Xavier Aptitude Test 2025 Paper

Concept:Use the logarithm base-change rule to simplify the equation, then examine the discrminant of the resulting quadratic equation.Explanation:For $a > 0$ and $a \neq 1$, we have $\frac{\log_a b}{\log_a c} = \log_c b$.$$\frac{\log_a 16}{\log_a 32} = \log_{32}16 = \frac45$$$$\frac{\log_a 64}{\log_a 32} = \log_{32}64 = \frac65$$Substitute these values into the given equation:$$\frac45 x^2 - x - \frac65 = 0$$Multiply the whole equation by $5$:$$4x^2 - 5x - 6 = 0$$Find the discriminant:$$D = (-5)^2 - 4(4)(-6) = 25 + 96 = 121 > 0$$Since $D > 0$, this quadratic has two distinct real roots for every valid base $a$.However, the question only states $a > 0$. If $a = 1$, then $\log_a$ is not defined, so no real solution exists.Thus, the number of real values of $x$ changes depending on whether $a = 1$ or $a \neq 1$.Answer:Depends on the value of $a$.