JEE Mathematics Complex Numbers MCQs Set D

Question: Find the value of [i]198

• a) –1
• b) 1
• c) 0
• d) i

Answer: –1

Question: Find the value of iⁿ + iⁿ⁺¹+ iⁿ⁺² + iⁿ⁺³

• a) 0
• b) i
• c) –1
• d) 1

Answer: 0

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

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 circle

Question:

• a) square
• b) rectangle
• c) parallelogram
• d) rhombus

Answer: square

Question:

• a) 1√2
• b) 1√3
• c) None of these
• d) 1

Answer: 1√2

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: R(z) > 3

Question:

• a) x axis
• b) x – y = 0
• c) Circle passing through origin
• d) y axis

Answer: x axis

Question: The amplitude of a +ib/a -ib is equal to-

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

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: 3, 1

Question:

• a) purely imaginary
• b) zero or purely imaginary
• c) purely real
• d) None of these

Answer: purely imaginary

Question: √-2  √-3   is equal to -

• a) -√6
• b) √6
• c) 6
• d) None of these

Answer: -√6

Question: The sum of series i² + i⁴ + i⁶ + .......up to (2n + 1) terms is -

• a) – 1
• b) 1
• c) 0
• d) n

Answer: – 1

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: x = 5/13, y = 14/13

Question: The polar form of – 1 + i is –

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

More Questions..................................

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: 1/25(3 + 4i)

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: a² + b² = 1

Question:

• a) i √3
• b) 1+ i √3
• c) -1+ i √3
• d) -i √3

Answer: i √3 

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: (2/5, -4/5)

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: For any two non real complex numbers z₁, z₂ ; if z₁ + z₂ and z₁z₂ are real numbers, then

• a)

  

• b) z₁ = z₂
• c) z₁ = 1/z₂
• d) All of these

Answer: 

Question:

• a) 1, 2 and 3 are correct
• b) 2 and 4 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 1, 2 and 3 are correct

Question:

(1) 1 / i (2) – i
(3) 1     (4) i

• 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: 1 and 2 are correct

Question:

(1) i⁴    (2) 1
(3) 1/2 (4) –1/2

• 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: 1 and 2 are correct

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: Statement -1 is False, Statement-2 is True.

Question:

• 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: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

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: Statement -1 is True, Statement-2 is False.

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: 0

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 line

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: z₁ + z₂ + z₃ = 0

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: (a₂ + b₂)(|z₁ |² + |z₂ |²)

Question:

• a) | z | = 5
• b) | z | = 3
• c) | z | = 4
• d) | z | = 2

Answer: | z | = 5

Question:

• a)

  

• b)

  

• c)

  

• d) None of these

Answer:

Question:

• a) – 1
• b) 0
• c) 1
• d) None of these

Answer: – 1

Question:

• a) 5
• b) 3
• c) 2
• d) 6

Answer: 5

Question:

• a) 3 √2
• b) 2 √2
• c) 4√2
• d) √2

Answer: 3 √2

Question: The polar form of complex number

• a)

  

• b)

  

• c)

  

• d) None of these

Answer:

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: Square root of – 8 – 6i is –

• a) ± (1 – 3i)
• b) ± (3 + i)
• c) ± (1 + 3i)
• d)

  

Answer: ± (1 – 3i)

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: z₁ + z₃ = z₂ + z₄

Question: The complex numbers sin x + i cos 2x and cos x – i sin 2x are conjugate to each other when –

• a) no value of x
• b) x = 0
• c)

  

• d)

  

Answer: no value of x

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: collinear

Question:

• a) 4
• b) – 2
• c) 2
• d) 0

Answer: 4

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: 12 + 16i

Question:

• a)

  

• b)

  

• c)

  

• d) 0 

Answer:

Question:

• a) i
• b) 0
• c) – 1
• d) – i

Answer: i

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: z₀²/3

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: 25/4

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: 1 and 2 are correct

Question: If z₁, z₂, z₃, z₄ are the roots of the equation

(1) 0  (2) 2
(3) 3  (4) 1

• a) 1, 2 and 3 are correct
• b) 2 and 4 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 1, 2 and 3 are correct

Question: Locus of the point z satisfying the equation

• a) A circle
• b) A parabola
• c) A straight line
• d) An ellipse

Answer: A circle

Question:

• 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: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Question:

• 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: Statement -1 is False, Statement-2 is True

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: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.