Question

please answer these questions

Answer 1

Hey

Here is your answer,

5) The given polynomial is

f (x) = 2x² + 5x + k

Since, α and β are the zeroes of f (x),

α + β = - 5/2 ...(i)

αβ = k/2 ...(ii)

Given that,

α² + β² + αβ = 21/4

⇒ (α + β)² - 2αβ + αβ = 21/4
⇒ (α + β)² - αβ = 21/4
⇒ (- 5/2)² - k/2 = 21/4, by (i) and (ii)
⇒ 25/4 - k/2 = 21/4
⇒ k/2 = 25/4 - 21/4
⇒ k/2 = (25 - 21)/4
⇒ k/2 = 4/4
⇒ k/2 = 1
⇒ k = 2

∴ The value of k is 2.

6) For the polynomial x²-(k-3)x+(5k-3)

Sum of the roots = -b/a = (k+3)

Product of the roots = c/a = (5k-3)

As per the question, 
Sum of zeroes = 1/4( product of zeroes)
k+3 = 1/4(5k-3)
4k+12 = 5k - 3
k = 15.

Hope it helps you!