If the image of $\left(\frac{- 7}{6}, \frac{- 6}{5}\right) \mathrm{in} \mathrm{a} \mathrm{line} \mathrm{is} \left(1, 2\right), \mathrm{then} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{is}$

• 4x + 3y = 1

• 3x - y = 0

• 4x - y = 0

• 3x + 4y = 1

If a line l passes through (k, 2k), (3k, 3k) and (3, 1), k $\ne$ 0, then the distance from the origin to the line is

• $\frac{1}{\sqrt{5}}$

• $\frac{4}{\sqrt{5}}$

• $\frac{3}{\sqrt{5}}$

• $\frac{2}{\sqrt{5}}$

The area (in sq. units) of the triangle formed by the lines x² - 3y + y = 0 and x + y + 1 = 0, is

• $\frac{2}{\sqrt{3}}$

• $\frac{\sqrt{3}}{2}$

• $5\sqrt{2}$

• $\frac{1}{2\sqrt{5}}$

If ${\mathrm{x}}^2 + {\mathrm{\alpha y}}^2 +\hspace{0.17em}2\mathrm{\beta y} = {\mathrm{\alpha }}^2$ represents a pair of perpendicular lines, then p equals to

• 4a

• a

• 2a

• 3a

The condition for the lines lx + my + n = 0 and l₁a + m₁y + n₁ = 0 to be conjugate with respect to the circle x² + y² = r², is

• r²(ll₁ + mm₁) = nn₁

• r²(ll₁ - mm₁) = nn₁

• r²(ll₁ + mm₁) + nn₁ = 0

• r²(lm₁ + l₁m) = nn₁

If the line joining A(1, 3, 4) and B is divided by the point    (- 2, 3, 5) in the ratio 1 : 3, then B is

• (- 11, 3 , 8)

• (- 11, 3 , - 8)

• (8, 12 , 20)

• (13, 6 , - 13)

If the direction cosines of two lines are given by l + m + n = 0 and l² - 5m² + n² = 0, then the angle between them is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

If A(3, 4, 5), B(4, 6, 3), C( - 1, 2, 4) and D(1, 0, 5) are such that the angle between the lines DC and AB is $\left(\mathrm{\theta }\right)$ , then cos$\left(\mathrm{\theta }\right)$ is equal to

• $\frac{7}{9}$

• $\frac{2}{9}$

• $\frac{4}{9}$

• $\frac{5}{9}$

If the point P(1, 3) undergoes the following transformations successively.

(i) Reflection with respect to the line y = x.

(ii) Translation through 3 units along the positive direction of the X-axis.

(iii) Rotation through an angle of $\frac{\mathrm{\pi }}{6}$ about the origin in the clockwise direction.Then, the final position of the point P is

• $\left(\frac{6\sqrt{3} + 1}{2}, \frac{\sqrt{3}{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle 6}}{2}\right)$

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

• $\left(\frac{6 + \sqrt{3}}{2}, \frac{1{\displaystyle }{\displaystyle -} {\displaystyle }{\displaystyle 6\sqrt{3}}}{2}\right)$

• $\left(\frac{6 + \sqrt{3} - 1}{2}, \frac{6 + \sqrt{3}}{2}\right)$

$$\begin{gathered} \mathrm{The} \mathrm{shortest} \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{skew} \mathrm{lines} \\ \hat{\mathrm{r}} = \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) \mathrm{and} \hat{\mathrm{r}} = \left(4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right) + \mathrm{t}\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) \mathrm{is} \end{gathered}$$

• $\sqrt{6}$

• 3

• $2\sqrt{3}$

• $\sqrt{3}$