CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. 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 3 : For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is

Video Explanation

Explanatory Answer

\( x + y = 4 \)
\( ( a + 5 ) x + \left( b ^ { 2 } - 15 \right) y = 8 b \)
if \( a _ { 1 } x + b _ { 1 } y + c _ { 1 } = 0 , a _ { 2 } x + b _ { 2 } y + c _ { 2 } = 0 \)have ∞ solutions, then the equations are multiples of one another.
That is, \( \frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } = \frac { c _ { 1 } } { c _ { 2 } } = k \)
So, we have, \( \frac { a + 5 } { 1 } = \frac { b ^ { 2 } - 15 } { 1 } = \frac { 8 b } { 4 } = k \)
\( a + 5 = b ^ { 2 } - 15 = 2 b \)
\( b ^ { 2 } - 15 = 2 b \)
\( b ^ { 2 } - 2 b - 15 = 0 \)
\( b ^ { 2 } - 5 b + 3 b - 15 = 0 \)
\( b ( b - 5 ) + 3 ( b - 5 ) = 0 \)
\( b = 5 \) or \( b = - 3 \)
\( a + 5 = 2 b \)
if \( b = 5 \);
\( a + 5 = 10 \)
\( a = 5 \)
\( a b = 25 \)
if \( b = - 3 \)
\( a + 5 = - 6 \)
\( a = - 11 \)
\( a b = 33 \)
So, the maximum possible value of \( a b = 33 \)

The question is " or some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is "

The answer is '33'

Choice B is the correct answer.