| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question covering completing the square, domain/range, and finding an inverse function. All parts follow routine procedures taught in P1 with no novel problem-solving required. The completing the square is straightforward, and finding the inverse involves standard algebraic manipulation. Slightly easier than average due to the step-by-step scaffolding through familiar techniques. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2x^2 - 12x + 13 = 2(x - 3)^2 - 5\) | \(3 \times\) B1 [3] | Allow even if \(a\), \(b\), \(c\) not specifically quoted |
| (ii) Symmetrical about \(x = 3\). \(A = 6\) | B1√ [1] | For \(2 \times\) his (\(-b\)) |
| (iii) One limit is \(-5\) | B1√ | For his \(c\) |
| Other limit is 13 | B1 [2] | co |
| (iv) Inverse since 1:1 (\(4 > 3\)) | B1 [1] | Valid argument |
| (v) Makes \(x\) the subject of the equation | M1 | Attempts to change the formula |
| Order of operations correct | DM1 | "\(+5\)", \(+2\), \(N\), \(+3\). Allow for simple algebraic slips such as \(-5\) for \(+5\) etc. |
| \(\rightarrow \sqrt{\frac{x+5}{2}} + 3\) | A1 [3] | co – as a function of \(x\), not \(y\). condone \(\pm\) |
(i) $2x^2 - 12x + 13 = 2(x - 3)^2 - 5$ | $3 \times$ B1 [3] | Allow even if $a$, $b$, $c$ not specifically quoted
(ii) Symmetrical about $x = 3$. $A = 6$ | B1√ [1] | For $2 \times$ his ($-b$)
(iii) One limit is $-5$ | B1√ | For his $c$
Other limit is 13 | B1 [2] | co
(iv) Inverse since 1:1 ($4 > 3$) | B1 [1] | Valid argument
(v) Makes $x$ the subject of the equation | M1 | Attempts to change the formula
Order of operations correct | DM1 | "$+5$", $+2$, $N$, $+3$. Allow for simple algebraic slips such as $-5$ for $+5$ etc.
$\rightarrow \sqrt{\frac{x+5}{2}} + 3$ | A1 [3] | co – as a function of $x$, not $y$. condone $\pm$10 The function f is defined by $\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 13$ for $0 \leqslant x \leqslant A$, where $A$ is a constant.\\
(i) Express $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
(ii) State the value of $A$ for which the graph of $y = \mathrm { f } ( x )$ has a line of symmetry.\\
(iii) When $A$ has this value, find the range of f .
The function g is defined by $\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 13$ for $x \geqslant 4$.\\
(iv) Explain why $g$ has an inverse.\\
(v) Obtain an expression, in terms of $x$, for $\mathrm { g } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2009 Q10 [10]}}