Complex numbers and quadratic equations is a segment of maths that deals with crucial theorems and concepts along with various formulae. It comprises of linear and quadratic equations along with roots related to the complex number's set (known as complex roots)..
Q3. If α,β,γ are the roots of x^3 -x^2 -1=0 then the value of (1+α)/(1-α)+(1+β)/(1-β)+(1+γ)/(1-γ) is equal to
Solution
(a) ∑ α=1,∑ αβ=0, αβγ = 1
∑(1+α) / (1-α) = -∑(-α+1-2)/(1-α) = ∑(2/(1-α)-1)
= 2∑〖1/(1-α)-3〗
Now,
1/((x-α)) +1/((x-β)) +1/((1-γ)) = ( 3x^2- 2x)/( x^3 -x^2 -1)
⇒ 1/(1-α) +1/(1-β) +1/(1-γ) =(3-2)/(1-1-1)=-1
⇒ (1+α)/(1-α)=-5
(a) ∑ α=1,∑ αβ=0, αβγ = 1
∑(1+α) / (1-α) = -∑(-α+1-2)/(1-α) = ∑(2/(1-α)-1)
= 2∑〖1/(1-α)-3〗
Now,
1/((x-α)) +1/((x-β)) +1/((1-γ)) = ( 3x^2- 2x)/( x^3 -x^2 -1)
⇒ 1/(1-α) +1/(1-β) +1/(1-γ) =(3-2)/(1-1-1)=-1
⇒ (1+α)/(1-α)=-5
Q4. If the equation |x^2+bx+c|=k has four roots, then
Q6. If a(p+q)^2+2bpq+c=0 and a(p+r)^2+2bpr+c=0 (a≠0), then
Solution
(b) Given,
a(p+q)^2+2bpq+c=0 and a(p+r)^2+2bpr+c=0
⇒q and r satisfy the equation a(p+x)^2+2bpx+c=0
⇒q and r are the roots of ax^2 + 2(ap +bp )x + c+ ap^2 = 0
⇒qr= product of roots =(c+ ap^2)/a = p^2 + c/a
(b) Given,
a(p+q)^2+2bpq+c=0 and a(p+r)^2+2bpr+c=0
⇒q and r satisfy the equation a(p+x)^2+2bpx+c=0
⇒q and r are the roots of ax^2 + 2(ap +bp )x + c+ ap^2 = 0
⇒qr= product of roots =(c+ ap^2)/a = p^2 + c/a
Q7.
The value of m for which one of the roots of x^2 -3x+ 2m = 0 is double of one of the roots of x^2-x+m=0 is
Solution
(a) Let α be the root of x^2-x+m=0 and 2α be the root of x^2-3x+2m=0.
Then, α^2-α+m=0 and 4α^2-6α+2m=0
Eliminating α,m^2=-2m
⇒m=0,m=-2
(a) Let α be the root of x^2-x+m=0 and 2α be the root of x^2-3x+2m=0.
Then, α^2-α+m=0 and 4α^2-6α+2m=0
Eliminating α,m^2=-2m
⇒m=0,m=-2
Q9.Total number of integral values of ‘a’ so that x^2-(a+1)x+a-1=0 has integral roots is equal to
Solution
(a) x^2-(a+1)x+a-1=0
⇒(x-a)(x-1)=1
Now, a∈I and we want x to be an integer.
Hence, x-a=1,x-1=1 or x-a=-1,x-1=-1
⇒a=1 in both cases
(a) x^2-(a+1)x+a-1=0
⇒(x-a)(x-1)=1
Now, a∈I and we want x to be an integer.
Hence, x-a=1,x-1=1 or x-a=-1,x-1=-1
⇒a=1 in both cases
Q10. z_1,z_2,z_3,z_4 are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If z_1-z_4=z_2-z_3 and arg〖[(z_4-z_1)/(z_2-z_1)]〗=Ï€/2, then the quadrilateral is: