**Question 1 :**

If one root of the equation

2x^{2}-ax+64 = 0

is twice the other, then find the value of a.

**Solution :**

Roots of quadratic equation will be α and β.

α = 2β

By comparing the given equation with general form of quadratic equation, we get

a = 2, b = -a and c = 64

Sum of roots : α+β = -b/a = -(-a)/2 α+β = a/5 ---(1) |
Product of roots : αβ = c/a = 64/2 αβ = 32 ---(2) |

By applying the value of α in (1), we get

2β+β = a/5

3β = a/5

By applying the value of α in (2), we get

αβ = 32

2β (β) = 32

2β^{2} = 32

β = √16

β = √4 x 4

β = 4

Then,

3β = a/5

3(4) = a/5

a = 60

**Question 2 :**

If α and β are the roots of

5x^{2}-px+1 = 0

and α - β = 1, then find p.

**Solution:**

From the given quadratic equation, we get

a = 5, b = -p and c = 1

Sum of the roots : α+β = -b/a = -(-p)/5 = p/5 ----(1) |
Product of roots : αβ = c/a αβ = 1/5 ----(2) |

Given that :

α-β = 1

α-β = √(α+β)^{2}-4αβ

By applying the values from (1) and (2), we get

(p/5)^{2}-4(1/5) = 1

p^{2}/25 - 4/5 = 1

(p^{2}-20)/25 = 1

p^{2}-20 = 25

p^{2} = 45

p = 3√5

**Question 3 :**

If one root of the equation

3x^{2}+kx-81 = 0

is the square of the other, find k.

**Solution :**

α = β^{2}

a = 3, b = k and c = -81

Sum of the roots : α+β = -b/a α+β = -k/3 ---(1) |
Product of roots : α β = c/a = -81/2 αβ = -27 ---(2) |

By applying the value of α in(1), we get

β^{2}+β = -k/3 ---- (1)

Applying the value of α in (2), we get

β^{2}β = -27

β^{3} = (-3)^{3}

β = -3

β^{2}+β = -k/3

6 = k/3

18 = k

So, the value of k is 18.

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