31.

A particle is acted upon by constant forces  which displace it from a point  The work done in standard units by the forces is given by 

• 40

• 30

• 25

• 25

• all values of λ

• all except one value of λ

• all except two values of λ

• all except two values of λ

Let  be such that  If the projection   is equal to that of  are perpendicular to each other then  equals

• 2

• 
• 
• 

Let  be non-zero vectors such that  If θ is the acute angle between the vectors  then sin θ equals

• 1/3

• 

• 2/3

• 2/3

Let $\overrightarrow{u}$ be a vector coplanar with the vectors a $\overrightarrow{a} = 2\hat{i} + 3\hat{j} -\hat{k} and \overrightarrow{b} = \hat{j} +\hat{k}.$ $if \overrightarrow{u} is perpendicular to \overrightarrow{a} and \overrightarrow{u}.\overrightarrow{b} = 24, then \left|\overrightarrow{u}\right| is equal to:$

• 84

• 336

• 315

• 256

The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane, x + y + z = 7 is:

• $\sqrt{\frac{2}{3}}$

• $\frac{2}{\sqrt{3}}$

• 2/3

• 1/3

For any vector x, where i, j, k have their usual meanings the value of ${\left(\mathrm{x} \times \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{x} \times \hat{\mathrm{k}}\right)}^2 \mathrm{where} \hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ have their usual meanings, is equal to

• ${\left|\mathrm{x}\right|}^2$

• 2${\left|\mathrm{x}\right|}^2$

• 3${\left|\mathrm{x}\right|}^2$

• 4${\left|\mathrm{x}\right|}^2$

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

• $\sqrt{2}$ units

• 2 units

• $\sqrt{3}$ units

• $\sqrt{5}$ units

For non-zero vectors a and b, if $\left|\mathrm{a} + \mathrm{b}\right| < \left|\mathrm{a} - \mathrm{b}\right|$,  then a and b are

• collinear

• perpendicular to each other

• inclined at an acute angle

• inclined at an obtuse angle

Which of the following is not always true?

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 = {\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2$, if a and b are perpendicular to each other

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$,  if a and b are perpendicular to each other

• ${\left|\mathrm{a} + \mathrm{b}\right|}^2 +\hspace{0.17em}{\left|\mathrm{a} - \mathrm{b}\right|}^2 = 2\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2\right)$

• $\left|\mathrm{a} + \mathrm{\lambda b}\right| \ge \left|\mathrm{a}\right| \mathrm{for} \mathrm{all} \mathrm{\lambda } \in \mathrm{R}$, if a is parallel to b