Question

Please help in solving this question.......

Answer 1

Answer:

(4) 19

Step-by-step explanation:

Given equation is x² - x(λ - 2) + (λ² + 3λ + 5)=0

Since, α & β are real roots of the given equation,

So, the equation's discriminant will be greater than or equal to 0

so,

(λ - 2)² - 4(λ² + 3λ + 5) ≥ 0

=> λ² - 4λ + 4 - 4λ² - 12λ - 20 ≥ 0

=> -3λ² - 16λ - 16 ≥ 0

=> 3λ² + 16λ + 16 ≤ 0

=> 3λ² + 12λ + 4λ + 16 ≤ 0

=> 3λ (λ + 4) + 4 (λ + 4) ≤ 0

=> (λ + 4)(3λ + 4) ≤ 0

=> λ € ( - 4, - 4/3 )

Now, we have,

α + β = (λ - 2) & αβ = (λ² + 3λ + 5)

α² + β² = (α + β)² - 2αβ

= (λ -2)² - 2(λ² + 3λ + 5)

= λ² - 4λ + 4 - 2λ² - 6λ - 10

= - λ² - 10λ - 6

= - (λ² + 10λ +25 - 19)

= - (λ + 5)² + 19

Since, (a+b)²≥0,

Hence the maximum value of α² + β² is 19