### Complex Numbers Questions with Answers:

**Ques. **Triangle ABC, A(z_{1}), B(z_{2}), C(z_{3}) 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_{d}^{2} = z_{2}z_{3}

(B) z_{d}^{2} = z_{1}z_{3
}(C) z_{d}^{2} = z_{2}z_{1}

(D) none of these

Ans. (a)

**Ques. **If the complex numbers z_{1}, z_{2}, z_{3 }represent the vertices of an equilateral triangle such that |z_{1}| = |z_{2}| = |z_{3}|, then z_{1} + z_{2} + z_{3 }=

(a) 0

(b) 1

(c) –1

(d) None of these

Ans: (a)

**Ques. **If z_{1}, z_{2}, z_{3} are vertices of an equilateral triangle with z_{0} its centroid, then z_{1}^{2} + z_{2}^{2} + z_{3}^{2} =

(a) z_{0}^{2}

(b) 9z_{0}^{2
}(c) 3z_{0}^{2}

(d) 2z_{0}^{2
}Ans. (c)

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**Ques. **If (1 + x + x^{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + … + a_{r}x^{r} + … + a_{2n}x^{2n}, then a_{0} + a_{3} + a_{6} + =

(a) 3^{n – 1}

(b) 3^{n
}(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 z_{1} and z_{2} 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 z^{3} – 2z^{2} + 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 |z_{1}/z_{2}| = 1 and arg (z_{1} z_{2}) = 0, then

(A) z_{1} = z_{2}

(B) |z_{2}|^{2} = z_{1}z_{2}

(C) z_{1}z_{2} = 1

(D) none of these

Ans. (b)

**Ques. **Number of non-zero integral solutions to (3 + 4i)^{n} = 25^{n} 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 – z_{1}|^{2} + | z – z_{2}|^{2} = k represents the equation of a circle, where z_{1} º 2+ 3i, z_{2} 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 (z^{2}) = 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 (a^{2} + b^{2})

(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 |z_{1}| = 4, |z_{2}| = 4, then |z_{1} + z_{2} + 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 z_{1}, z_{2}, z_{3} 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 z_{1}, z_{2}, z_{3} and z_{4} are the vertices of a rhombus, taken in order, then for a non-zero real number k

(A) z_{1} – z_{3} = i k( z_{2} –z_{4})

(B) z_{1} – z_{2} = i k( z_{3} –z_{4})

(C) z_{1} + z_{3} = k( z_{2} +z_{4})

(D) z_{1} + z_{2} = k( z_{3} +z_{4})

Ans. (a)

**Question: **If *z* is a complex number, then |3*z* – 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 z^{n} = (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 z_{1} and z_{2} be the complex roots of the equation 3z^{2} + 3z+ b = 0. If the origin, together with the points represented by z_{1} and z_{2} 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 x^{4} – 4x^{3} + 7x^{2} – 6x + 3 is

(A) –1

(B) 1

(C) 2

(D) None of these

Ans. (b)

**Ques. **For all complex numbers z_{1}, z_{2 }satisfying |z_{1}| = 12 and |z_{2} – 3 – 4i| = 5, the minimum value of |z – z_{2}| is

(a) 4

(b) 3

(c) 1

(d) 2

Ans. (d)

**Ques. **If two non-zero complex numbers are such that |z_{1 }+ z_{2}| = |z_{1} | – |z_{2}| then z_{1}/z_{2} is;

(a) a positive real number

(b) a negative real number

(c) a purely imaginary number

(d) none of these

Ans. (b)