JEE Advanced 2020 · Paper 2 · Mathematics Q11

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MathematicsMultiple correct+4 / -2Question 47 of 54

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ} = a\hat i + b\hat j$ and $\overrightarrow{PS} = a\hat i - b\hat j$ are adjacent sides of a parallelogram $PQRS$ . Let $\vec u$ and $\vec v$ be the projection vectors of $\vec w = \hat i + \hat j$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$ , respectively. If $|\vec u| + |\vec v| = |\vec w|$ and if the area of the parallelogram $PQRS$ is 8, then which of the following statements is/are TRUE?

a + b = 4

a - b = 2

The length of the diagonal

PR

of the parallelogram

PQRS

is 4

\vec w

is an angle bisector of the vectors

\overrightarrow{PQ}

and

\overrightarrow{PS}

Answer and worked solution

Official answer: A, C

Projections.
$|\vec u|=\frac{a+b}{\sqrt{a^2+b^2}},\qquad|\vec v|=\frac{|a-b|}{\sqrt{a^2+b^2}}.$

The condition $|\vec u|+|\vec v|=\sqrt2$ :
$\frac{2\max(a,b)}{\sqrt{a^2+b^2}}=\sqrt2\implies2\max(a,b)^2=a^2+b^2\implies a=b.$

Area. $|\overrightarrow{PQ}\times\overrightarrow{PS}|=2ab=8$ , so $a=b=2$ .

The statements.

(A) $a+b=4$ . True.
(B) $a-b=0\ne2$ . False.
(C) $\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{PS}=(4,0)$ , of length 4. True.
(D) The bisector of $(2,2)$ and $(2,-2)$ is along $\hat i$ , while $\vec w=\hat i+\hat j$ is along $\overrightarrow{PQ}$ itself. False.

(A), (C)

Topics:Dot product Cross product

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JEE Advanced 2020 · Paper 2, Mathematics question 11