SATHEE: Complex Numbers 4 Question 5

Complex Numbers 4 Question 5

5. If $0 < α < \frac{π}{2}$ is a fixed angle. If $P = (\cos θ, \sin θ)$ and $Q = \cos (α - θ), \sin (α - θ)$ , then $Q$ is obtained from $P$ by

(2002, 2M)

(a) clockwise rotation around origin through an angle $α$

(b) anti-clockwise rotation around origin through an angle $α$

(d) reflection in the line through origin with slope $\tan \frac{α}{2}$

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Answer:

Correct Answer: 5. (d)

Solution:

In the Argand plane, $P$ is represented by $e^{i 0}$ and $Q$ is represented by $e^{i (α - θ)}$

Now, rotation about a line with angle $α$ is given by $e^{θ} \to e^{(α - θ)}$ . Therefore, $Q$ is obtained from $P$ by reflection in the line making an angle $α / 2$ .

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Complex Numbers 4 Question 5

5. If 0<α<π2 is a fixed angle. If P=(cos⁡θ,sin⁡θ) and Q=cos⁡(α−θ),sin⁡(α−θ), then Q is obtained from P by

Answer:

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5. If $0 < α < \frac{π}{2}$ is a fixed angle. If $P = (\cos θ, \sin θ)$ and $Q = \cos (α - θ), \sin (α - θ)$ , then $Q$ is obtained from $P$ by