| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Topic | Roots of polynomials |
| Type | Substitution to find new equation |
| Difficulty | Standard +0.8 This is a Further Maths question requiring systematic manipulation of polynomial roots through substitution. Students must eliminate x between two equations (the original cubic and y = 1 + x²) to derive a new cubic in y, which requires algebraic skill beyond standard A-level but is a recognizable technique in Further Maths. The multi-step nature and need for careful algebraic manipulation place it moderately above average difficulty. |
| Spec | 4.05b Transform equations: substitution for new roots |
Subst. $y = 1 + x^2$
$x(x^2 + 1) + 2(x^2 + 1) - 9 = 0$ **M1** Complete substitution
$y\sqrt{y-1} = 9 - 2y$ **M1** Re-arranging to isolate the $\sqrt{y-1}$'s
$y^2(y-1) = 81 - 36y + 4y^2$ **M1** Squaring (genuinely)
$y^3 - 5y^2 + 36y - 81 = 0$ **A1 cao** (integer multiples ok)
Allow a direct approach:
**B1** for $\Sigma\alpha = -2$, $\Sigma\alpha\beta = 1$ and $\Sigma\alpha\beta\gamma = 7$
**B1** for $\Sigma\alpha' = 3 + \Sigma\alpha^2 = 5$ using $\Sigma\alpha^2 = (\Sigma\alpha)^2 - 2\Sigma\alpha\beta$
**B1** for $\Sigma\alpha'\beta' = 3 + 2\Sigma\alpha^2 + \Sigma\alpha^2\beta^2 = 29$ using $\Sigma\alpha^2\beta^2 = (\Sigma\alpha\beta)^2 - 2\alpha\beta\gamma\Sigma\alpha$
**B1** for $\alpha'\beta'\gamma' = 1 + \Sigma\alpha^2 + \Sigma\alpha^2\beta^2 + (\alpha\beta\gamma)^2 = 81$ and correct equation
**[4]**1 The equation $x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0$ has roots $\alpha , \beta$ and $\gamma$. Use the substitution $y = 1 + x ^ { 2 }$ to find an equation, with integer coefficients, whose roots are $1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }$ and $1 + \gamma ^ { 2 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q1 [4]}}