| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |linear| = constant |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic understanding of modulus, function composition, and inverse functions. Part (i) requires simple substitution and solving a linear equation. Part (ii) involves finding the inverse of a composite linear function (routine). Part (iii) requires splitting cases for the modulus but is a standard textbook exercise with minimal problem-solving demand. All parts are mechanical applications of well-practiced techniques. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: Attempt solution of at least one linear eq'n of form \(ax + b = 12\) | M1 | |
| Obtain \(\frac{1}{4}\) | A2 | 3 marks: and (finally) no other answer |
| Or: Attempt solution of 3-term quadratic eq'n obtained by squaring attempt at \(g(x + 2)\) on LHS and squaring 12 or -12 on RHS | M1 | |
| Obtain \(\frac{1}{4}\) | A2 | (3) and (finally) no other answer |
| (ii) Either: Obtain \(3(3x + 5) + 5\) for h | B1 | |
| Attempt to find inverse function | M1 | of function of form \(ax + b\) |
| Obtain \(\frac{1}{3}(x - 20)\) | A1 | 3 marks: or equiv in terms of \(x\) |
| Or: State or imply \(g^{-1}\) is \(\frac{1}{3}(x - 5)\) | B1 | |
| Attempt composition of \(g^{-1}\) with \(g^{-1}\) | M1 | |
| Obtain \(\frac{1}{3}(x - 5) - \frac{5}{3}\) | A1 | (3) or more simplified equiv in terms of \(x\) |
| (iii) State \(x \leq 0\) | B2 | 2 marks: give B1 for answer \(x < 0\) |
**(i)** Either: Attempt solution of at least one linear eq'n of form $ax + b = 12$ | M1 |
Obtain $\frac{1}{4}$ | A2 | 3 marks: and (finally) no other answer
Or: Attempt solution of 3-term quadratic eq'n obtained by squaring attempt at $g(x + 2)$ on LHS and squaring 12 or -12 on RHS | M1 |
Obtain $\frac{1}{4}$ | A2 | (3) and (finally) no other answer
**(ii)** Either: Obtain $3(3x + 5) + 5$ for h | B1 |
Attempt to find inverse function | M1 | of function of form $ax + b$
Obtain $\frac{1}{3}(x - 20)$ | A1 | 3 marks: or equiv in terms of $x$
Or: State or imply $g^{-1}$ is $\frac{1}{3}(x - 5)$ | B1 |
Attempt composition of $g^{-1}$ with $g^{-1}$ | M1 |
Obtain $\frac{1}{3}(x - 5) - \frac{5}{3}$ | A1 | (3) or more simplified equiv in terms of $x$
**(iii)** State $x \leq 0$ | B2 | 2 marks: give B1 for answer $x < 0$
---7 The functions $\mathrm { f } , \mathrm { g }$ and h are defined for all real values of $x$ by
$$\mathrm { f } ( x ) = | x | , \quad \mathrm { g } ( x ) = 3 x + 5 \quad \text { and } \quad \mathrm { h } ( x ) = \mathrm { gg } ( x ) .$$
(i) Solve the equation $\mathrm { g } ( x + 2 ) = \mathrm { f } ( - 12 )$.\\
(ii) Find $\mathrm { h } ^ { - 1 } ( x )$.\\
(iii) Determine the values of $x$ for which
$$x + \mathrm { f } ( x ) = 0 .$$
\hfill \mbox{\textit{OCR C3 2011 Q7 [8]}}