Complex numbers questions

Questions to attempt

6 questions on Complex numbers, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-2.1 · Solve any quadratic equation with real coefficients. Solve cubic or quartic equations with real coefficients.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   Given that $x=2$ is a root of $x^3-2x^2+5x-10=0$, solve the equation completely.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

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2. CP-2.2 · Add, subtract, multiply and divide complex numbers in the form x + iy with x and y real. Understand and use the terms 'real part' and 'imaginary part'.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   Express $\dfrac{4+i}{2-i}$ in the form $x+iy$.

   (3)

   (Total for Question 2 is 3 marks)

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3. CP-2.3 · Understand and use the complex conjugate. Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   The polynomial $p(x)=x^3-x^2-7x+15$ has real coefficients, and $2+i$ is a root. Solve $p(x)=0$ completely.

   (4)

   (Total for Question 3 is 4 marks)

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4. CP-2.4 · Use and interpret Argand diagrams.

   Q4

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   4.

   The points $A$, $B$ and $C$ are represented by the complex numbers $1+i$, $5+i$ and $1+4i$, respectively. Determine the exact area of triangle $ABC$.

   (3)

   (Total for Question 4 is 3 marks)

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5. CP-2.5 · Convert between the Cartesian form and the modulus-argument form of a complex number.

   Q5

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   5.

   Express $-2-2i$ in modulus-argument form using its principal argument.

   (3)

   (Total for Question 5 is 3 marks)

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6. CP-2.6 · Multiply and divide complex numbers in modulus-argument form.

   Q6

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   6.

   Express $\dfrac{12e^{5\pi i/6}}{3e^{-\pi i/4}}$ in the form $re^{i\theta}$ using the principal argument.

   (3)

   (Total for Question 6 is 3 marks)

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CP-2 Complex numbers revision notes