Complex Numbers Practice Test (Mathematics)

Complex Numbers Questions with Answers:

Ques. Triangle ABC, A(z₁), B(z₂), C(z₃) is inscribed in the circle |z| = 2. If internal bisector of the angle A meets its circumcircle again at D(z_d) then
(A) z_d² = z₂z₃
(B) z_d² = z₁z₃
(C) z_d² = z₂z₁
(D) none of these
Ans. (a)

Ques. If the complex numbers z₁, z₂, z₃ represent the vertices of an equilateral triangle such that |z₁| = |z₂| = |z₃|, then z₁ + z₂ + z₃ =
(a) 0
(b) 1
(c) –1
(d) None of these
Ans: (a)

Ques. If z₁, z₂, z₃ are vertices of an equilateral triangle with z₀ its centroid, then z₁² + z₂² + z₃² =
(a) z₀²
(b) 9z₀²
(c) 3z₀²
(d) 2z₀²
Ans. (c)

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Ques. If (1 + x + x²)ⁿ = a₀ + a₁x + a₂x² + … + a_rx^r + … + a_2nx²ⁿ, then a₀ + a₃ + a₆ + =
(a) 3^{n – 1}
(b) 3ⁿ
(c) –3^r
(d) 3^{r – 1}
Ans. (a)

Ques. Which of the following is correct?
(A) 6 + i > 8 – i
(B) 6 + i > 4 – i
(C) 6 + i > 4 + 2i
(D) None of these
Ans. (d)

Ques. Number of solutions to the equation (1 –i)^x = 2^x is
(a)

1
(B) 2
(C) 3
(D) no solution
Ans. (a)

Ques. If z₁ and z₂ be the n^th roots of unity which subtend right angle at the origin. Then n must be of the form
(a) 4k + 1
(b) 4k + 2
(C) 4k + 3
(D) 4k
Ans. (d)

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Ques. If z³ – 2z² + 4z – 8 = 0 then
(A) |z| = 1
(B) |z| = 2
(C) |z| = 3
(D) None
Ans. (b)

Ques. If z  be  any  complex  number  such  that |3z –2| + |3z +2| = 4,  then  locus  of  z is
(A)  an ellipse
(B) a circle
(C)  a  line-segment
(D)  None of these
Ans. (c)

Ques. For  a  complex  number   z ,  | z – 1| + |z +1| = 2. Then z lies on a
(A) parabola
(B) line segment
(C) circle
(D) none of these
Ans. (b)

Ques. If  |z₁/z₂| = 1 and arg (z₁ z₂) = 0, then
(A) z₁ =  z₂
(B) |z₂|² = z₁z₂
(C) z₁z₂ =  1
(D) none of these
Ans. (b)

Ques. Number of non-zero integral solutions to (3 + 4i)ⁿ = 25ⁿ is
(A) 1
(B) 2
(C) finitely many
(D) none of these
Ans. (d)

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Ques. If |z| < 4, then  | iz +3 – 4i| is less than
(A) 4
(B) 5
(C) 6
(D) 9
Ans. (d)

Ques. If the equation |z – z₁|² + | z – z₂|² = k represents the equation of a circle, where z₁ º 2+ 3i, z₂ º 4 + 3i are the extremities of a diameter, then the value of k is
(A) ¼
(B) 4
(C) 2
(D) None of these
Ans. (b)

Ques. If z = x + iy satisfies the equation arg (z – 2) = arg(2z + 3i), then 3x – 4y is equal to
(A) 5
(B) –3
(C) 7
(D) 6
Ans. (d)

Ques. Number of solutions of Re (z²) = 0 and |Z| = aÖ2, where z is a complex number and a > 0, is
(A) 1
(B) 2
(C) 4
(D) 8
Ans. (a)

Ques. If (x – iy) ^1/3 = a – ib, then x/a + y/b equals
(A) -2 (a² + b²)
(B) 4 (a + b)
(C) 4 (a – b)
(D) 4 ab
Ans. (a)

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Ques. If |z| = 1, then |z – 1| is
(A) < |arg z|
(B) >|arg z|
(C) = |arg z|
(D) None of these
Ans. (a)

Ques. The  locus  of  z  which  satisfied  the  inequality log_0.5|z – 2| > log_0.5|z – i| is  given  by
(A) x+ 2y > 1
(B) x – y < 0
(C) 4x – 2y >  3
(D) none  of these
Ans. (c)

Ques. If |z₁| = 4, |z₂| = 4, then |z₁ + z₂ + 3 + 4i| is less than
(A) 2
(B) 5
(C) 10
(D) 13
Ans. (d)

Ques. If  |z +1|  = z + 1 , where z is a  complex  number, then  the locus  of z  is
(A) a straight line
(B) a ray
(C) a circle
(D) an arc of a circle
Ans. (b)

Ques. If the complex numbers z₁, z₂, z₃ are in A.P., then they lie on a
(A) circle
(B) parabola
(C) line
(D) ellipse
Ans. (c)

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Ques. If  points  corresponding  to  the complex  numbers z₁, z₂, z₃ and z₄ are  the  vertices of a  rhombus, taken in  order,  then  for a non-zero  real number k
(A)  z₁ – z₃ = i k( z₂ –z₄)
(B) z₁ – z₂ = i k( z₃ –z₄)
(C) z₁ + z₃ = k( z₂ +z₄)
(D) z₁ + z₂ = k( z₃ +z₄)
Ans. (a)

Question: If z is a complex number, then |3z – 1| = 3|z – 2| represents
(a) y-axis
(b) a circle
(c) x-axis
(d) a line parallel to y-axis
Ans: (d)

Ques. The roots of equation zⁿ = (z +1)ⁿ
(A) are vertices of regular  polygon
(B) lie on a circle
(C) are collinear
(D) none of these
Ans. (c)

Ques. Let z₁ and z₂ be the complex roots of the equation 3z² + 3z+ b = 0. If the origin, together with the points represented by z₁ and z₂ form an equilateral triangle then the value of b is
(A) 1
(B) 2
(C) 3
(D) None of these
Ans. (a)

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Ques. If x = 1 + i, then the value of the expression    x⁴ – 4x³ + 7x² – 6x + 3 is
(A) –1
(B) 1
(C) 2
(D) None of these
Ans. (b)

Ques. For all complex numbers z₁, z₂ satisfying |z₁| = 12 and |z₂ – 3 – 4i| = 5, the minimum value of |z – z₂| is
(a) 4
(b) 3
(c) 1
(d) 2
Ans. (d)

Ques. If two non-zero complex numbers are such  that   |z₁ + z₂|  = |z₁ |  – |z₂|  then z₁/z₂ is;
(a) a positive real number
(b) a negative real number
(c) a purely imaginary number
(d) none of these
Ans. (b)