Complex Number & Quadratic Equation Quiz-12
Dear Readers,
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)..
Q1. If |z2-1|=|z|2+1, then z lies on
Solution
(d)
Let z = x+iy. Then,
|z2-1|=|z|2+1
⇒ |(x2-y2-1)+2ixy|=x2+y2+1
⇒ (x2-y2-1)2+4x2 y2=(x2+y2+1)2 ⇒ x=0 Hence, z lies on imaginary axis
(d)
Let z = x+iy. Then,
|z2-1|=|z|2+1
⇒ |(x2-y2-1)+2ixy|=x2+y2+1
⇒ (x2-y2-1)2+4x2 y2=(x2+y2+1)2 ⇒ x=0 Hence, z lies on imaginary axis
Q2.The locus of point z satisfying Re(1/z) = k, where k is a non-zero real number, is
Solution
Q3. The number of integral values of x satifying √(–x2+10x-16)
Solution
Q4. Let α , β be the roots of the equation x2-px+r=0 and α/2 , 2β be the roots of the equation x2-qx+r=0. Then the value of r is
Solution
(d)
Since, the equation x^2-px+r=0 has roots (α,β) and the equation x2-qx+r=0 has roots (α/2,2β)
∴ α+β = p and r = αβ and α/2 + 2β=q
⇒ β=(2q-p)/3 and α=(2(2p-q))/3
∴ αβ=r=2/9 (2q-p)(2p-q)
(d)
Since, the equation x^2-px+r=0 has roots (α,β) and the equation x2-qx+r=0 has roots (α/2,2β)
∴ α+β = p and r = αβ and α/2 + 2β=q
⇒ β=(2q-p)/3 and α=(2(2p-q))/3
∴ αβ=r=2/9 (2q-p)(2p-q)
Q5.For x2-(a+3) |x|+4=0 to have real solutions, the range of a is
Solution
(d)
a = (x2+4)/(|x|)-3
= |x| + 4/|x| - 3 = (√(|x| ) - 2/√(|x| ))+1
⇒ a ≥ 1
(d)
a = (x2+4)/(|x|)-3
= |x| + 4/|x| - 3 = (√(|x| ) - 2/√(|x| ))+1
⇒ a ≥ 1
Q6. If z=3/(2+ cosθ + isinθ), then locus of z is
Solution
Q7. The number of real solutions of the equation (9/10)x=-3+x-x2 is
Solution
(b) Let f(x)=-3+x-x2. Then f(x) < 0 for all x, because coefficient of x2 is less than 0 and D< 0. Thus, L.H.S. of the given equation is always positive whereas the R.H.S. is always less than zero. Hence, there is no solution
(b) Let f(x)=-3+x-x2. Then f(x) < 0 for all x, because coefficient of x2 is less than 0 and D< 0. Thus, L.H.S. of the given equation is always positive whereas the R.H.S. is always less than zero. Hence, there is no solution
Q8.
If p,q,r are +ve and are in A.P., in the roots of quadratic equation px2+qx+r=0 are all real for
Solution
(b) For real roots, q2-4pr≥0 ⇒((p+r)/2)2-4pr≥0 (∵p,q,r are in A.P.) ⇒p2+r2-14pr≥0 ⇒p2/r2 -14 p/r+1≥0 ⇒(p/r-7)2-48≥0 ⇒|p/r-7|≥4√3
(b) For real roots, q2-4pr≥0 ⇒((p+r)/2)2-4pr≥0 (∵p,q,r are in A.P.) ⇒p2+r2-14pr≥0 ⇒p2/r2 -14 p/r+1≥0 ⇒(p/r-7)2-48≥0 ⇒|p/r-7|≥4√3
Q9.
If arg(z)< 0 , then arg(-z) - arg(z)=
Solution
(a)
arg(-z)- arg(z) = arg((-z)/z) = arg(-1)=Ï€
(a)
arg(-z)- arg(z) = arg((-z)/z) = arg(-1)=Ï€
Q10. If α , β , γ , σ are the roots of the equation x4+4x3-6x2+7x-9=0, then the value of (1+α2 )(1+β2 )(1+γ2 )(1+σ2) is
Solution
