Moderate -0.3 This is a straightforward application of standard relationships between roots and coefficients (Vieta's formulas) combined with the factor theorem. Given three conditions about the roots, students systematically use α+β+γ=-p/2, αβγ=-r/2, and substitution of x=4 to find three unknowns. While it requires careful algebraic manipulation, it's a routine FP1 exercise with no conceptual surprises—slightly easier than average due to its mechanical nature.
3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).
OR M1 using \(\sum\alpha\beta\) OR use of remainder after division
\(\Rightarrow q = 11\)
A1
for q, cao
[6]
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{-p}{2} = 6 \Rightarrow p = -12$ | M1, M1 | M1 use of $\sum\alpha$ for p and M1 use of $\alpha\beta\gamma$ for r - allow one sign error; 2 sign errors is M1 M0 |
| $\frac{-r}{2} = -10 \Rightarrow r = 20$ | A1, A1 | for p, cao; for r, cao |
| **OR:** $\alpha + \beta + 4 = 6$, $4\alpha\beta = -10$ | OR | |
| Implies $\alpha, \beta$ satisfy $2x^2 - 4x - 5 = 0$ | M1 | Valid method to create a quadratic equation |
| Roots $1 \pm \frac{\sqrt{14}}{2}$ | M1 | Attempt to solve a 3-term quadratic |
| $-\frac{p}{2} = 1 + \frac{\sqrt{14}}{2} + 1 - \frac{\sqrt{14}}{2} + 4 = 6 \Rightarrow p = -12$ | A1 | for p, cao |
| Product of roots $= -10 = -\frac{r}{2} \Rightarrow r = 20$ | A1 | for r, cao |
| **THEN** | THEN | |
| EITHER $x = 4$ is a root, so $2\times64 + 16p + 4q + r = 0$ | M1 | Substitution and attempt to solve for coefficient of $x^2$, or for the remaining unknown. Allow making q the subject if p and r not found |
| OR $\alpha + \beta + 4 = 6 \Rightarrow \alpha + \beta = 2$ | | |
| $4\alpha\beta = -10 \Rightarrow \alpha\beta = -\frac{10}{4}$ | | |
| $\frac{q}{2} = 4\alpha + 4\beta + \alpha\beta = 4\times2 - \frac{5}{2}$ | | OR M1 using $\sum\alpha\beta$ OR use of remainder after division |
| $\Rightarrow q = 11$ | A1 | for q, cao |
| | **[6]** | |
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3 The equation $2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0$ has a root at $x = 4$. The sum of the roots is 6 and the product of the roots is - 10 . Find $p , q$ and $r$.
\hfill \mbox{\textit{OCR MEI FP1 2015 Q3 [6]}}