What is the equation of the straight line cutting off an intercept 2 from the negative direction of y-axis and inclined at 30° with the positive direction of x-axis ?

• $\mathrm{x} - 2\sqrt{3}\mathrm{y} - 3\sqrt{2}$

• $\mathrm{x} + 2\sqrt{3}\mathrm{y} - 3\sqrt{2} = 0$

• $\mathrm{x} + \sqrt{3}\mathrm{y} - 2\sqrt{3} = 0$

• $\mathrm{x} - \sqrt{3}\mathrm{y} - 2\sqrt{3} = 0$

What is the equation of the line passing through the point of intersection of the lines x + 2y - 3 = 0 and 2x - y + 5 = 0 and parallel to the line y - x + 10 = 0

• 7x - 7y + 18 = 0

• 5x - 7y + 18 = 0

• 5x - 5y + 18 = 0

• x - y + 5 = 0

Consider the following statements :

1. The length p of the perpendicular from the origin to the line ax + by = c satisfies the relation ${\mathrm{p}}^2 = \frac{{\mathrm{c}}^2}{{\mathrm{a}}^2 + {\mathrm{b}}^2}$
2. The length p of the perpendicular from the origin to the line $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} = 1$ satisfies the relation $\frac{1}{{\mathrm{p}}^2} = \frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2}$
3. The length p of the perpendicular from the origin to the line y = mx + c satisfies the relation $\frac{1}{{\mathrm{p}}^2} = \frac{1 + {\mathrm{m}}^2 + {\mathrm{c}}^2}{{\mathrm{c}}^2}$

Which of the above is/are correct ?

• 1, 2 and 3

• 1 only

• 1 and 2 only

• 2 only

14.

What is the equation of the straight line passing through the point (2, 3) and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis ?

• $2\mathrm{x} + \mathrm{y} = 5$

• 2x + y = 7

• x + 2y = 7

• 2x - y = 1

The perpendiculars that fall from any point of the straight line 2x + 11y = 5 upon the two straight lines 24x + 7y = 20 and 4x - 3y = 2 are :

• 2 and 4 respectively

• 11 and 5 respectively

• Equal to each other

• Not equal to each other

The equation of the line, when the portion of it intercepted between the axes is divided by the point(2, 3) in the ratio of 3 : 2, is

• Either x + y = 4 or 9x + y = 12

• $\mathrm{Either} \mathrm{x} + \mathrm{y} = 5 \mathrm{or} 4\mathrm{x} + 9\mathrm{y} = 30$

• Either x + y = 4 or x + 9y = 12

• Either x + y = 5 or 9x + 4y = 30

What is the distance between the straight lines 3x + 4y = 9 and 6x + 8y = 15 ? 

• $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} - 6\mathrm{y} - 8\mathrm{z} = 4$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} + 6\mathrm{y} + 8\mathrm{z} = 4$

Which one of the following is correct in respect of the cube roots of unity ?

• They are collinear

• They lie on a circle of radius $\sqrt{3}$

• They form an equilateral triangle

• None of the above

The top of a hill observed from the top and bottom of a building of height h is at angles of elevation $\frac{\mathrm{\pi }}{6}$ and $\frac{\mathrm{\pi }}{3}$ respectively. What is the height of the hill ?

• 2h

• $\frac{3\mathrm{h}}{2}$

• h

• $\frac{\mathrm{h}}{2}$

Consider the following statements :

$$\begin{gathered} 1. \mathrm{The} \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{lines} \mathrm{y} = \mathrm{mx} + {\mathrm{c}}_1 \mathrm{and} \mathrm{y} = \mathrm{mx} + {\mathrm{c}}_2 \mathrm{is} \frac{\left|{\mathrm{c}}_1 - {\mathrm{c}}_2\right|}{\sqrt{1 + {\mathrm{m}}^2}} \\ 2. \mathrm{The} \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{lines} \\ \mathrm{ax} + \mathrm{by} + {\mathrm{c}}_1 = 0 \mathrm{and} \mathrm{ax} + \mathrm{by} + {\mathrm{c}}_2 = 0 \mathrm{is} \frac{\left|{\mathrm{c}}_1 - {\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}} \\ 3. \mathrm{The} \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{lines} \mathrm{x} = {\mathrm{c}}_1 \mathrm{and} \mathrm{x} = {\mathrm{c}}_2 \mathrm{is} \left|{\mathrm{c}}_1 - {\mathrm{c}}_2\right| \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{statements} \mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3