| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Moderate -0.3 This is a straightforward composite function question requiring students to form fg(x) and gf(x) by substitution, then solve a resulting equation. Part (i) is routine manipulation, and part (ii) involves expanding, simplifying to get a cubic equation that factors nicely (x³ - 2x² - 4 = 24 leads to x³ - 2x² - 28 = 0). While it requires multiple steps and careful algebra, it's a standard textbook-style question with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{fg}(x) = f(x^2 + 2) = (x^2 + 2)^3\) | B1 1.1 [1] | — |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{gf}(x) = g(x^3) = (x^3)^2 + 2 (= x^6 + 2)\) | B1 1.1 [1] | No simplification required |
| Answer | Marks | Guidance |
|---|---|---|
| \((x^2+2)^3 = (x^2)^3 + 3(x^2)^2(2) + 3(x^2)(2)^2 + 2^3\) | M1 1.1 | Binomial expansion of \((x^2+2)^3\) — correct powers and coefficients; allow one slip |
| \(\text{fg}(x) = x^6 + 6x^4 + 12x^2 + 8\) | A1 1.1 | — |
| \(\text{fg}(x) - \text{gf}(x) = 24 \Rightarrow 6x^4 + 12x^2 - 18 = 0\) | A1 2.1 | — |
| \(x^4 + 2x^2 - 3 = 0 \Rightarrow (x^2-1)(x^2+3) = 0\) | M1 1.1 | Correct method for solving quadratic in \(x^2\); if M0 next two marks become B marks |
| \(x^2 + 3 = 0\) has no real solutions | A1 2.4 | \(x^2 + 3 \neq 0\) acceptable |
| \(x^2 - 1 = 0 \Rightarrow x = \pm 1\) | A1 2.2a [6] | — |
## Question 4(i)(a):
$\text{fg}(x) = f(x^2 + 2) = (x^2 + 2)^3$ | B1 1.1 [1] | —
---
## Question 4(i)(b):
$\text{gf}(x) = g(x^3) = (x^3)^2 + 2 (= x^6 + 2)$ | B1 1.1 [1] | No simplification required
---
## Question 4(ii):
**DR**
$(x^2+2)^3 = (x^2)^3 + 3(x^2)^2(2) + 3(x^2)(2)^2 + 2^3$ | M1 1.1 | Binomial expansion of $(x^2+2)^3$ — correct powers and coefficients; allow one slip
$\text{fg}(x) = x^6 + 6x^4 + 12x^2 + 8$ | A1 1.1 | —
$\text{fg}(x) - \text{gf}(x) = 24 \Rightarrow 6x^4 + 12x^2 - 18 = 0$ | A1 2.1 | —
$x^4 + 2x^2 - 3 = 0 \Rightarrow (x^2-1)(x^2+3) = 0$ | M1 1.1 | Correct method for solving quadratic in $x^2$; if M0 next two marks become B marks
$x^2 + 3 = 0$ has no real solutions | A1 2.4 | $x^2 + 3 \neq 0$ acceptable
$x^2 - 1 = 0 \Rightarrow x = \pm 1$ | A1 2.2a [6] | —
---4 In this question you must show detailed reasoning.\\
The functions f and g are defined for all real values of $x$ by
$$\mathrm { f } ( x ) = x ^ { 3 } \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + 2 .$$
(i) Write down expressions for
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { fg } ( x )$,
\item $\operatorname { gf } ( x )$.\\
(ii) Hence find the values of $x$ for which $\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2018 Q4 [8]}}