Quadratic Equations
Integer Roots — Divisibility Constraint on λ
nta_pyq_2024_jan
Grade 11
Question:
Let $\alpha,\beta\in\mathbb{N}$ be roots of equation $x^2-70x+\lambda=0$, where $\dfrac{\lambda}{2},\dfrac{\lambda}{3}\notin\mathbb{N}$. If $\lambda$ assumes the minimum possible value, then $\dfrac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to:
Step-by-Step Solution
Key Concept: $\alpha+\beta=70$, $\alpha\beta=\lambda$. Need $\lambda$ not divisible by 2 or 3, minimize $\lambda=\alpha(70-\alpha)$. Try $\alpha=5$: $\lambda=5\times65=325$ (not div by 2 or 3). Check $\alpha=1$: $\lambda=69$ divisible by 3. $\alpha=7$: $\lambda=441$ divisible by 3. $\alpha=5$: $\lambda=325=5^2\times13$, not div by 2 or 3. ✓ $\beta=65$.
$\alpha=5,\beta=65,\lambda=325$. $\frac{(2+8)(360)}{60}=60$.
Correct Answer: 60