| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Interpret or complete given sketch of two |linear| functions |
| Difficulty | Easy -1.2 This is a straightforward modulus question requiring basic understanding of graph transformations and simple algebraic verification. Part (i) is immediate recognition that the minimum of |x-2|+1 occurs at (2,1). Part (ii) involves solving |x|=|x-2|+1, which splits into cases but requires only routine algebraic manipulation to verify the given y-coordinate. This is easier than average A-level work as it's heavily scaffolded with a diagram and the answer provided for verification. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| M1: Identify minimum point of \(y = | x - 2 | + 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| M1: Set up equation \( | x | = |
(i) Write down the coordinates of P. [1]
M1: Identify minimum point of $y = |x - 2| + 1$
A1: P = (2, 1)
(ii) Verify that the y-coordinate of Q is $1\frac{1}{2}$. [4]
M1: Set up equation $|x| = |x - 2| + 1$
M1: Solve for case $x \geq 2$: $x = x - 2 + 1$ (no solution)
M1: Solve for case $0 \leq x < 2$: $x = -(x - 2) + 1 = -x + 3$, giving $x = \frac{3}{2}$
A1: Substitute $x = \frac{3}{2}$ to get $y = \frac{3}{2}$ or verify $y = 1\frac{1}{2}$
---1 Fig. 1 shows the graphs of $y = | x |$ and $y = | x - 2 | + 1$. The point P is the minimum point of $y = | x - 2 | + 1$, and Q is the point of intersection of the two graphs.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-2_490_844_493_607}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
(i) Write down the coordinates of P .\\
(ii) Verify that the $y$-coordinate of Q is $1 \frac { 1 } { 2 }$.
\hfill \mbox{\textit{OCR MEI C3 2007 Q1 [5]}}