Quadratic Equation Question 4
Question 4 - 25 January - Shift 1
Let
$S={\alpha: \log _2(9^{2 \alpha-4}+13)-\log _2(\frac{5}{2} \cdot 3^{2 \alpha-4}+1)=2}$.
Then the maximum value of $\beta$ for which the equation $x^{2}-2(\sum _{\alpha \in S} \alpha)^{2} x+\sum _{\alpha \in S}(\alpha+1)^{2} \beta=0$ has real roots, is
Show Answer
Answer: 25
Solution:
Formula: Nature of Roots (ii)
$\log _2(9^{2 \alpha-4}+13)-\log _2(\frac{5}{2} \cdot 3^{2 \alpha-4}+1)=2$
$\Rightarrow \frac{9^{2 \alpha-4}+13}{\frac{5}{2} 3^{2 \alpha-4}+1}=4$
$\Rightarrow \alpha=2 \quad$ or $\quad 3$
$\sum _{\alpha \in S} \alpha=5$ and $\sum _{\alpha \in S}(\alpha+1)^{2}=25$
$\Rightarrow x^{2}-50 x+25 \beta=0$ has real roots
$\Rightarrow \beta \leq 25$
$\Rightarrow \beta _{\max }=25$
