JEE Main 2026 Jan 24 Shift 2, Mathematics Q23: Introduction to Complex Numbers And Their Conjugates

JEE Main2026Jan 24, Shift 2Mathematics

Let $z = (1 + i) (1 + 2 i) (1 + 3 i) \dots (1 + ni)$ , where $i = - 1$ . If $∣ z ∣^{2} = 44200$ , then $n$ is equal to ________.

Your answer

Solution

$∣ z ∣^{2} = k = 1 \prod n ∣1 + k i ∣^{2} = k = 1 \prod n (1 + k^{2})$ .

Tabulate partial products of $(1 + k^{2})$ starting from $k = 1$ : $2, 5, 10, 17, 26, \dots$

$2 \cdot 5 = 10$ ; $\cdot 10 = 100$ ; $\cdot 17 = 1700$ ; $\cdot 26 = 44200$ . ✓

So 5 factors are needed: $n = 5$ .

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