CAT 2025 Question Paper | Quant Slot 1

CAT Questions

CAT 2025 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 13 : The \((x, y)\) coordinates of vertices \(\mathrm{P}, \mathrm{Q}\) and R of a parallelogram PQRS are \((-3,-2),(1,-5)\) and \((9,1)\), respectively. If the diagonal SQ intersects the \(x\)-axis at \((a, 0)\), then the value of \(a\) is

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Video Explanation

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Explanatory Answer

In any parallelogram \(PQRS\), the diagonals \(PR\) and \(QS\) bisect each other at the same midpoint.

Midpoint of \(PR\):
\[ \text{Midpoint}=\left(\frac{-3+9}{2},\frac{-2+1}{2}\right) =\left(3,-\frac{1}{2}\right) \]

Line Equation using \(Q\) and \(M\):
\(Q(1,-5)\)
\(M(3,-\frac{1}{2})\)

Slope (\(m\)) of line \(QM\):
\[ m=\frac{-\frac{1}{2}-(-5)}{3-1} =\frac{\frac{9}{2}}{2} =\frac{9}{4} \]

Equation of line \(QM\):
\(y-(-5)=\frac{9}{4}(x-1)\)
\(y+5=\frac{9}{4}(x-1)\)

Set \(y=0\):
\(0+5=\frac{9}{4}a-1\)
\(5=\frac{9}{4}a-1\)
\(a-1=\frac{20}{9}\)
\(a=1+\frac{20}{9}=\frac{29}{9}\)

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The answer is '\(\frac{29}{9}\)'

Choice 2 is the correct answer.

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