CAT 2023 Slot 3 QA Question 1: For some real numbers a and b, the system of equations $x + y = 4$ and $(a + 5)x + (b^2 - 15)y = 8b$ has infinitely many solutions for x and y. Then, the maximum possible value of ab is

Question

For some real numbers a and b, the system of equations $x + y = 4$ and $(a + 5)x + (b^2 - 15)y = 8b$ has infinitely many solutions for x and y. Then, the maximum possible value of ab is

Accepted Answer

For two linear equations to have infinitely many solutions, the ratios of the corresponding coefficients and constants must be equal.

$\dfrac{a + 5}{1} = \dfrac{b^2 - 15}{1} = \dfrac{8b}{4}$
$a + 5 = b^2 - 15 = 2b$

From $b^2 - 15 = 2b$,
$b^2 - 2b - 15 = 0$
$(b - 5)(b + 3) = 0$
$b = 5$ or $b = -3$

For some real numbers a and b, the system of equations x+y=4x + y = 4x+y=4 and (a+5)x+(b2−15)y=8b(a + 5)x + (b^2 - 15)y = 8b(a+5)x+(b2−15)y=8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is

For some real numbers a and b, the system of equations $x + y = 4$ and $(a + 5)x + (b^2 - 15)y = 8b$ has infinitely many solutions for x and y. Then, the maximum possible value of ab is