm{\pi }}{3}$

The locus of the centroid of the triangle with vertices at (acos(θ), asin(θ)), (bsin(θ), - bcos(θ)) and (1, 0) is (here, θ is a parameter)

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$