GCE Advanced Level (Sri Lanka) · Combined Mathematics

Pure Maths: Polynomials, Complex Numbers and De Moivre's Theorem

6 practice questions, each with the answer and an explanation.

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All 6 questions

If $α$ and $β$ are the roots of the quadratic equation $x^{2} - 4 x + 2 = 0$ , find the quadratic equation whose roots are $α^{2}$ and $β^{2}$ .
Using De Moivre's Theorem, evaluate $(cos \frac{π}{6} + i sin \frac{π}{6})^{6}$ .
Find the modulus and principal argument of the complex number $z = - 1 - i 3$ .
The polynomial $P (x) = x^{3} + a x^{2} + b x - 6$ has $(x - 1)$ and $(x + 2)$ as factors. Find the values of $a$ and $b$ .
If $w$ is a complex cube root of unity ( $w \neq = 1$ ), find the value of $(1 - w + w^{2}) (1 + w - w^{2})$ .
Find the condition for the cubic equation $x^{3} + 3 p x + q = 0$ to have two equal real roots.

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