A straight line with direction cosines (0,1, 0) is

• parallel to x-axis

• parallel to y-axis

• parallel to z-axis

• equally inclined to all the axes

(0, 0, 0), (a. 0, 0) (0, b, 0) and (0, 0, c) are four distinct points. What are the coordinates of the point which is equidistant from the four point ?

• $\left(\frac{\mathrm{a} + \mathrm{b} + \mathrm{c}}{3}, \frac{{\displaystyle \mathrm{a} + \mathrm{b} + \mathrm{c}}}{{\displaystyle 3}}, \frac{{\displaystyle \mathrm{a} + \mathrm{b} + \mathrm{c}}}{{\displaystyle 3}}\right)$

• (a, b, c)

• $\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{b}}{2}, \frac{\mathrm{c}}{2}\right)$

• $\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{b}}{3}, \frac{\mathrm{c}}{3}\right)$

The line passing through the points (1, 2, - 1) and ( 3, - 1, 2) meets the yz-plane at which one of the following points ?

• $\left(0, - \frac{7}{2}, \frac{5}{2}\right)$

• $\left(0, \frac{7}{2}, \frac{1}{2}\right)$

• $\left(0, - \frac{7}{2}, \frac{5}{2}\right)$

• $\left(0, \frac{7}{2}, - \frac{5}{2}\right)$

Under which one of the following conditions are the lines   x = ay + b; z = cy + d and x = ey + f; z= gy + h perpendicular?

• ae + cg - 1 = 0

• ae + bf - 1 = 0

• ae + cg + 1 = 0

• ae + ce + 1 = 0

The points P(3, 2, 4), Q(4, 5, 2), R(6, 8. 0) and S(2, - 1, 6) are

• vertices of a rhombus which is not a square

• non-coplanar

• collinear

• coplanar but not collinear

The angles of elevation of the top of a tower from the top and foot of a pole-are respectively 30° and 45°. If h_T is the height of the tower and h_P is the height of the pole, then which of the following are correct ?

$1. \frac{2{\mathrm{h}}_{\mathrm{P}}{\mathrm{h}}_{\mathrm{T}}}{3 + \sqrt{3}} = {{\mathrm{h}}_{\mathrm{P}}}^2$

$2 . \frac{{\mathrm{h}}_{\mathrm{T}} - {\mathrm{h}}_{\mathrm{P}}}{\sqrt{3} + 1} = \frac{{\mathrm{h}}_{\mathrm{P}}}{2}$

$3. \frac{2\left({\mathrm{h}}_{\mathrm{T}} + {\mathrm{h}}_{\mathrm{P}}\right)}{{\mathrm{h}}_{\mathrm{P}}} = 4 + \sqrt{3}$

Select the correct answer using the code given below :

• 1 and 3 only

• 2 and 3 only

• 1 and 2 only

• 1, 2 and 3

The points (a, b), (0, 0), (- a, - b) and (ab, b²) are :

• the vertices of a parallelogram

• the vertices of a rectangle

• the vertices of a square

• collinear

38.

The length of the normal from origin to the plane x + 2y - 2z = 9 is equal to :

• 2 units

• 3 units

• 4 units

• 5 units

The angle between the lines x + y - 3 = 0 and x - y + 3 = 0 is a and the acute angle between the lines  $\mathrm{x} - \sqrt{3\mathrm{y}} + 2\sqrt{3} \mathrm{and} \sqrt{3\mathrm{x}} - \mathrm{y} + 1 = 0 \mathrm{is} \mathrm{\beta }$. Which one of the following is correct ?

• $\mathrm{\alpha } = \mathrm{\beta }$

• $\mathrm{\alpha } > \mathrm{\beta }$

• $\mathrm{\alpha } < \mathrm{\beta }$

• $\mathrm{\alpha } = 2\mathrm{\beta }$

Let $\overrightarrow{\mathrm{\alpha }} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{\beta }} = 2\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}6\hat{\mathrm{k}}$ be the three vectors. If

$\overrightarrow{\mathrm{\alpha }} \mathrm{and} \overrightarrow{\mathrm{\beta }}$ are perpendicular to the vector $\overrightarrow{\mathrm{\delta }} \mathrm{and} \overrightarrow{\mathrm{\delta }}. \overrightarrow{\mathrm{\gamma }} = 10, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{magnitude} \mathrm{of} \overrightarrow{\mathrm{\delta }}$ ?

• $\sqrt{3}$ units

• $2\sqrt{3}$ units

• $\frac{\sqrt{3}}{2}$ units

• $\frac{1}{\sqrt{3}}$ units