2025 SJI P1 Q8

2025 SJI P1 Q8

IB Year 5 | Grade 11
14 marks
    1. Show that \(\mathrm e^{\mathrm i(k\theta)}+\mathrm e^{-\mathrm i(k\theta)}=2\cos(k\theta)\).
    2. Hence write down a similar expression for \(\mathrm e^{\mathrm i9\theta/2}+\mathrm e^{-\mathrm i9\theta/2}\).[3]
  2. Show that \(\mathrm e^{\mathrm i\theta}-\mathrm e^{\mathrm i2\theta}+\mathrm e^{\mathrm i3\theta}-\cdots+\mathrm e^{\mathrm i9\theta}=\dfrac{\mathrm e^{\mathrm i(10\theta)}+\mathrm e^{\mathrm i\theta}}{\mathrm e^{\mathrm i\theta}+1}\).[3]
  3. Hence show that \[\sin\theta-\sin2\theta+\sin3\theta-\cdots+\sin9\theta=\frac{\sin5\theta\cos(9\theta/2)}{\cos(\theta/2)}.\][5]
  4. Hence evaluate \(\displaystyle\sum_{r=1}^9(-1)^{r-1}\sin\left(\dfrac\pi3r\right)\).[3]

Solution:

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Answer:(a)(i) Shown, (ii) \(2\cos(9\theta/2)\). (b), (c) Shown. (d) \(0\)

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