CAT 2025 Question Paper | Quant Slot 2

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 16 : The equations \( 3 x ^ { 2 } - 5 x + p = 0 \) and \( 2 x ^ { 2 } - 2 x + q = 0 \) have one common root. The sum of the other roots of these two equations is

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Video Explanation

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Explanatory Answer

Let \(k\) be the common root.

For \(3x^2-5x+p=0\), let roots be \(k\) and \(\alpha\):
\(k+\alpha=\frac{5}{3}\)

For \(2x^2-2x+q=0\), let roots be \(k\) and \(\beta\):
\(k+\beta=\frac{2}{2}=1\)

Since \(k\) satisfies both equations:
\(2k+\alpha+\beta=\frac{5}{3}+1=\frac{8}{3}\)
\(\alpha+\beta=\frac{8}{3}-2k\)

Subtracting the two equations to eliminate \(k^2\):
\(3k^2-5k+p=0\Rightarrow6k^2-10k+2p=0\)
\(2k^2-2k+q=0\Rightarrow6k^2-6k+3q=0\)

\((6k^2-6k+3q)-(6k^2-10k+2p)=0\)
\(4k+3q-2p=0\Rightarrow2k=\frac{2p-3q}{2}\)

Compute the sum of the other roots \((\alpha+\beta)\):
\[ \alpha+\beta=\frac{8}{3}-\frac{2p-3q}{2} =\frac{8}{3}-p+\frac{3q}{2} \]

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The question is " The equations \( 3 x ^ { 2 } - 5 x + p = 0 \) and \( 2 x ^ { 2 } - 2 x + q = 0 \) have one common root. The sum of the other roots of these two equations is "

Hence, the answer is '\( \frac { 8 } { 3 } - p + \frac { 3 } { 2 } q \)'

Choice 4 is the correct answer.

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