JEE PYQ: Vector Algebra - Question ID 965bb7fd0f37 (JEE Main 2021)

Let the position vectors of $A,B,C,D,O$ be $\vec a,\vec b,\vec c,\vec d,\vec 0$ respectively.

Check Assertion (A)

$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD} =(\vec b-\vec a)+(\vec c-\vec a)+(\vec d-\vec a) =\vec b+\vec c+\vec d-3\vec a$

Also,

$4\overrightarrow{AO}+\overrightarrow{OB}+\overrightarrow{OC} =4(\vec 0-\vec a)+(\vec b-\vec 0)+(\vec c-\vec 0) =-4\vec a+\vec b+\vec c$

Equating both sides gives

$\vec b+\vec c+\vec d-3\vec a=-4\vec a+\vec b+\vec c \;\Rightarrow\; \vec d=-\vec a$

But $\vec d=-\vec a$ means $O$ is the midpoint of $AD$, i.e. $AD$ is a diameter. Since $A$ and $D$ are the endpoints of the given semicircle (as in the figure), this is true.

So Assertion A is correct (and it doesn’t actually need $|AB|=|BC|=|CD|$).

Check Reason (R)

By the polygon law along the path $A\to B\to C\to D$,

$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{AD}$

Hence

$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{AD} =2\overrightarrow{AD}\neq 2\overrightarrow{AO}$

So Reason R is incorrect.

Correct option

Option A: A is correct but R is not correct.