Given two vectors $\hat{\mathrm{i}} - \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, the unit vector coplanar with the two vectors and perpendicular to first, is

• $\frac{1}{\sqrt{2}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\frac{1}{\sqrt{5}}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\pm \frac{1}{\sqrt{2}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• None of these

If a and b are unit vectors and $\mathrm{\theta }$ is the angle between them, then the value of $\left|\cos \left(\frac{\mathrm{\theta }}{2}\right)\right|$ is

• $\frac{1}{2}\left|\mathrm{a} + \mathrm{b}\right|$

• $\frac{1}{2}\left|\mathrm{a} - \mathrm{b}\right|$

• $\frac{\left|\mathrm{a} - \mathrm{b}\right|}{\left|\mathrm{a} + \mathrm{b}\right|}$

• $\frac{\left|\mathrm{a} + \mathrm{b}\right|}{\left|\mathrm{a} - \mathrm{b}\right|}$

If $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2& 1 + {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& 1 + {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& 1 + {\mathrm{c}}^3\end{array}\right|$ and vectors (1, a, a²), (1, b, b²) and (1, c, c²) are non-coplanar, then the product abc equals

• 2

• - 1

• 1

• 0

If the length of perpendicular drawn from origin on a plane is 7 unit and its direction ratios are - 3, 2 and 6, then that plane is

• - 3x + 2y + 6z - 7 = 0

• - 3x + 2y + 6z - 49 = 0

• 3x - 2y + 6z + 7 = 0

• - 3x + 2y - 6z - 49 = 0

655.

The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x² + y² + z² + 4x - 2y - 6z = 155,is

• 26

• $11\frac{4}{13}$

• 13

• 39

The acute angle between the line joining the points (2, 1, - 3), (- 3, 1, 7)and a line parallel to $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y}}{4} = \frac{\mathrm{z} + 3}{5}$, through the point (- 1, 0, 4), is

• ${\cos }^{-1}\left(\frac{7}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{1}{\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{3}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{1}{5\sqrt{10}}\right)$

Two systems ofrectangular axis have the same origin. If a plane cuts them at distances a, b, c and d', b', c' from the origin, then

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} + \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}}$ = 0

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

• $\frac{1}{{\mathrm{a}}^2} - \frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2} + \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

• $\frac{1}{{\mathrm{a}}^2} + \frac{1}{{\mathrm{b}}^2} + \frac{1}{{\mathrm{c}}^2} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{a}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{b}{\text{'}}^2}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{c}{\text{'}}^2}} = 0$

If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} - 1}{0} = \frac{\mathrm{z} - 1}{4}$ then its perpendicular distance from the origin is

• $\frac{3}{4}$

• $\frac{4}{3}$

• $\frac{7}{5}$

• 1

The intersection of the spheres x² + y² + z² + 7x - 2y - z = 13 and x² + y² + z² - 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane

• x - y - z = 1

• x - 2y - z = 1

• x - y - 2z = 1

• 2x - y - z = 1

If $\mathrm{\theta }$ is the angle between the vectors a = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and b = $6\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then

• $\cos \left(\mathrm{\theta }\right) = \frac{4}{21}$

• $\cos \left(\mathrm{\theta }\right) = \frac{3}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{2}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{5}{21}$