Question

The real values of a for which the sum of the squares of the roots of the equation x2 – (a – 2) x – a – 1 = 0 assume the least value is

Solution

Correct option is

1

 

Discriminant of the equation is (a – 2)2 + 4(a + 1)

              = a2 – 4a + 4 + 4a + 4

              = a2 + 8 > 0 as a Ïµ R.

∴ Roots of the given equation are real. Let these roots be α and β

Then α + β = a – 2, αβ = – (α + 1).

We have

         α2 + β 2 = (α + β)2 - 2αβ = (α - 2)2 + 2 (α + 1)

                        = a2 – 4a + 4 + 2a + 2 = a2 – 2a + 6

                        = (a – 1)2 + 5

Thus, α2 + β 2 is least when a =1.

SIMILAR QUESTIONS

Q1

If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 +px + q = 0 has equal roots, the value of q is

Q2

If x2 + px + 1 is a factor of ax2 + bx + c, then

Q3

If 8, 2 are the roots of x2 + ax + β = 0 and 3, 3 are the roots of x2 + α xb = 0 then the roots of x2 + ax + b = 0 are

Q4

The number of real roots of 

Q5

The product of real of the equation  

              is

Q6

Sum of the non – real roots of 

 

Q7

If tan A and tan B are the roots of the quadratic equation x2 – px + q = 0, then value of sin2 (A + B) is

Q8

If x Ïµ R, the number of solution of  = 1 is

Q9

If lmn are real, l + m ≠ 0, then the roots of the equation

         (l + m)x2 – 3(l – m)x – 2 (l + m) = 0 are

Q10

If p and q are distinct primes and x2 – px + = 0 has distinct positive roots, then p + equals