Question

JEE ADVANCED MATHS QUESTION SEPTEMBER 2020

Answer 1

Answer in the attachment.

Answer 2

Let a and b be positive real numbers, suppose PQ = ai + bj and PS = ai - bj are adjacent sides of parallelogram PQRS. Let u and v be the projection vectors of w = i + j along PQ and PS, respectively. if |u| + |v| = |w| and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE ?

solution : projection vector u = |(i + j).PQ|

= |(i + j).(ai + bj)/√(a² + b²)|

= (a + b)/√(a² + b²)

similarly, projection vector v = |(i + j).PS|

= |(i + j).(ai - bj)/√(a² + b²)|

= (a - b)/√(a² + b²)

now, |u| + |v| = |w|

⇒|(a + b)|+ |(a - b)|/√(a² + b²) = √(1² + 1²) = √2

⇒2a = √2(√(a² + b²))

⇒4a² = 2a² + 2b²

⇒a² = b²

⇒a = b ........(1)

now area of parallelogram = |PQ × PS|

⇒8 = |(ai + bj) × (ai - bj)|

⇒8 = 2ab

⇒ab = 4

⇒a = b = 2

so a + b = 4 , option (A) is correct choice.

length of diagonal is PR = PQ + QR = (ai + bj) + (ai - bj) = 2ai

so, |PR| = 2a = 4 , option (C) is also correct choice.

Therefore options (A) and (C) are correct choices.