Paper 1, Section I, B

Vectors and Matrices | Part IA, 2015

(a) Describe geometrically the curve

αz+βzˉ=αβ(z+zˉ)+(αβ)2,|\alpha z+\beta \bar{z}|=\sqrt{\alpha \beta}(z+\bar{z})+(\alpha-\beta)^{2},

where zCz \in \mathbb{C} and α,β\alpha, \beta are positive, distinct, real constants.

(b) Let θ\theta be a real number not equal to an integer multiple of 2π2 \pi. Show that

m=1Nsin(mθ)=sinθ+sin(Nθ)sin(Nθ+θ)2(1cosθ)\sum_{m=1}^{N} \sin (m \theta)=\frac{\sin \theta+\sin (N \theta)-\sin (N \theta+\theta)}{2(1-\cos \theta)}

and derive a similar expression for m=1Ncos(mθ)\sum_{m=1}^{N} \cos (m \theta).

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