| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | March |
| 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 (basic algebra), identifying where a quadratic is decreasing (vertex knowledge), finding an inverse function (standard procedure), and composing functions. All parts are textbook exercises requiring recall and mechanical application rather than problem-solving or insight. Slightly easier than average due to straightforward nature and guided structure. |
| 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 |
|---|---|
| 8(i) | ( )2 +[ ] |
| Answer | Marks | Guidance |
|---|---|---|
| | B1 DB1 | 2nd B1 dependent on ±2 in 1st bracket |
| Answer | Marks | Guidance |
|---|---|---|
| 8(ii) | Largest k is 2 Accept k(cid:45)2 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(iii) | y=( x−2 )2 +3 ⇒ x−2=(±) y−3 | M1 |
| Answer | Marks |
|---|---|
| ⇒f−1 x x−3 for x > 4 | A1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 8(iv) | 2 2 |
| Answer | Marks | Guidance |
|---|---|---|
| x2 −4x+7−1 ( x−2 )2 +2 | B1 | Either form |
| Since f ( x )>4⇒gf ( x )<2/3 (or since x<1 etc) | M1A1 | 2/3 in answer implies M1 www |
| range of gf(x) is 0 < gf(x)( < 2/3) | B1 | Accept 0 < y < 2/3, (0, 2/3) but 0 < x < 2/3 is |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | ( )2 +[ ]
x−2 3
| B1 DB1 | 2nd B1 dependent on ±2 in 1st bracket
2
--- 8(ii) ---
8(ii) | Largest k is 2 Accept k(cid:45)2 | B1 | Must be in terms of k
1
--- 8(iii) ---
8(iii) | y=( x−2 )2 +3 ⇒ x−2=(±) y−3 | M1
( )=2−
⇒f−1 x x−3 for x > 4 | A1B1
3
Question | Answer | Marks | Guidance
--- 8(iv) ---
8(iv) | 2 2
gf(x) = =
x2 −4x+7−1 ( x−2 )2 +2 | B1 | Either form
Since f ( x )>4⇒gf ( x )<2/3 (or since x<1 etc) | M1A1 | 2/3 in answer implies M1 www
range of gf(x) is 0 < gf(x)( < 2/3) | B1 | Accept 0 < y < 2/3, (0, 2/3) but 0 < x < 2/3 is
SCM1A1B0
4
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Express $x^2 - 4x + 7$ in the form $(x + a)^2 + b$. [2]
\end{enumerate}
The function $f$ is defined by $f(x) = x^2 - 4x + 7$ for $x < k$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item State the largest value of $k$ for which $f$ is a decreasing function. [1]
\end{enumerate}
The value of $k$ is now given to be $1$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$. [3]
\item The function $g$ is defined by $g(x) = \frac{2}{x-1}$ for $x > 1$. Find an expression for $gf(x)$ and state the range of $gf$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q8 [10]}}