If ∣ w ∣ = 2 , then the set of points z = w − w 1 is contained in or equal to the set of points z satisfying

We have, ∣ w ∣ = 2 ⇒ w = 2 ( cos θ + i sin θ ) ∴ z = w − w 1 = 2 ( cos θ + i sin θ ) − 2 1 ( cos θ − i sin θ ) ⇒ z = 2 3 cos θ + 2 5 i sin θ ⇒ ∣ z ∣ = 4 9 + 4 25 = 2 17 < 3 Hence, option (d) is correct.

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MathematicsComplex NumberGeneral(Modulus,Argument,Conjugate)Hard2 minPYQ_2022

MathematicsHardsingle choice

If $∣ w ∣ = 2$ , then the set of points $z = w - \frac{1}{w}$ is contained in or equal to the set of points $z$ satisfying

Options:

Answer:

Solution:

We have, $∣ w ∣ = 2 \Rightarrow w = 2 (cos θ + i sin θ)$ $∴ z = w - \frac{1}{w}$ $= 2 (cos θ + i sin θ) - \frac{1}{2} (cos θ - i sin θ)$ $\Rightarrow z = \frac{3}{2} cos θ + \frac{5}{2} i sin θ$ $\Rightarrow ∣ z ∣ = \frac{9}{4} + \frac{25}{4} = \frac{17}{2} < 3$ Hence, option (d) is correct.

Stream:BITSATSubject:MathematicsTopic:Complex NumberSubtopic:General(Modulus,Argument,Conjugate)

⏱ 2mℹ️ Source: PYQ_2022

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