| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then find vertex/turning point |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise with a standard follow-up question. Part (i) is routine algebraic manipulation requiring only the formula (x+a)² = x²+2ax+a², and part (ii) is direct application recognizing that 1/f(x) is maximized when f(x) is minimized. Both parts are below average difficulty for A-level, being standard C1 textbook exercises with no problem-solving insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 + 4x + ... = (x+2)^2 + ...\) | M1 | |
| \(\Rightarrow x^2 + 4x + 14 = (x+2)^2 + 14 - 4 = (x+2)^2 + 10\) | A1 A1 | For each value. Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Greatest value of \(\frac{1}{x^2+4x+14}\) = least value of \(x^2+4x+14=10\) | F1 | |
| i.e. Greatest value of \(\frac{1}{x^2+4x+14} = \frac{1}{10}\) | Total: 1 |
# Question 2:
## Part (i):
$x^2 + 4x + ... = (x+2)^2 + ...$ | M1 |
$\Rightarrow x^2 + 4x + 14 = (x+2)^2 + 14 - 4 = (x+2)^2 + 10$ | A1 A1 | For each value. **Total: 3**
## Part (ii):
Greatest value of $\frac{1}{x^2+4x+14}$ = least value of $x^2+4x+14=10$ | F1 |
i.e. Greatest value of $\frac{1}{x^2+4x+14} = \frac{1}{10}$ | | **Total: 1**
---2 (i) Find the constants $a$ and $b$ such that, for all values of $x$,
$$x ^ { 2 } + 4 x + 14 = ( x + a ) ^ { 2 } + b$$
(ii) Write down the greatest value of $\frac { 1 } { x ^ { 2 } + 4 x + 14 }$.
\hfill \mbox{\textit{OCR MEI C1 Q2 [4]}}