JEE Main 2021MathematicsComplex NumberMediumNumerical

JEE Main 2021 — Complex Number Question with Solution

JEE Main 2021 (27 Jul Shift 2)

Question

If the real part of the complex number z=3+2icosθ1-3icosθ,θ∈0,π2 is zero, then the value of sin23θ+cos2θ is equal to ______.

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Correct answer
1

Step-by-step explanation

We have, z=3+2icosθ1-3icosθ,θ∈0,π2

⇒z=3+2icosθ1-3icosθ×1+3icosθ1+3icosθ

⇒z=3+2icosθ1+3icosθ1+9cos2θ

⇒z=3-6cos2θ+11icosθ1+9cos2θ

Now, Rez=3-6cos2θ1+9cos2θ=0

⇒3-6cos2θ=0

⇒cos2θ=36

⇒cos2θ=12

⇒cosθ=±12

⇒θ=π4, ∵θ∈0,π2

Hence, sin23θ+cos2θ

=sin23π4+cos2π4

=sinπ-π42+cosπ42

=sinπ42+cosπ42=1

Since, sin2θ+cos2θ=1

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About this question

This is a previous-year question from JEE Main 2021, covering the Complex Number chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.