| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question on completing the square and inverse functions, slightly easier than average. Part (a) is routine completing the square, (b) requires identifying the vertex, (c) is standard inverse function technique, and (d) applies the inverse to solve a composite function equation. All parts follow textbook procedures with no novel insight required, though the composite function in (d) adds minor complexity. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a) | 3(x−2)2 +2 or a=−2, b=2 | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(b) | 2 or k =2 or k2 | B1FT |
| Answer | Marks |
|---|---|
| 8(c) | y−2 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | M1 | Using their completed square form. |
| Answer | Marks |
|---|---|
| 3 | DM1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | A1 | 3x−6 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 8(d) | Finding f−1(29) [= 5] | M1 |
| Finding f−1(their 5) | M1 | Or solving f(x) = their 5. |
| x=3 | A1 | If using f(x) method, x = 1 must be discarded. |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3(x−2)2 +2 −2)2 +2=29 using their completed square form | M1 | Or 3 ( 3x2 −12x+14 )2 −12 ( 3x2 −12x+14 ) +14=29. |
| Answer | Marks | Guidance |
|---|---|---|
| Solving as far as 9(x−2)4 =9 or x2 −4x+3=0 | DM1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| x=3 only | A1 | WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(a) ---
8(a) | 3(x−2)2 +2 or a=−2, b=2 | B1 B1
2
--- 8(b) ---
8(b) | 2 or k =2 or k2 | B1FT | FT on their a.
Do not accept x = 2 or x ⩾ 2.
1
--- 8(c) ---
8(c) | y−2
3(x−2)2 +14−12= y (x−2)2 =
3 | M1 | Using their completed square form.
y−2
x= + 2
3 | DM1
x−2
f−1(x)= +2
3 | A1 | 3x−6
OE, e.g. y= +2.
3
3
Question | Answer | Marks | Guidance
--- 8(d) ---
8(d) | Finding f−1(29) [= 5] | M1 | Or solving f(x) = 29 [using their completed square form, OE].
Finding f−1(their 5) | M1 | Or solving f(x) = their 5.
x=3 | A1 | If using f(x) method, x = 1 must be discarded.
Alternative solution for Question 8(d)
( )
3 3(x−2)2 +2 −2)2 +2=29 using their completed square form | M1 | Or 3 ( 3x2 −12x+14 )2 −12 ( 3x2 −12x+14 ) +14=29.
Allow if the '=29' appears later in the working.
Solving as far as 9(x−2)4 =9 or x2 −4x+3=0 | DM1 | OE
Or 27 ( x4 −8x3+24x2 −32x+15 ) =0.
x=3 only | A1 | WWW
Only dependent on the first M1.
3
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\alph*)]
\item Express $3x^2 - 12x + 14$ in the form $3(x + a)^2 + b$, where $a$ and $b$ are constants to be found. [2]
\end{enumerate}
The function f(x) = $3x^2 - 12x + 14$ is defined for $x \geqslant k$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the least value of $k$ for which the function $\text{f}^{-1}$ exists. [1]
\end{enumerate}
For the rest of this question, you should assume that $k$ has the value found in part (b).
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find an expression for $\text{f}^{-1}(x)$. [3]
\item Hence or otherwise solve the equation $\text{f f}(x) = 29$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q8 [9]}}