De Moivre's theorem

De Moivre's theorem Quiz

Spec B8 · AQA A-Level Further Maths: Quiz

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5 questions, one answer each

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  1. 1. Find (cos⁡π8+isin⁡π8)4\left(\cos\frac{\pi}{8}+\mathrm{i}\sin\frac{\pi}{8}\right)^4.

    • A. 11
    • B. i\mathrm{i}
    • C. −1-1
    • D. −i-\mathrm{i}

  2. 2. Find (1−i)4(1-\mathrm{i})^4.

    • A. 4i4\mathrm{i}
    • B. −4i-4\mathrm{i}
    • C. 44
    • D. −4-4

  3. 3. Which expression equals cos⁡3θ\cos3\theta?

    • A. 4cos⁡3θ−3cos⁡θ4\cos^3\theta-3\cos\theta
    • B. 3cos⁡θ−4cos⁡3θ3\cos\theta-4\cos^3\theta
    • C. cos⁡3θ−3cos⁡θ\cos^3\theta-3\cos\theta
    • D. 3cos⁡3θ−4cos⁡θ3\cos^3\theta-4\cos\theta

  4. 4. If z=cos⁡θ+isin⁡θz=\cos\theta+\mathrm{i}\sin\theta, what is z3+1z3z^3+\frac{1}{z^3}?

    • A. 2isin⁡3θ2\mathrm{i}\sin3\theta
    • B. 2cos⁡θ2\cos\theta
    • C. 2cos⁡3θ2\cos3\theta
    • D. cos⁡3θ\cos3\theta

  5. 5. Why can ∑rkcos⁡kθ\sum r^k\cos k\theta be found using a geometric series?

    • A. Because cos⁡kθ\cos k\theta is the modulus of rkr^k
    • B. Because rkcos⁡kθr^k\cos k\theta is the real part of (r(cos⁡θ+isin⁡θ))k\left(r(\cos\theta+\mathrm{i}\sin\theta)\right)^k
    • C. Because cos⁡kθ\cos k\theta is a multiple of cos⁡θ\cos\theta
    • D. Because rkcos⁡kθr^k\cos k\theta is the argument of rkr^k

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