| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine if inverse exists |
| Difficulty | Standard +0.3 This is a standard C3 composite functions question with routine parts: solving fg(x)=8 requires basic log manipulation, identifying which function has an inverse is straightforward (f does, g doesn't due to symmetry), differentiation using chain rule is standard, and Simpson's rule is a bookwork numerical method. All parts are textbook exercises requiring no novel insight, though slightly above average difficulty due to multiple techniques across four parts. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07q Product and quotient rules: differentiation1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\ln(x^2 + 8) = 8\) | B1 | or equiv such as \(x^2 + 8 = e^8\) |
| Attempt solution involving \(e^8\) | M1 | by valid (exact) method at least as far as \(x^2 = ...\) |
| Obtain \(\sqrt{e^8 - 8}\) | A1 | 3 or exact equiv; and no other answer |
| (ii) State f only | B1 | |
| State \(e^x\) or \(e^y\) | B1 | or equiv; allow if g, or f and g, chosen however expressed |
| Indicate domain is all real numbers | B1 | 3 howsoever expressed |
| (iii) Attempt use of chain rule | M1 | whether applied to gf or fg; or equiv such as use of product rule on \((\ln x)(\ln x) + 8\) |
| Obtain \(\frac{2\ln x}{x}\) | A1 | or equiv |
| Obtain \(6e^{-3}\) | A1 | 3 or exact equiv but not including \(\ln 1\) remaining |
| (iv) Attempt evaluation using y attempts | M1 | with coeffs 1, 4 and 2 occurring at least once each; whether fg or gf any constant \(k\) |
| Obn \(k(\ln(24 + 4\ln 12 + 2\ln 8 + 4\ln 12 + \ln 24)\) | A1 | Use \(k = \frac{1}{3}\) and obtain 20.3 |
**(i)** State $\ln(x^2 + 8) = 8$ | B1 | or equiv such as $x^2 + 8 = e^8$
Attempt solution involving $e^8$ | M1 | by valid (exact) method at least as far as $x^2 = ...$
Obtain $\sqrt{e^8 - 8}$ | A1 | 3 or exact equiv; and no other answer
**(ii)** State f only | B1 |
State $e^x$ or $e^y$ | B1 | or equiv; allow if g, or f and g, chosen however expressed
Indicate domain is all real numbers | B1 | 3 howsoever expressed
**(iii)** Attempt use of chain rule | M1 | whether applied to gf or fg; or equiv such as use of product rule on $(\ln x)(\ln x) + 8$
Obtain $\frac{2\ln x}{x}$ | A1 | or equiv
Obtain $6e^{-3}$ | A1 | 3 or exact equiv but not including $\ln 1$ remaining
**(iv)** Attempt evaluation using y attempts | M1 | with coeffs 1, 4 and 2 occurring at least once each; whether fg or gf any constant $k$
Obn $k(\ln(24 + 4\ln 12 + 2\ln 8 + 4\ln 12 + \ln 24)$ | A1 | Use $k = \frac{1}{3}$ and obtain 20.3 | A1 | 3 or greater accuracy (20.26...) but must round to 20.3
[Note that use of Simpson's rule between 0 and 4 with two strips, coeffs 1, 4, 1, followed by doubling of result is equiv; SC: Use of Simpson's rule between 0 and 4 with four strips followed by doubling of result - allow 3/3 - answer is 20.2 (20.2327...)]
---7 The function f is defined for $x > 0$ by $\mathrm { f } ( x ) = \ln x$ and the function g is defined for all real values of $x$ by $\mathrm { g } ( x ) = x ^ { 2 } + 8$.\\
(i) Find the exact, positive value of $x$ which satisfies the equation $\operatorname { fg } ( x ) = 8$.\\
(ii) State which one of f and g has an inverse and define that inverse function.\\
(iii) Find the exact value of the gradient of the curve $y = \operatorname { gf } ( x )$ at the point with $x$-coordinate $\mathrm { e } ^ { 3 }$.\\
(iv) Use Simpson's rule with four strips to find an approximate value of
$$\int _ { - 4 } ^ { 4 } \mathrm { fg } ( x ) \mathrm { d } x$$
giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C3 2011 Q7 [12]}}