JEE Main 2022 · 30 June · morning · Single correct · +4 / −1

The real part of the complex number ((1 + 2i)⁸ . (1 - 2i)²)/((3 + 2i) . (4 - 6i)̅) is equal to :

  1. (a) 500/13
  2. (b) 110/13
  3. (c) 55/6
  4. (d) 550/13
Answer and solution

Answer (d) 550/13

Solution

Maths sir · 4 steps

Pair each (1-2i) with a (1+2i) to make 5, and the messy eighth power shrinks to a sixth.

  1. Pair conjugates. Since (1+2i)(1-2i) = 1 + 4 = 5, the numerator is (1+2i)⁶ × 5² = 25(1+2i)⁶.
  2. Build the power. (1+2i)² = 1 + 4i - 4 = -3 + 4i. Squaring again gives (-3+4i)² = 9 - 24i - 16 = -7 - 24i. Then (-7-24i)(-3+4i) = 21 - 28i + 72i + 96 = 117 + 44i.
  3. Denominator. 4-6i̅ = 4+6i, so (3+2i)(4+6i) = 12 + 18i + 8i - 12 = 26i, a purely imaginary number.

    25(117+44i)/26i = 25(117+44i)(-i)/26 = 25(44 - 117i)/26

  4. So. The real part is (25 × 44)/26 = 1100/26 = 550/13, which is option (d).

The idea to keep. When a number and its conjugate both appear, pair them first to get a real modulus squared. Dividing by a pure imaginary ki is the same as multiplying by -i/k.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Complex Numbers

Asked about 160 times in JEE Main since 2020. This question is from 2022.