| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| 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 on completing the square and inverse functions. Part (i) is standard algebraic manipulation, part (ii) requires identifying the vertex, part (iii) involves standard inverse function technique by swapping and rearranging, and part (iv) asks for the range. All techniques are textbook exercises with no novel problem-solving required, making it 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 |
|---|---|---|
| 9(i) | ( 3x−1 )2 +5 | B1B1B1 |
| Answer | Marks |
|---|---|
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(ii) | Smallest value of p is 1/3 seen. (Independent of (i)) | B1 |
| Total: | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(iii) | y=( )2 +5⇒3x−1=(±) | |
| 3x−1 y−5 | B1 FT | 2 2 |
| Answer | Marks | Guidance |
|---|---|---|
| x=(±) ⅓ y−5+⅓ OE | B1 FT | Both starts require 2 operations for each mark. FT for their values from part |
| Answer | Marks |
|---|---|
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(iv) | q<5 CAO | B1 |
| Total: | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | ( 3x−1 )2 +5 | B1B1B1 | First 2 marks dependent on correct ( ax+b )2 form. OR a=3, b=−1, c=5
e.g. from equating coefs
Total: | 3
--- 9(ii) ---
9(ii) | Smallest value of p is 1/3 seen. (Independent of (i)) | B1 | Allow p(cid:46)1/3 or p=1/3 or 1/3 seen. But not in terms of x.
Total: | 1
--- 9(iii) ---
9(iii) | y=( )2 +5⇒3x−1=(±)
3x−1 y−5 | B1 FT | 2 2
OR y= 9 x− 1 +5⇒ ( y−5 ) /9= x− 1 (Fresh start)
3 3
x=(±) ⅓ y−5+⅓ OE | B1 FT | Both starts require 2 operations for each mark. FT for their values from part
(i)
Total: | 4
--- 9(iv) ---
9(iv) | q<5 CAO | B1
Total: | 1
Alt 9(iii) For start ((cid:1853)(cid:1876)−(cid:1854))(cid:2870)+(cid:1855) or (cid:1853)((cid:1876)−(cid:1854))(cid:2870)+(cid:1855) (a≠ 0) ft for their a, b, c
For start ((cid:1876)−(cid:1854))(cid:2870)+(cid:1855) ft but award only B1 for 3 correct operations
For start (cid:1853)((cid:1854)(cid:1876)−(cid:1855))(cid:2870)+(cid:1856) ft but award B1 for first2 operations correct and B1 for the next 3 operations correct
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Express $9x^2 - 6x + 6$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are constants. [3]
\end{enumerate}
The function f is defined by $\text{f}(x) = 9x^2 - 6x + 6$ for $x \geqslant p$, where $p$ is a constant.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item State the smallest value of $p$ for which f is a one-one function. [1]
\item For this value of $p$, obtain an expression for $\text{f}^{-1}(x)$, and state the domain of $\text{f}^{-1}$. [4]
\item State the set of values of $q$ for which the equation $\text{f}(x) = q$ has no solution. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q9 [9]}}