For any three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ + $\overrightarrow{\mathrm{b}} \times \left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right)$ + $\overrightarrow{\mathrm{c}} \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$ is :

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) . \overrightarrow{\mathrm{c}}$

The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin is :

• 3x + 2y + z + 1 = 0

• 3x + 2y + z = 0

• 2x + 3y + z = 0

• x + y + z = 0

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are any three vectors, then $\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right]$ is equal to

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• ${\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}^2$

24.

The volume of the parallelopiped whose coterminus edges are $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, 2\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is :

• 4 cu unit

• 3 cu unit

• 2 cu unit

• 8 cu unit

If $\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)$ are the direction cosines of a line, then the value of n is

• $\frac{\sqrt{23}}{6}$

• $\frac{23}{36}$

• $\frac{2}{3}$

• $\frac{3}{2}$

For any vector $\overrightarrow{\mathrm{a}}, \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{j}} \times \left(\stackrel{\to }{\mathrm{a}}\times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ is equal to:

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}}$

• $2\overrightarrow{\mathrm{a}}$

• $3\overrightarrow{\mathrm{a}}$

The equation of the plane passing through (2, 3, 4) and parallel to the plane 5x - 6y + 7z = 3 is :

• 5x - 6y + 7z + 20 = 0

• 5x - 6y + 7z - 20 = 0

• - 5x + 6y - 7z + 3 = 0

• 5x + 6y + 7z + 3 = 0

If the vectors $3\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 8\hat{\mathrm{k}}$ are perpendicular, then $\mathrm{\lambda }$ is :

• - 14

• 7

• 14

• $\frac{1}{7}$

The projection of the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ along the vector $\hat{\mathrm{j}}$ is:

• 1

• 0

• 2

• - 1

The differential equation for, which y = a cos(x) + b sin(x) is a solution, is :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \left(\mathrm{a} + \mathrm{b}\right)\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = \left(\mathrm{a} + \mathrm{b}\right)\mathrm{y}$