| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Standard +0.3 Part (i) requires reading the range from a decreasing square root function (straightforward). Part (ii) is a routine composite function evaluation: f(f(4)) = f(0) = 2. Part (iii) requires understanding absolute values and analyzing when |f(x)| = k has two solutions, involving consideration of the function's behavior and symmetry about the x-axis. This is slightly above average due to the absolute value analysis in part (iii), but still a standard C3 question with no novel insights required. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(y \leq 2\) | B1 | 1 Or equiv; allow \(<\); allow any letter or none |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show correct process for composition of functions | M1 | Numerical or algebraic |
| Obtain 0 and hence 2 | A1 | 2 And no other value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State a range of values with 2 as one end-point | M1 | Continuous set, not just integers |
| State \(0 < k \leq 2\) | A1 | 2 With correct \(<\) and \(\leq\) now |
## Question 4(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $y \leq 2$ | B1 | **1** Or equiv; allow $<$; allow any letter or none |
## Question 4(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show correct process for composition of functions | M1 | Numerical or algebraic |
| Obtain 0 and hence 2 | A1 | **2** And no other value |
## Question 4(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State a range of values with 2 as one end-point | M1 | Continuous set, not just integers |
| State $0 < k \leq 2$ | A1 | **2** With correct $<$ and $\leq$ now |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701}
The function f is defined by $\mathrm { f } ( x ) = 2 - \sqrt { x }$ for $x \geqslant 0$. The graph of $y = \mathrm { f } ( x )$ is shown above.\\
(i) State the range of f.\\
(ii) Find the value of $\mathrm { ff } ( 4 )$.\\
(iii) Given that the equation $| \mathrm { f } ( x ) | = k$ has two distinct roots, determine the possible values of the constant $k$.
\hfill \mbox{\textit{OCR C3 2006 Q4 [5]}}