Question

The value of 'a' so that the sum of squares of the roots of the equation $x^2 - (a - 2)x - a + 1 = 0$ assumes the least value is

• A. 2
• B. 1
• C. 4
• D. 0

Hint:

To minimize a quadratic expression $Ax^2+Bx+C$, find the vertex at $x = -B/(2A)$.

Answer 1

The Correct Option is B

Step 1: Concept
Let roots be $\alpha, \beta$. Sum of squares is $S = \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.

Step 2: Meaning
$\alpha + \beta = a - 2$ and $\alpha\beta = -(a - 1) = 1 - a$.
$S = (a-2)^2 - 2(1-a) = a^2 - 4a + 4 - 2 + 2a = a^2 - 2a + 2$.

Step 3: Analysis
$S = a^2 - 2a + 2$ is a parabola opening upwards. Its minimum occurs at $a = -B/2A$.

Step 4: Conclusion
$a = -(-2) / 2(1) = 1$. Final Answer: (B)