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AIEEE 2010

JEE Main 2010 Mathematics question on Vector Algebra

Let PP, QQ, RR and SS be the points on the plane with position vectors −2i^−j^-2\hat{i} - \hat{j}, 4i^4\hat{i}, 3i^+3j^3\hat{i} + 3\hat{j} and −3i^+2j^-3\hat{i} + 2\hat{j}, respectively. The quadrilateral PQRSPQRS must be a
  1. A.
    parallelogram, which is neither a rhombus nor a rectangle
  2. B.
    square
  3. C.
    rectangle, but not a square
  4. D.
    rhombus, but not a square
Show answer and solution

Correct option: A

Solution

Find midpoints of diagonals PR and QS.

Midpoint of PR: ((-2+3)/2, (-1+3)/2) = (1/2, 1)
Midpoint of QS: ((4-3)/2, (0+2)/2) = (1/2, 1)

Diagonals bisect each other → PQRS is a parallelogram.

Now check sides:
PQ = (4-(-2), 0-(-1)) = (6,1), |PQ| = √37
QR = (3-4, 3-0) = (-1,3), |QR| = √10

Since |PQ| ≠ |QR|, it is not a rhombus.

Check if right angle: PQ·QR = (6)(-1)+(1)(3) = -3 ≠ 0, so not a rectangle.

Therefore PQRS is a parallelogram that is neither a rhombus nor a rectangle.

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