In a triangle ABC, if taken in order, consider the following statements :

$$\begin{gathered} 1. \overrightarrow{\mathrm{AB}} + \overrightarrow{\mathrm{BC}} + \overrightarrow{\mathrm{CA}} = \overrightarrow{0} \\ 2. \overrightarrow{\mathrm{AB}} + \overrightarrow{\mathrm{BC}} - \overrightarrow{\mathrm{CA}} = \overrightarrow{0} \\ 3. \overrightarrow{\mathrm{AB}} - \overrightarrow{\mathrm{BC}} + \overrightarrow{\mathrm{CA}} = \overrightarrow{0} \\ 4. \overrightarrow{\mathrm{BA}} - \overrightarrow{\mathrm{BC}} + \overrightarrow{\mathrm{CA}} = \overrightarrow{0} \end{gathered}$$

How many of the above statements are correct ?

• One

• Two

• Three

• Four

$$\begin{gathered} \mathrm{A} \mathrm{spacecraft} \mathrm{located} \mathrm{at} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} +3\hat{\mathrm{k}} \mathrm{is} \mathrm{subjected} \mathrm{to} \mathrm{aforce} \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{by} \mathrm{firing} \mathrm{a} \mathrm{rocket}. \\ \mathrm{The} \mathrm{spacecraft} \mathrm{is} \mathrm{subjected} \mathrm{to} \mathrm{a} \mathrm{moment} \mathrm{of} \mathrm{magnitude} \end{gathered}$$

• $\mathrm{\lambda }$

• $\sqrt{3}\mathrm{\lambda }$

• $\sqrt{5}\mathrm{\lambda }$

• None of the above

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \mathrm{m}\hat{\mathrm{j}} + \mathrm{n}\hat{\mathrm{k}} \mathrm{are} \mathrm{colanar} \mathrm{vectors} \mathrm{and} \\ \left|\overrightarrow{\mathrm{c}}\right| = \sqrt{6}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• $\mathrm{m} = 2 \mathrm{and} \mathrm{n} = \pm 1$

• $\mathrm{m} = \pm 2 \mathrm{and} \mathrm{n} = - 1$

• $\mathrm{m} = 2 \mathrm{and} \mathrm{n} = - 1$

• $\mathrm{m} = \pm 2 \mathrm{and} \mathrm{n} = 1$

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin. What is $\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OB}} + \overrightarrow{\mathrm{OC}} + \overrightarrow{\mathrm{OD}} = ?$

• $2\overrightarrow{\mathrm{OP}}$

• $4\overrightarrow{\mathrm{OP}}$

• $6\overrightarrow{\mathrm{OP}}$

• $8\overrightarrow{\mathrm{OP}}$

ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct ?

• $\overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CD}} = \overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{DB}}$

• $\overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CD}} = \overrightarrow{\mathrm{BD}} + \overrightarrow{\mathrm{CA}}$

• $\overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CD}} = \overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{BD}}$

• $\overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CD}} = \overrightarrow{\mathrm{BC}} + \overrightarrow{\mathrm{AD}}$

$\mathrm{If} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{a}} , \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}} \mathrm{are} \mathrm{orthogonal} \mathrm{in} \mathrm{pairs} \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = \overrightarrow{\left|\mathrm{c}\right|} \mathrm{and} \left|\overrightarrow{\mathrm{b}}\right| = 1$

• $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}} \mathrm{are} \mathrm{non}-\mathrm{orthogonal} \mathrm{to} \mathrm{each} \mathrm{other}$

• $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}} \mathrm{are} \mathrm{orthogonal} \mathrm{in} \mathrm{pairs} \mathrm{but} \left|\overrightarrow{\mathrm{a}}\right| \ne \overrightarrow{\left|\mathrm{c}\right|}$

• $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}} \mathrm{are} \mathrm{orthogonal} \mathrm{in} \mathrm{pairs} \mathrm{but} \left|\overrightarrow{\mathrm{b}}\right| \ne 1$

$\mathrm{If} \overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} - \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{are} \mathrm{perpendicular}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\lambda } ?$

• 2

• 3

• 4

• 5

If $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are the angles which the vector $\overrightarrow{\mathrm{OP}}$(0 being the origin) makes with positive direction of the coordinate axes, then which of the following are correct ?

$$\begin{gathered} 1. {\cos }^2\left(\mathrm{\alpha }\right) + {\cos }^2\left(\mathrm{\beta }\right) = {\sin }^2\left(\mathrm{\gamma }\right) \\ 2. {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) = {\cos }^2\left(\mathrm{\gamma }\right) \\ 3. {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right) = 2 \\ \mathrm{Select} \mathrm{the} \mathrm{correct} \mathrm{answer} \mathrm{using} \mathrm{the} \mathrm{code} \mathrm{given} \mathrm{below} : \end{gathered}$$

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

29.

If $\hat{\mathrm{a}} \mathrm{and} \hat{\mathrm{b}}$ are two unit vectors, then the vectors, then the vector $\left(\hat{\mathrm{a}} + \hat{\mathrm{b}}\right) \times \left(\hat{\mathrm{a}} \times \hat{\mathrm{b}}\right)$ is parallel to :

• $\left(\hat{\mathrm{a}} - \hat{\mathrm{b}}\right)$

• $\left(\hat{\mathrm{a}} + \hat{\mathrm{b}}\right)$

• $\left(2\hat{\mathrm{a}} - \hat{\mathrm{b}}\right)$

• $\left(2\hat{\mathrm{a}} + \hat{\mathrm{b}}\right)$

A force $\overrightarrow{\mathrm{F}} = \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ acts on a particle to displace it from the point $\mathrm{A}\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ to the point $\mathrm{B}\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$. The work done by the force will be :

• 5 units

• 7 units

• 9 units

• 10 units