| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a routine multi-part question testing standard techniques: completing the square (straightforward with coefficient 4), reading vertex coordinates, determining range from a restricted domain, and finding an inverse function. All steps are mechanical applications of well-practiced methods with no problem-solving insight required, making it easier than average but not trivial due to the multiple parts. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(4(x-3)^2 - 25\) | B1B1B1 | Or \(a = 4, b = 3, c = -25\). It to their \((b, c)\). Accept if not 'hence' |
| Vertex is \((3, -25)\) | B1 | [4] |
| (ii) range is \((\)g(x)\() \geq -9\). Allow \(>\) | B1B1 | B1 for \(\geq\), B1 for \(-9\). Accept e.g. \([-9, \infty)\) |
| (iii) \((x-3)^2 = \frac{1}{4}(y+25)\) | M1 | |
| \(x - 3 = (\pm)\frac{1}{2}\sqrt{y+25}\) | DM1 | Attempt to square root both sides |
| \(g^{-1}(x) = 3 - \frac{1}{2}\sqrt{x+25}\) | A1 | cao |
| Domain is \(x \geq -9\) | B1 | It from their (ii) |
**(i)** $4(x-3)^2 - 25$ | B1B1B1 | Or $a = 4, b = 3, c = -25$. It to their $(b, c)$. Accept if not 'hence'
Vertex is $(3, -25)$ | B1 | [4]
**(ii)** range is $($g(x)$) \geq -9$. Allow $>$ | B1B1 | B1 for $\geq$, B1 for $-9$. Accept e.g. $[-9, \infty)$ | [2]
**(iii)** $(x-3)^2 = \frac{1}{4}(y+25)$ | M1 |
$x - 3 = (\pm)\frac{1}{2}\sqrt{y+25}$ | DM1 | Attempt to square root both sides
$g^{-1}(x) = 3 - \frac{1}{2}\sqrt{x+25}$ | A1 | cao
Domain is $x \geq -9$ | B1 | It from their (ii) | [4]10 The function f is defined by $\mathrm { f } ( x ) = 4 x ^ { 2 } - 24 x + 11$, for $x \in \mathbb { R }$.\\
(i) Express $\mathrm { f } ( x )$ in the form $a ( x - b ) ^ { 2 } + c$ and hence state the coordinates of the vertex of the graph of $y = \mathrm { f } ( x )$.
The function g is defined by $\mathrm { g } ( x ) = 4 x ^ { 2 } - 24 x + 11$, for $x \leqslant 1$.\\
(ii) State the range of g .\\
(iii) Find an expression for $\mathrm { g } ^ { - 1 } ( x )$ and state the domain of $\mathrm { g } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE P1 2012 Q10 [10]}}