## Complex Number and Quadratic Equations Quiz-13

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.  The equation x-2/(x-1)=1-2/(x-1) has
•  No root
•  One root
•  Two equals roots
•  Infinitely many roots
Solution
Q2.  Let Î±,Î² be the roots of the equation (x-a)(x-b)=c,c≠0. Then the roots of the equation (x-Î±)(x-Î²)+c=0 are
•  a,c
•  b,c
•  a,b
•  a+c,b+c
Solution
(c)
Î±,Î² are roots of the equation (x-a)(x-b)=c,c≠0
∴ (x-a)(x-b)-c=(x-Î±)(x-Î²)
⇒(x-Î±)(x-Î²)+c=(x-a)(x-b)
Hence, the roots of (x-Î±)(x-Î²)+c=0 are a and b

Q3.  Number of complex numbers z such that |z| =1 and |z/z ̅+z ̅/z|=1 is (arg(z) ∈ [0,2Ï€))
•  4
•  6
•  8
•  More than 8
Solution

Q4. If x=1+i is a root of the equation x3-ix+1-i=0, then the other real root is
•  0
•  1
•  -1
•  None of these
Solution
(c)
Clearly, x=-1 satisfies the equation

Q5. If (mr,1/mr),r=1,2,3,4 be four pairs of values of x and y that satisfy the equation x^2+y^2+2gx+2fy+c=0, then value of m1 m2 m3 m4 is
•  0
•  1
•  -1
•  None of these
Solution

Q6. If k+|k+z2 |=|z|2 (k ∈R- ), then possible argument of z is
•  0
•  Ï€
• Ï€/2
•  None of these
Solution

Q7. Let a≠0 and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and –a when divided respectively by x+a and x-a, then remainder when p(x) is divided by x2-a2 is
•  2x
•  -2x
•  x
•  -x
Solution

Q8. If z is a complex number lying in the fourth quadrant of Argand plane and |[kz/(k+1)]+2i|>√2 for all real value of k(k≠-1), then range of arg⁡〖(z)〗 is
•  (-Ï€/8,0)
•  T(-Ï€/6,0)
•  (-Ï€/4,0)
•  None of these
Solution

Q9.  The largest interval for which x12-x9+x4-x+1>0 is
•  -4 < x ≤ 0
•  0 < x < 1
•  -100 < x < 100
•  -∞ < x < ∞
Solution

Q10. If |2z-1|=|z-2| and z1,z2,z3 are complex numbers such that |z1-Î±| < Î±,|z2-Î²| < Î², then |(z1+z2)/(Î±+Î²)|
•  <|z|
•  <2|z|
•  >|z|
• >2|z|
Solution

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