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class 10 question

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Answer 1

Answer:

9x² - 28x + 3

Step-by-step explanation:

Given Equation is 3x² - 4x + 1.

On comparing with ax² + bx + c = 0,we get

a = 3, b =-4, c = 1.

(i)

Sum of zeroes = -b/a

⇒ α + β = -(-4)/3

⇒ α + β = 4/3

(ii)

Product of zeroes = c/a

⇒ αβ = 1/3.

Now,

Given zeroes are α²/β, β²/α.

(i)

Sum of zeroes = (α²/β) + (β²/a)

                        = (α³ + β³)/αβ

                        = (α + β)³ -3αβ(α + β)/αβ

                        = [(4/3)³ - 3(1/3)(4/3)]/1/3

                        = [64/27 - 4/3) * 3

                        = 28/9.

(ii)

Product of zeroes = (α²/β)(β²/α)

                              = αβ

                              = 1/3

Required Quadratic polynomial:

⇒ x² - (Sum of Zeroes)x + (Product of Zeroes) = 0

⇒ x² - (28/9)x + (1/3) = 0

⇒ 9x² - 28x + 3.

Hope it helps!

Answer 2

Step-by-step explanation:

For the quadratic equation with roots, α ,β

Sum of roots = 4/3

Product of roots = 1/3 .

Now, The quadratic equation with roots α²/β ,

β²/ α

Now,

Sum of roots = α²/β + β²/ α = α³+β³/αβ

= (α + β)³-3αβ(α+β)/αβ

= (4/3)³-3(1/3)(4/3) / 1/3

= 64/27 - 4/3 / 1/3

= 64/27-36/27 / 1/3

= 28/24 * 3/1

= 28/8

= 14/4

= 7/2 .

Product of roots = (α²/β * β²/ α) = (αβ ) = 1/3 .

The quadratic equation with required roots = x²-7/2x + 1/3 = 6x² - 21x + 2