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Find a+b When x²=3+4i and x³=2+11i | Complex Numbers

JEE Maths question with a full step-by-step solution.

Question

x = a + ib

is a complex number such that

x^2 = 3 + 4i

and

x^3 = 2 + 11i

, where

i = \sqrt{-1}

, then

a + b

1

2

3

correct

4

Solution

Step 1: Recover

x

itself from the two given powers. Since

x = \dfrac{x^3}{x^2}

x = \frac{2 + 11i}{3 + 4i}

Step 2: Rationalise by multiplying numerator and denominator by the conjugate

3 - 4i

x = \frac{(2 + 11i)(3 - 4i)}{(3 + 4i)(3 - 4i)}

Step 3: Expand the numerator:

(2 + 11i)(3 - 4i) = 6 - 8i + 33i - 44i^2 = 6 + 25i + 44 = 50 + 25i

Compute the denominator:

(3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9 + 16 = 25

Step 4: Divide:

x = \frac{50 + 25i}{25} = 2 + i

Step 5: Identify

a = 2

b = 1

, so

a + b = 3

. Correct answer: (3)

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