FP1: Roots of quadratic equationsEdexcel International A Level Further Maths: Topic test

1. 1
   The roots of the equation $2x^2-9x+5=0$ are $\alpha$ and $\beta$.

   (a)

   Find $\alpha+\beta$.

   [1 mark]

   • A$-\frac92$
   • B$\frac92$
   • C$9$
   • D$\frac52$

   (b)

   Find $\alpha^2+\beta^2$.

   [1 mark]

   • A$\frac{81}{4}$
   • B$\frac{71}{4}$
   • C$\frac{61}{4}$
   • D$\frac{41}{4}$

   (c)

   Find a quadratic equation, with integer coefficients, that has roots $2\alpha$ and $2\beta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The roots of the equation $x^2+5x-7=0$ are $\alpha$ and $\beta$.

   (a)

   Find $\alpha^3+\beta^3$.

   [1 mark]

   • A$-230$
   • B$-20$
   • C$230$
   • D$-125$

   (b)

   Find $\dfrac1\alpha+\dfrac1\beta$.

   [1 mark]

   • A$-\frac57$
   • B$\frac75$
   • C$-\frac75$
   • D$\frac57$

   (c)

   Find a quadratic equation, with integer coefficients, that has roots $\dfrac1\alpha$ and $\dfrac1\beta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The roots of the equation $3x^2-11x+2=0$ are $\alpha$ and $\beta$.

   (a)

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

   [3 marks]

   (b)

   Find a quadratic equation, with integer coefficients, that has roots $\alpha+\dfrac2\beta$ and $\beta+\dfrac2\alpha$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The roots of the equation $x^2-7x+3=0$ are $\alpha$ and $\beta$.

   (a)

   (i) Show that $\alpha^3+\beta^3=280$.
   (ii) Find a quadratic equation, with integer coefficients, that has roots $\alpha^3$ and $\beta^3$.

   [6 marks]

   (b)

   Find a quadratic equation, with integer coefficients, that has roots $\alpha+\dfrac3\beta$ and $\beta+\dfrac3\alpha$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The roots of the equation $x^2-12x+c=0$ are $\alpha$ and $\beta$, where $\alpha=3\beta$.

   (a)

   Find the value of $\beta$.

   [1 mark]

   • A$4$
   • B$9$
   • C$6$
   • D$3$

   (b)

   Find the value of $c$.

   [1 mark]

   • A$27$
   • B$36$
   • C$9$
   • D$81$

   (c)

   Find a quadratic equation, with integer coefficients, that has roots $\dfrac1\alpha$ and $\dfrac1\beta$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The roots of the equation $5x^2+2x-4=0$ are $\alpha$ and $\beta$.

   (a)

   Find $\alpha^2+\beta^2$.

   [1 mark]

   • A$-\frac{36}{25}$
   • B$\frac{44}{25}$
   • C$\frac{4}{25}$
   • D$\frac{44}{5}$

   (b)

   Find $\dfrac1\alpha+\dfrac1\beta$.

   [1 mark]

   • A$-\frac12$
   • B$2$
   • C$\frac12$
   • D$\frac45$

   (c)

   Find a quadratic equation, with integer coefficients, that has roots $\alpha^2$ and $\beta^2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The roots of the equation $2x^2-8x+3=0$ are $\alpha$ and $\beta$, where $\alpha>\beta$.

   (a)

   Find the exact value of $\alpha-\beta$.

   [3 marks]

   (b)

   Find a quadratic equation, with integer coefficients, that has roots $\dfrac{1}{2\alpha}$ and $\dfrac{1}{2\beta}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The roots of the equation $x^2-4x+k=0$ are $\alpha$ and $\beta$, where $k$ is a constant and $\alpha^3+\beta^3=28$.

   (a)

   (i) Show that $k=3$.
   (ii) Find a quadratic equation, with integer coefficients, that has roots $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$.

   [6 marks]

   (b)

   (i) Given that $k=3$, show that $\alpha^4+\beta^4=82$.
   (ii) Find a quadratic equation, with integer coefficients, that has roots $\alpha^4$ and $\beta^4$.

   [6 marks]

   Total for question 8: 12 marks

End of questions