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Question

Let P,Q,R and S be the points on the plane with position vectors 2^i^j,4^i,3^i+3^j and 3^i+2^jrespectively. The quadrilateral PQRS must be a

A
Parallelogram, which is neither a rhombus nor a rectangle
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B
Rectangle, but not a square
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C
Square
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D
Rhombus, but not a square
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Solution

The correct option is A Parallelogram, which is neither a rhombus nor a rectangle
Let O be the origin,
then OP=2^i^j
OQ=4^i
OR=3^i+3^j
OS=3^i+2^j
Here we have
PQ=6^i+^j
QR=^i+3^j
RS=6^i^j
and PS=^i+3^j
PR=5^i+4^j
QS=7^i+2^j
PR.QS=35+8=270
Diagonals are not perpendicular
and PQ=RS,QR=PS
Hence PQRS is a parallelogram which is neither a rhombus nor a rectangle.

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