Answer:
- 11) k = -1
- 12) (i) 4, (ii) 6.5
Step-by-step explanation:
11. ......................................................
Given equation
- x² + (k - 1)x - k = 0
- The roots are equal
To find
Solution
When the roots are equal, the discriminant equals zero
Substituting the values and solving for k:
- (k - 1)² - 4*1*(-k) = 0
- k² - 2k + 1 - 4k = 0
- k² + 2k + 1 = 0
- (k + 1)² = 0
- k + 1 = 0
- k = -1
12. ......................................................
Given equation
- 2x² - 6x + 1 = 0
- The roots are α and β
To find
The values of
- (i) 2αβ² + 2α²β + 2αβ
- (ii) α² - 3αβ + β²
Solution
The sum and the product of the roots is:
- α + β = - b/a = - ( - 6)/2 = 3
- αβ = c/a = 1/2
(i)
- 2αβ² + 2α²β + 2αβ =
- 2αβ ( α + β + 1) =
- 2(1/2)(3 + 1) =
- 1(4) =
- 4
(ii)
- α² - 3αβ + β² =
- (α + β)² - 5αβ =
- 3² - 5(1/2) =
- 9 - 5/2 =
- 6.5
