| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Quadratic with surd roots, exact form |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic quadratic theory (sum/product of roots, forming equations from roots, quadratic formula, and surd conjugates). All parts are routine textbook exercises requiring only standard recall and simple algebraic manipulation, making it easier than average for A-level. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks |
|---|---|
| \((x-p)(x-q)=x^2-(p+q)x+pq\) | B1 |
| Answer | Marks |
|---|---|
| \(p+q=4,\ pq=2^2-3=1\) | B1 B1 |
| \(\Rightarrow x^2-(p+q)x+pq=x^2-4x+1\) | M1 A1 |
| i.e. \(x^2-4x+1=0\) | Total: 4 |
| Answer | Marks |
|---|---|
| \(x=\frac{-5\pm\sqrt{25+28}}{2}=\frac{-5-\sqrt{53}}{2}\) and \(\frac{-5+\sqrt{53}}{2}\) | M1 A1 A1 |
| Answer | Marks |
|---|---|
| \(f(1)=1+2-3=0\) | B1 |
| \(f(x)=(x-1)(x^2+x+3)=0\) | M1 |
| For \(x^2+x+3=0\), \(b^2-4ac=1-12<0\) so no roots | A1 |
| Answer | Marks |
|---|---|
| \(3-\sqrt{5}\) | B1 |
## Question 11(i):
$(x-p)(x-q)=x^2-(p+q)x+pq$ | B1 |
**Total: 1**
## Question 11(ii):
$p+q=4,\ pq=2^2-3=1$ | B1 B1 |
$\Rightarrow x^2-(p+q)x+pq=x^2-4x+1$ | M1 A1 |
i.e. $x^2-4x+1=0$ | **Total: 4**
## Question 11(iii):
$x=\frac{-5\pm\sqrt{25+28}}{2}=\frac{-5-\sqrt{53}}{2}$ and $\frac{-5+\sqrt{53}}{2}$ | M1 A1 A1 |
**Total: 3**
## Question 11(iv):
$f(1)=1+2-3=0$ | B1 |
$f(x)=(x-1)(x^2+x+3)=0$ | M1 |
For $x^2+x+3=0$, $b^2-4ac=1-12<0$ so no roots | A1 |
**Total: 3**
## Question 11(v):
$3-\sqrt{5}$ | B1 |
**Total: 1**
---11 (i) Multiply out $( x - p ) ( x - q )$.\\
(ii) You are given that $p = 2 + \sqrt { 3 }$ and $q = 2 - \sqrt { 3 }$ are the roots of a quadratic equation. Find $p + q$ and $p q$ and hence find the quadratic equation with roots $x = p$ and $x = q$.\\
(iii) Solve the quadratic equation $x ^ { 2 } + 5 x - 7 = 0$ giving the roots exactly.\\
(iv) Show that $x = 1$ is the only root of the equation $x ^ { 3 } + 2 x - 3 = 0$.\\
(v) A quadratic equation $x ^ { 2 } + r x + s = 0$, where $r$ and $s$ are integers, has two roots. One root is $x = 3 + \sqrt { 5 }$. Without finding $r$ or $s$, write down the other root.
\hfill \mbox{\textit{OCR MEI C1 Q11 [12]}}