Sum and product of rootsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The roots of the equation $2x^2-7x+4=0$ are $\alpha$ and $\beta$.

   (a)

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

   [1 mark]

   • A$\frac72$
   • B$-\frac72$
   • C$2$
   • D$7$

   (b)

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

   [1 mark]

   • A$\frac{41}{4}$
   • B$\frac{65}{4}$
   • C$\frac{33}{4}$
   • D$\frac{49}{4}$

   (c)

   Find the value of $\dfrac1\alpha+\dfrac1\beta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The roots of the equation $x^2+6x+7=0$ are $\alpha$ and $\beta$.

   (a)

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

   [1 mark]

   • A$-342$
   • B$-90$
   • C$90$
   • D$-216$

   (b)

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

   [1 mark]

   • A$36$
   • B$64$
   • C$22$
   • D$8$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The roots of the equation $3x^2-5x+1=0$ are $\alpha$ and $\beta$.

   (a)

   Show that $\alpha^2+\beta^2=\frac{19}{9}$.

   [3 marks]

   (b)

   Find a quadratic equation, with integer coefficients, whose roots are $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The roots of the equation $x^2+kx+12=0$, where $k$ is a constant, are $\alpha$ and $\beta$, and $\alpha^2+\beta^2=25$.

   (a)

   (i) Show that $k^2=49$.
   (ii) Given that $k>0$, find the value of $\alpha^3+\beta^3$.

   [6 marks]

   (b)

   Given that $k=7$, find a quadratic equation, with integer coefficients, whose roots are
   (i) $\alpha+2$ and $\beta+2$,
   (ii) $\alpha^3$ and $\beta^3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions