MHTCET 15 May 2024 Shift 2 Paper

Concept:A quadrilateral is a parallelogram if both pairs of opposite sides are equal and parallel. For a parallelogram, equal diagonals mean rectangle, and perpendicular diagonals mean rhombus.Explanation:The vertices are $P(-2,-1)$, $Q(4,0)$, $R(3,3)$ and $S(-3,2)$.Side vectors are:$\overrightarrow{PQ} = (6,1)$, $\overrightarrow{QR} = (-1,3)$, $\overrightarrow{RS} = (-6,-1)$, $\overrightarrow{SP} = (1,-3)$.Since $\overrightarrow{PQ} = -\overrightarrow{RS}$ and $\overrightarrow{QR} = -\overrightarrow{SP}$, opposite sides are equal and parallel.Hence, $PQRS$ is a parallelogram.Now check diagonals: $\overrightarrow{PR} = (5,4)$ and $\overrightarrow{QS} = (-7,2)$.So $PR = \sqrt{41}$ and $QS = \sqrt{53}$, which are not equal; therefore it is not a rectangle.Also, adjacent sides are $PQ = \sqrt{37}$ and $QR = \sqrt{10}$, which are not equal; therefore it is not a rhombus.Thus, $PQRS$ is a parallelogram that is neither a rhombus nor a rectangle.Answer:Option A: parallelogram, which is neither a rhombus nor a rectangle.