| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a routine multi-part question on completing the square and inverse functions, both standard P1 topics. Part (i) is mechanical algebra, part (ii) requires knowing that a quadratic is one-one on one side of its vertex (straightforward concept), and part (iii) is standard inverse function procedure. All parts follow textbook methods with no novel problem-solving required, making this easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-(1)(x-3)^2 + 4\) | B1B1B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Smallest \((m)\) is \(3\) | B1\(\checkmark\) | Accept \(m \geqslant 3\), \(m = 3\). Not \(x \geqslant 3\). Ft *their b* |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x-3)^2 = 4 - y\) | M1 | Or \(x/y\) transposed. Ft *their a, b, c* |
| Correct order of operations | M1 | |
| \(f^{-1}(x) = 3 + \sqrt{4-x}\) cao | A1 | Accept \(y =\) if clear |
| Domain is \(x \leqslant 0\) | B1 | |
| Total: 4 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-(1)(x-3)^2 + 4$ | B1B1B1 | |
| **Total: 3** | | |
---
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Smallest $(m)$ is $3$ | B1$\checkmark$ | Accept $m \geqslant 3$, $m = 3$. **Not** $x \geqslant 3$. Ft *their b* |
| **Total: 1** | | |
---
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x-3)^2 = 4 - y$ | M1 | Or $x/y$ transposed. Ft *their a, b, c* |
| Correct order of operations | M1 | |
| $f^{-1}(x) = 3 + \sqrt{4-x}$ cao | A1 | Accept $y =$ if clear |
| Domain is $x \leqslant 0$ | B1 | |
| **Total: 4** | | |
---9 (i) Express $- x ^ { 2 } + 6 x - 5$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
The function $\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5$ is defined for $x \geqslant m$, where $m$ is a constant.\\
(ii) State the smallest value of $m$ for which f is one-one.\\
(iii) For the case where $m = 5$, find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-16_771_636_260_756}
The diagram shows a cuboid $O A B C P Q R S$ with a horizontal base $O A B C$ in which $A B = 6 \mathrm {~cm}$ and $O A = a \mathrm {~cm}$, where $a$ is a constant. The height $O P$ of the cuboid is 10 cm . The point $T$ on $B R$ is such that $B T = 8 \mathrm {~cm}$, and $M$ is the mid-point of $A T$. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O C$ and $O P$ respectively.\\
\hfill \mbox{\textit{CAIE P1 Q9 [8]}}