If α, β are the roots of 3x^2 +5x-1=0, Construct equations whose root are (i) 5α, 5β (ii) α^2, β^2 (iii) 1/α, 1/β (iv) α+1/β, β+1/α

It's given that α, β are the roots of equation 3x^2 + 5x - 1 = 0. By applying the sum and product of roots formulae, we'll get

α + β = -5/3
αβ = -1/3

We also know that any quadratic equation is actually x^2 - (sum of its roots)x + (product of its roots) = 0. Like here, we have the equation x^2 -(-5/3)x + (-1/3) = 0 => 3x^2 + 5x - 1 = 0

(i) We have to find the equation whose roots are 5α, 5β

Sum of these roots = 5α + 5β = 5(α + β) = 5(-5/3) = -25/3
Product of these roots = 5α . 5β = 25αβ = -25/3

Hence, the equation will be

x^2 - (sum of roots)x + (product of roots) = 0
=> x^2 -(-25/3)x + (-25/3) = 0
=> 3x^2 + 25x - 25 = 0

(ii) The roots are α^2, β^2

Their sum = α^2 + β^2
= (α + β)^2 - 2αβ
= (-5/3)^2 - 2(-1/3)
= 25/9 + 2/3
= 31/9

Their Product = α^2.β^2
= (αβ)^2
= 1/9

Hence, the equation:
x^2 - (31/9)x + 1/9 = 0
=> 9x^2 -31x + 1 = 0

(iii) The roots are 1/α, 1/β

Their sum = 1/α +  1/β
= (α + β)/αβ
= (-5/3)/(-1/3) = 5

Their product = (1/α)(1/β) = 1/αβ = -3

Hence, the equation:
x^2 - (5)x + (-3) = 0
=> x^2 -5x - 3 = 0

(iv) The roots are (α + 1/β), (β + 1/α)

Their sum
= (α + 1/β)+ (β + 1/α)

= α + β + 1/α + 1/β

= (α + β) + (α + β)/αβ

= -5/3  + (-5/3)/(-1/3)

= -5/3 + 5

= 10/3

Their Product

= (α + 1/β) (β + 1/α)

= αβ + 1 + 1 + 1/αβ

= -1/3 + 2  +(-3)

= -4/3

Hence, the equation:

x^2 -(-10/3)x + (-4/3) = 0

=> 3x^2 + 10x - 4 = 0