KVPY Sample Paper KVPY Stream-SX Model Paper-11

Let \[\vec{a}=\hat{i}+\sqrt{2}\hat{k},\] \[\vec{b}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+\sqrt{2}\hat{k}\] and \[\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k},\] be three vectors such that the projection vector of \[\vec{b}\] on \[\vec{a}\] is \[\vec{a}\]. If \[\vec{a}+\vec{b}\] is perpendicular to \[\vec{c},\] then \[\left| {\vec{b}} \right|\]is equal to:

A) \[\sqrt{22}\]

B) 4

C) \[\sqrt{32}\]

D) 6

Correct Answer: D

Solution :

Projection of \[\vec{b}\]on \[\vec{a}\] \[=\frac{\vec{a}.\vec{b}}{\left\| {\vec{a}} \right\|}=\left\| {\vec{a}} \right\|\]
\[\Rightarrow \] \[{{b}_{1}}+{{b}_{2}}=-10\]	? (1)
and \[\left( \vec{a}+\vec{b} \right)\bot \vec{c}\] \[\Rightarrow \] \[\left( \vec{a}+\vec{b} \right).\vec{c}=0\]
\[\Rightarrow \] \[5{{b}_{1}}+{{b}_{2}}=-10\]		? (2)
from [a] and (2)
\[\Rightarrow \] \[{{b}_{1}}=-\,3\] and \[{{b}_{2}}=5\]
then, \[\left\| {\vec{b}} \right\|=\sqrt{b_{1}^{2}+b_{2}^{2}+6.}\]