Quadratic Equations
Discriminant condition for real roots
nta_pyq_2023_jan
Grade 11
Question:
Let $S = \left\{\alpha : \log_2(9^{2\alpha-4}+13) - \log_2\left(\dfrac{5}{2} \cdot 3^{2\alpha-4}+1\right) = 2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2\left(\displaystyle\sum_{\alpha \in S} \alpha\right)^2 x + \displaystyle\sum_{\alpha \in S}(\alpha+1)^2 \beta = 0$ has real roots, is ______.
Step-by-Step Solution
Key Concept: First solve for $S$ from the logarithmic equation (substituting $u = 3^{2\alpha-4}$). Then use the discriminant $\geq 0$ condition on the quadratic in $x$.
The log equation gives $\frac{9^{2\alpha-4}+13}{\frac{5}{2}\cdot 3^{2\alpha-4}+1} = 4$, yielding $\alpha = 2$ or $\alpha = 3$. So $\sum \alpha = 5$ and $\sum(\alpha+1)^2 = 9+16 = 25$. Equation: $x^2 - 50x + 25\beta = 0$. For real roots: $\Delta \geq 0 \Rightarrow 2500 - 100\beta \geq 0 \Rightarrow \beta \leq 25$. Maximum $\beta = 25$.
Correct Answer: 25