Manipulating expressions in the rootsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The roots of the equation $x^2-6x+4=0$ are $\alpha$ and $\beta$.

   (a)

   Find the value of $\alpha^2+\beta^2$.

   [1 mark]

   • A$32$
   • B$36$
   • C$28$
   • D$44$

   (b)

   Find the value of $\frac{1}{\alpha}+\frac{1}{\beta}$.

   [1 mark]

   • A$\frac{2}{3}$
   • B$\frac{3}{2}$
   • C$6$
   • D$-\frac{3}{2}$

   (c)

   Find the value of $(\alpha+1)(\beta+1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The roots of the equation $2x^2+5x-3=0$ are $\alpha$ and $\beta$.

   (a)

   Find the value of $\alpha\beta$.

   [1 mark]

   • A$\frac{3}{2}$
   • B$-3$
   • C$-\frac{5}{2}$
   • D$-\frac{3}{2}$

   (b)

   Find the value of $\alpha^2+\beta^2$.

   [1 mark]

   • A$\frac{37}{4}$
   • B$\frac{13}{4}$
   • C$\frac{25}{4}$
   • D$\frac{31}{4}$

   (c)

   Find the exact value of $(\alpha-\beta)^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The roots of the equation $x^2-5x+2=0$ are $\alpha$ and $\beta$.

   (a)

   Find the value of $\alpha^3+\beta^3$.

   [3 marks]

   (b)

   Find the value of $\alpha^4+\beta^4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The equation $x^2-8x+k=0$, where $k$ is a constant, has roots $\alpha$ and $\beta$, and $\alpha^2+\beta^2=40$.

   (a)

   (i) Find the value of $k$.
   (ii) Find the value of $\alpha^3+\beta^3$.

   [6 marks]

   (b)

   Using your answers to part (a), find the exact value of
   (i) $\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha}$,
   (ii) $\alpha^4+\beta^4$,
   (iii) $|\alpha-\beta|$.

   [6 marks]

   Total for question 4: 12 marks

End of questions