| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.3 This is a standard application of Newton's identities/symmetric functions requiring students to use Vieta's formulas (α+β+γ=-2, αβ+βγ+γα=3) and the identity (α+β+γ)²=α²+β²+γ²+2(αβ+βγ+γα) to find α²+β²+γ²=(-2)²-2(3)=-2. The negative result indicates complex roots. While this requires knowing the technique, it's a routine textbook exercise with straightforward algebra, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
**Question 2**
$\Sigma\alpha = -2$ and $\Sigma\alpha\beta = 3$ **B1** Both ($\alpha\beta\gamma = -7$ not required)
$\alpha^2 + \beta^2 + \gamma^2 = (\Sigma\alpha)^2 - 2\Sigma\alpha\beta = -2$ **M1A1** FT
1 real and 2 complex (conjugate) roots **B1** Accept any comment that "not all roots are real"
**Alternative:**
Form an equation with roots $\alpha^2, \beta^2, \gamma^2$: $y^3 + 2y^2 - 19y - 49 = 0$ **M1A1**
$\Sigma\alpha^2 = -\dfrac{b}{a} = -2$ **B1** FT
1 real and 2 complex (conjugate) roots **B1** Accept any comment that "not all roots are real"
**Total: 4 marks**2 The equation $x ^ { 3 } + 2 x ^ { 2 } + 3 x + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$. Evaluate $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ and use your answer to comment on the nature of these roots.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2017 Q2 [4]}}