Forming quadratic equations with new rootsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The roots of the equation $2x^2-7x+3=0$ are $\alpha$ and $\beta$.

   (a)

   A new equation has roots $\frac1\alpha$ and $\frac1\beta$. Find the sum of its roots.

   [1 mark]

   • A$\frac{3}{7}$
   • B$\frac{7}{2}$
   • C$\frac{7}{3}$
   • D$\frac{2}{3}$

   (b)

   Which equation has roots $\frac1\alpha$ and $\frac1\beta$?

   [1 mark]

   • A$3x^2-7x+2=0$
   • B$2x^2-7x+3=0$
   • C$3x^2+7x+2=0$
   • D$x^2-\frac27x+\frac23=0$

   (c)

   Find an equation, with integer coefficients, that has roots $\alpha+1$ and $\beta+1$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

   • A$64$
   • B$88$
   • C$58$
   • D$40$

   (b)

   Which equation has roots $\alpha^3$ and $\beta^3$?

   [1 mark]

   • A$x^2+40x+8=0$
   • B$x^2-40x+8=0$
   • C$x^2-40x+64=0$
   • D$x^2-64x+8=0$

   (c)

   Find an equation, with integer coefficients, that has roots $\frac1\alpha$ and $\frac1\beta$.

   [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 an equation, with integer coefficients, that has roots $\frac{1}{2\alpha}$ and $\frac{1}{2\beta}$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Find an equation, with integer coefficients, that has roots $\alpha+\frac{2}{\beta}$ and $\beta+\frac{2}{\alpha}$.

   [6 marks]

   (b)

   Find an equation, with integer coefficients, that has roots $\alpha^3$ and $\beta^3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions