Complex Numbers Questions with Answers:
Q 1. 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(zd) then
A. zd2 = z2z3
B. zd2 = z1z3
C. zd2 = z2z1
D. none of these
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Answer:-A. zd2 = z2z3Q 2. 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
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Answer:-A. 0Q 3. 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
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Answer:-C. 3z02Q 4. If (1 + x + x2)n = a0 + a1x + a2x2 + … + arxr + … + a2nx2n, then a0 + a3 + a6 + =
A. 3n – 1
B. 3n
C. –3r
D. 3r – 1
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Answer:-A. 3n – 1Q 5. 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
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Answer:-D. None of theseQ 6. Number of solutions to the equation (1 –i)x = 2x is
A. 1
B. 2
C. 3
D. no solution
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Answer:-A. 1Q 7. If z1 and z2 be the nth 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
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Answer:-D. 4kQ 8. If z3 – 2z2 + 4z – 8 = 0 then
A. |z| = 1
B. |z| = 2
C. |z| = 3
D. None
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Answer:-B. |z| = 2Q 9. 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
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Answer:-C. a line-segmentQ 10. 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
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Answer:-B. line segmentQ 11. If |z1/z2| = 1 and arg (z1 z2) = 0, then
A. z1 = z2
B. |z2|2 = z1z2
C. z1z2 = 1
D. none of these
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Answer:-B. |z2|2 = z1z2Q 12. Number of non-zero integral solutions to (3 + 4i)n = 25n is
A. 1
B. 2
C. finitely many
D. none of these
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Answer:-D. none of theseRelated:
Q 13. If |z| < 4, then | iz +3 – 4i| is less than
A. 4
B. 5
C. 6
D. 9
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Answer:-D. 9Q 14. 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
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Answer:-B. 4Q 15. 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
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Answer:-B. –3Q 16. 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
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Answer:-A. 1Q 17. 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
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Answer:-A. -2 (a2 + b2)Related:
Q 18. If |z| = 1, then |z – 1| is
A. < |arg z|
B. >|arg z|
C. = |arg z|
D. None of these
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Answer:-A. < |arg z|Q 19. The locus of z which satisfied the inequality log0.5|z – 2| > log0.5|z – i| is given by
A. x+ 2y > 1
B. x – y < 0
C. 4x – 2y > 3
D. none of these
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Answer:-C. 4x – 2y > 3Q 20. If |z1| = 4, |z2| = 4, then |z1 + z2 + 3 + 4i| is less than
A. 2
B. 5
C. 10
D. 13
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Answer:-D. 13Q 21. 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
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Answer:-B. a rayQ 22. If the complex numbers z1, z2, z3 are in A.P., then they lie on a
A. circle
B. parabola
C. line
D. ellipse
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Answer:-C. lineRelated:
Q 23. 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)
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Answer:-A. z1 – z3 = i k( z2 –z4)Q 24. 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
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Answer:-D. a line parallel to y-axisQ 25. 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
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Answer:-C. are collinearQ 26. 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
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Answer:-A. 1Related:
Q 27. 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
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Answer:-B. 1Q 28. For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 – 3 – 4i| = 5, the minimum value of |z – z2| is
A. 1
B. 2
C. 3
D. 4
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Answer:-B. 2Q 29. 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
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