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 10 : Two tangents drawn from a point \( P \) touch a circle with center \( O \) at points \( Q \) and \( R \). Points \( A \) and \( B \) lie on \( P Q \) and \( P R \), respectively, such that \( A B \) is also a tangent to the same circle. If \( \angle A O B = 50 ^ { \circ } \), then \( \angle A P B \), in degrees, equals

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Video Explanation

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Explanatory Answer

Use tangency properties to relate angles

Let \(AB\) touch the circle at point \(C\).
Ray \(OA\) bisects \(\angle QOC\), so \(\angle QOA=\angle AOC=x\).
Ray \(OB\) bisects \(\angle COR\), so \(\angle COB=\angle BOR=y\).

Find total central angle \(\angle QOR\)
\(\angle AOB=x+y=50^\circ\)
\(\angle QOR=2x+2y=2(50^\circ)=100^\circ\)

Calculate \(\angle QPR\) from cyclic quadrilateral \(PQOR\)

In quadrilateral \(PQOR\), \(\angle PQO=90^\circ\) and \(\angle PRO=90^\circ\):
\(\angle QPR+\angle QOR=180^\circ\)
\(\angle QPR=180^\circ-100^\circ=80^\circ\)

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The answer is '80'

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