| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Perimeter or area constraint inequality |
| Difficulty | Moderate -0.8 This is a straightforward application question requiring translation of word problems into inequalities and basic algebraic manipulation. Part (i) is simple linear inequality setup, part (ii) involves expanding and factorizing a quadratic (with the factorization given), and part (iii) requires solving linear and quadratic inequalities—all standard C1 techniques with no novel problem-solving required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | Marks | Guidance |
| \(2[10 + x + x] > 64\) | B1 1 | \(20 + 4x > 64\) o.e. |
| (ii) | Marks | Guidance |
| \(x(x+10) < 299\) | B1 | |
| \(x^2 + 10x - 299 < 0\) | ||
| \((x-13)(x+23) < 0\) | B1 2 | Correctly shows \((x-13)(x+23) < 0\) |
| (iii) | Marks | Guidance |
| \(x > 11\) | B1 \(\sqrt{--}\) | \(x > 11\) ft from their (i) |
| \((x-13)(x+23) < 0\) | M2 | Correct method to solve \((x-13)(x+23) < 0\) eg graph |
| \(-23 < x < 13\) | A1 | \(-23 < x < 13\) seen in this form or as number line |
| SR if seen with no working B1 | ||
| \(\therefore 11 < x < 13\) | B1 5 | |
| 8 |
**(i)** | **Marks** | **Guidance**
---|---|---
$2[10 + x + x] > 64$ | B1 1 | $20 + 4x > 64$ o.e.
**(ii)** | **Marks** | **Guidance**
---|---|---
$x(x+10) < 299$ | B1 |
$x^2 + 10x - 299 < 0$ | |
$(x-13)(x+23) < 0$ | B1 2 | Correctly shows $(x-13)(x+23) < 0$ | **AG**
**(iii)** | **Marks** | **Guidance**
---|---|---
$x > 11$ | B1 $\sqrt{--}$ | $x > 11$ ft from their (i)
$(x-13)(x+23) < 0$ | M2 | Correct method to solve $(x-13)(x+23) < 0$ eg graph
| |
$-23 < x < 13$ | A1 | $-23 < x < 13$ seen in this form or as number line
| | **SR** if seen with no working B1
$\therefore 11 < x < 13$ | B1 5 |
| **8** |8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is $x$ metres.\\
(i) The perimeter of the playground is greater than 64 m . Write down a linear inequality in $x$.\\
(ii) The area of the playground is less than $299 \mathrm {~m} ^ { 2 }$. Show that $( x - 13 ) ( x + 23 ) < 0$.\\
(iii) By solving the inequalities in parts (i) and (ii), determine the set of possible values of $x$.
\hfill \mbox{\textit{OCR C1 2005 Q8 [8]}}