| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find minimum domain for inverse |
| Difficulty | Moderate -0.3 This is a standard three-part question on completing the square, finding domains for one-one functions, and finding inverse functions. Part (i) is routine algebraic manipulation, part (ii) requires identifying the vertex of a parabola, and part (iii) involves standard inverse function technique. All parts are textbook exercises requiring no novel insight, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-(1)(x-3)^2 + 4\) | B1B1B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Smallest \((m)\) is \(3\) | B1\(\checkmark\) [1] | Accept \(m \geqslant 3\), \(m=3\). Not \(x \geqslant 3\). Ft *their b* |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | 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 [4] |
## Question 9:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-(1)(x-3)^2 + 4$ | B1B1B1 [3] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Smallest $(m)$ is $3$ | B1$\checkmark$ [1] | Accept $m \geqslant 3$, $m=3$. **Not** $x \geqslant 3$. Ft *their b* |
### Part (iii):
| Answer/Working | 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 [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 $\mathrm { f } ^ { - 1 }$.\\[0pt]
[Questions 10 and 11 are printed on the next page.]
{www.cie.org.uk} after the live examination series.
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\includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_773_641_260_753}
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 2015 Q9 [8]}}