| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Perimeter or area constraint inequality |
| Difficulty | Moderate -0.3 This is a straightforward C1 inequality question requiring students to form and solve a quadratic inequality in part (i) and a compound linear inequality in part (ii). While it involves multiple steps and some algebraic manipulation, the techniques are standard and the problem-solving required is minimal—students simply translate the given constraints into algebraic form and solve using routine methods. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area of tile \(= 4x(x+3)\) | B1 | Correct expression for area of rectangle (may be unsimplified) |
| \(4x(x+3) < 112\) | B1\(\checkmark\) | Correct inequality for their expression |
| \(4x^2 + 12x - 112 < 0\) | Correct alternative forms for factorised inequality include: \((x+7)(4x-16) < 0\); \((4x+28)(x-4) < 0\); \((2x+14)(2x-8) < 0\) etc. | |
| \(4(x+7)(x-4) < 0\) | M1 | Correct method to solve a three term quadratic |
| M1 | Chooses correct region for the quadratic inequality i.e. lower root \(< x <\) higher root. (May be implied by correct final answer) | |
| \(-7 < x < 4\) | A1 | |
| \(\therefore 0 < x < 4\) | A1 [6] | Restricts range to positive values of \(x\) CWO. Do not allow \(\leq\) for final A mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Perimeter \(= 4y + (y+3) + 2y + y + 2y + 3\) | M1 | Clear attempt to add lengths of all 6 edges. Allow \(<\) or \(\leq\) throughout part (ii) |
| A1 | Correct perimeter simplified to \(10y + 6\) seen | |
| \(20 < 10y + 6 < 54\) | B1 FT | Correct inequalities for their expression. Can still be unsimplified here |
| M1 | Solving 2 linear equations or inequalities dealing with all 3 terms | |
| \(1.4 < y < 4.8\) | A1 [5] | Accept "\(1.4 < y\), \(y < 4.8\)", "\(1.4 < y\) and \(y < 4.8\)" but NOT "\(1.4 < y\) or \(y < 4.8\)". Do not ISW if contradictory incorrect form follows correct answer |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area of tile $= 4x(x+3)$ | B1 | Correct expression for area of rectangle (may be unsimplified) |
| $4x(x+3) < 112$ | B1$\checkmark$ | Correct inequality for their expression |
| $4x^2 + 12x - 112 < 0$ | | Correct alternative forms for factorised inequality include: $(x+7)(4x-16) < 0$; $(4x+28)(x-4) < 0$; $(2x+14)(2x-8) < 0$ etc. |
| $4(x+7)(x-4) < 0$ | M1 | Correct method to solve a three term quadratic |
| | M1 | Chooses correct region for the quadratic inequality i.e. lower root $< x <$ higher root. (May be implied by correct final answer) |
| $-7 < x < 4$ | A1 | |
| $\therefore 0 < x < 4$ | A1 [6] | Restricts range to positive values of $x$ **CWO**. **Do not allow $\leq$ for final A mark** |
---
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Perimeter $= 4y + (y+3) + 2y + y + 2y + 3$ | M1 | **Clear** attempt to add lengths of all **6** edges. **Allow $<$ or $\leq$ throughout part (ii)** |
| | A1 | Correct perimeter simplified to $10y + 6$ seen |
| $20 < 10y + 6 < 54$ | B1 FT | Correct inequalities for their expression. Can still be unsimplified here |
| | M1 | Solving 2 linear equations or inequalities dealing with all 3 terms |
| $1.4 < y < 4.8$ | A1 [5] | Accept "$1.4 < y$, $y < 4.8$", "$1.4 < y$ **and** $y < 4.8$" but **NOT** "$1.4 < y$ **or** $y < 4.8$". **Do not ISW** if contradictory incorrect form follows correct answer |
---9 (i) A rectangular tile has length $4 x \mathrm {~cm}$ and width $( x + 3 ) \mathrm { cm }$. The area of the rectangle is less than $112 \mathrm {~cm} ^ { 2 }$. By writing down and solving an inequality, determine the set of possible values of $x$.\\
(ii) A second rectangular tile of length $4 y \mathrm {~cm}$ and width $( y + 3 ) \mathrm { cm }$ has a rectangle of length $2 y \mathrm {~cm}$ and width $y \mathrm {~cm}$ removed from one corner as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662}
Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of $y$.
\hfill \mbox{\textit{OCR C1 2012 Q9 [11]}}