JEE Mathematics Complex Numbers MCQs Set D

Question. Find the value of [i]198
(a) –1
(b) 1
(c) 0
(d) i

Answer: B

Question. Find the value of iⁿ + iⁿ⁺¹+ iⁿ⁺² + iⁿ⁺³
(a) 0
(b) i
(c) –1
(d) 1

Answer: A

Question. If complex number z-1/z+1 is purely imaginary, then locus of z is -
(a) a circle
(b) a parabola
(c) a straight line
(d) None of these

Answer: A

Question. If for any complex number z, |z – 4| < |z – 2|, then
(a) R(z) > 3
(b) R(z) < 0
(c) R(z) > 2
(d) R(z) > 0

Answer: A

Question. If | z + 2i | ≤ 1, then greatest and least value of | z – √3 + i |are-
(a) 3, 1
(b) 1, 3
(c) None of these
(d) 0

Answer: A

Question. √-2  √-3   is equal to -
(a) -√6
(b) √6
(c) 6
(d) None of these

Answer: C

Question. The sum of series i² + i⁴ + i⁶ + .......up to (2n + 1) terms is -
(a) – 1
(b) 1
(c) 0
(d) n

Answer: B

Question. If (x + iy) (2 – 3i) = 4 + i, then-
(a) x = 5/13, y = 14/13
(b) x = 5/13, y = –14/13
(c) x = –14/13, y = 5/13
(d) x = 14/13, y = 5/13

Answer: A

Question. The conjugate of 1/3+ 4i is -
(a) 1/25(3 + 4i)
(b) (3 – 4i)
(c) 1/26(3 + 4i)
(d) None of these

Answer: A

Question. If x be real and 1-ix/1+ix= a – ib the relation in a and b is
(a) a² + b² = 1
(b) a² - b² = 1
(c) ab = 1
(d) None of these

Answer: A

Question. If z = (1/2, 1) , then the value of z^–1 is-
(a) (2/5, -4/5)
(b) (2/5, -5/5)
(c) (1/5, -5/5)
(d) (1/5, 3/5)

Answer: A

Question. Statement 1 : 3+ix²y and x² + y + 4i are conjugate numbers, then x² + y² = 3
Statement 2 : If sum and product of two complex numbers is real, then they are conjugate complex numbers.
(a) Statement -1 is False, Statement-2 is True.
(b) Statement -1 is True, Statement-2 is False
(c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A

Question. Statement-1 : If a, b, c are non-zero real numbers and the equation ax² + bx + c + i = 0 has purely imaginary roots then a = b²c .
Statement-2 : The roots of the equation must be conjugate of each other.
(a) Statement -1 is True, Statement-2 is False.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT b a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is correct explanation for Statement-1.

Answer: A

Question. If complex numbers z₁, z₂ and 0 are vertices of an equilateral triangle, then z₁² ₊ z₂² – z₁z₂ is equal to-
(a) 0
(b) z₁ – z₂
(c) z₁ + z₂
(d) 1

Answer: A

Question. If w =z-(1/ 5)i/z and | w | = 1, then complex number z lies on
(a) a line
(b) a parabola
(c) a circle
(d) None of these

Answer: A

Question. If complex numbers z₁, z₂, z₃ represent the vertices of an equilateral triangle such that |z_1| = |z₂| = |z₃| ; then-
(a) z₁ + z₂ + z₃ = 0
(b) I(z₁ + z₂ + z₃) = 0
(c) R(z₁ + z₂ + z₃) = 0
(d) None of these

Answer: A

Question. If z₁,z₂ are any two complex numbers and a, b are any two real numbers, then |az₁ – bz₂|² + |bz₁ + az₂|² is equal to-
(a) (a₂ + b₂)(|z₁ |² + |z₂ |²)
(b) a₂b₂(|z₁ |² + |z₂ |²)
(c) (a+b)²(|z₁ |² + |z₂ |²)
(d) None of these

Answer: A

Question. In a complex plane z₁, z₂, z₃, z₄ taken in order are vertices of parallelogram, if
(a) z₁ + z₃ = z₂ + z₄
(b) z₁ + z₂ = z₃ + z₄
(c) z₁ - z₂ = z₃ + z₄
(d) None of these

Answer: A

Question. If A, B and C are represented by the complex numbers 3 + 4i, 5 – 2i, – 1 + 16i respectively, then A, B, C are-
(a) collinear
(b) vertices of right-angle triangle
(c) vertices of isosceles triangle
(d) vertices of equilateral triangle

Answer: A

Question. The complex number z having least positive argument which satisfy the condition | z – 25i | ≤ 15 is -
(a) 12 + 16i
(b) 12 + 25i
(c) 25i
(d) 16 + 12i

Answer: A

Question. If z₀ is the circumcenter of an equilateral triangle with vertices z₁, z₂, z₃, then z₁ ² + z₂ ² + z₃ ² is equal to
(a) z₀²/3
(b) 2z₀²/3
(c) z₀²
(d) 3z₀²/3

Answer: C

Question. If |z₂ + i z₁| = | z₁ | + | z₂ |, | z₁ | = 3 & | z₂ | = 4, then area of triangle ABC, if A, B & C are represented by (z₁), (z₂) and (z₂-z₁ /1-i) respectively, is –
(a) 25/4
(b) 0
(c) 5/2
(d) 25/2

Answer: A

Question. If z is a complex number satisfying | z – i Re (z) | = |z–Im(z) | then z lies on – (1) y = x        (2) y = – x
(3) y = x + 1 (4) y = – x + 1
(a) 1 and 2 are correct
(b) 1 and 3 are correct
(c) 1, 2 and 3 are correct
(d) 2 and 4 are correct

Answer: A

Question. Let Z₁ and Z₂ be complex number such that | Z₁ + Z₂ | = | Z₁ | + | Z₂ |
Statement 1 : Z₁, Z₂ and origin are collinear and Z₁, Z₂ are on same side of origin Statement 2 : If Arg (Z₁/Z₂)= 0 , then origin, Z1 and Z2 are collinear.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A