Quadratic Equations
Roots Difference Bounded — Sum of Integer λ Values
nta_pyq_2026_jan
Grade None
Question:
Let $\alpha,\beta$ be the roots of the quadratic equation $12x^2-20x+3\lambda=0$, $\lambda\in\mathbb{Z}$. If $\dfrac{1}{2}\leq|\beta-\alpha|\leq\dfrac{3}{2}$, then the sum of all possible values of $\lambda$ is:
Step-by-Step Solution
Key Concept: $|\beta-\alpha|=\tfrac{\sqrt{25-9\lambda}}{3}$. Condition: $\tfrac{1}{2}\leq\tfrac{\sqrt{25-9\lambda}}{3}\leq\tfrac{3}{2}\Rightarrow\tfrac{9}{4}\leq25-9\lambda\leq\tfrac{81}{4}$.
$\lambda\in\{1,2\}$. Sum $=3$.
Correct Answer: 3