## Complex Number and Quadratic Equations Quiz-15

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. The sum of values of x satisfying the equation (31+8√15)(x2 - 3) + 1 = (32 + 8√15)(x2 - 3) is
•  3
•  0
•  2
•  None of these
Solution
Q2. Let a,b,c be real numbers, a≠0. If Î± is a root of a^2 x^2+bx+c=0. Î² is the root of a2x2-bx-c=0 and 0<Î±<Î², then the equation a2x2+2bx+2c=0 has a root Î³ that always satisfies
•  Î³=(Î±+Î²)/2
•  Î³=Î±+Î²/2
•  Î³=Î±
•  Î±<Î³<Î²
Solution

Q3. If Î±,Î² be the non-zero roots of ax2+bx+c=0 and Î±2,Î²2 be the roots of a2 x2+b2 x+c2=0, then a,b,c are in
•  G.P.
•  H.P.
•  A.P.
•  None of these
Solution
(a)

Q4. The equation 22x+(a-1) 2(x+1)+a=0 has roots of opposite signs then exhaustive set of values of a is
•  a∈(-1,0)
•  a<0
•  a∈(-∞,1/3)
•  a∈(0,1/3)
Solution

Q5. Let z=x+iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation zz ̅3+z ̅z3=350 is
•  48
•  32
•  40
•  80
Solution

Q6. If the quadratic equation 4x2-2(a+c-1)x+ac-b=0 (a>b>c)
•  Both roots are greater than a
•  Both roots are less than c
• Both roots lie between c/2 and a/2
•  Exactly one of the roots lies between c/2 and a/2
Solution

Q7. Let |zr-r|≤r,∀r=1,2,3,…n. Then |∑n(r=1)zr | is less than
•  n
•  2n
•  n(n+1)
•  n(n+1)/2
Solution

Q8. If (ax2+c)y+(a'x2+c')=0 and x is a rational function of y and ac is negative, then
•  ac'+a'c=0
•  a/a'=c/c'
•  a2+c2=a'2+c'2
•  aa'+cc'=1
Solution

Q9.  Let z and Ï‰ be two non-zero complex numbers such that |z|=|Ï‰| and arg⁡z=Ï€-arg⁡Ï‰, then z equals
•  Ï‰
•  -Ï‰
•  ¯Ï‰
•  -¯Ï‰
Solution

Q10. Suppose that f(x) is a quadratic expression positive for all real x. If g(x)=f(x)+f'(x)+f''(x), then for any real x (where f'(x) and f''(x) represent 1st and 2nd derivative respectively)
•  g(x)< 0
•  g(x)>0
•  g(x)=0
• g(x)≥0
Solution

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