CAT 2024 Slot 3 QA Question & Solution
AlgebraMedium
Question
The sum of all distinct real values of x that satisfy the equation $10^x + \cfrac{4}{10^x} = \cfrac{81}{2}$, is
Options
$2 \log_{10}2$
$4 \log_{10}2$
$\log_{10}2$
$3 \log_{10}2$
Solution
Taking $10^x=a$
we get $a+\frac{4}{a}=\frac{81}{2}$
This would give the quadratic equation: $2a^2-81a+8=0$
We want to find the sum of possible values of x, let the value of x be x1 and x2
these would correspond to log a1, and log a2
The sum of log a1 + log a2 would be log (a1 x a2)
From the quadratic equation we got above, we can see that the product of the possible values of a would-be 8/2 = 4
Threfore, the sum of values of x would be log (4) which would be $2\ \log_{10}2$
Therefore, Option A is the correct answer.