| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Topic | Integration by Substitution |
| Type | Volume of revolution with substitution |
| Difficulty | Challenging +1.2 Part (i) is a guided substitution integral with algebraic manipulation leading to a logarithmic result - standard A-level fare but requires careful execution. Part (ii) applies the result to find a volume of revolution, requiring students to set up the integral correctly (finding where y=4 intersects the curve) and use part (i). The multi-step nature and combination of techniques elevates this slightly above average difficulty, but it remains a structured question with clear guidance. |
| Spec | 1.08h Integration by substitution4.08e Mean value of function: using integral |
**Question 10(i) and 10(ii)**
**(i)**
- $\int -\frac{1}{2}\left(\frac{6-2u}{u}\right)^2 du$ **M1** Attempt integrand in terms of $u$. M0 if just $du = dx$
- $\int\left(-\frac{18}{u^2} + \frac{12}{u} - 2\right)du$ **M1** Attempt to simplify to three terms
- **A1** Obtain correct three term expression, with fractions simplified. Allow $\frac{12u}{u^2}$ if subsequently integrated as $k\ln u^2$
- $\frac{18}{u} + 12\ln u - 2u$ **M1** Integrate expression of form $au^{-2} + bu^{-1} + c$. Allow errors in coefficients only
- **A1** Correct integral in terms of $u$
- $(18 + 12\ln 1 - 2) - (6 + 12\ln 3 - 6)$ **M1** Dep on first M mark. Attempt $F(1) - F(3)$, must be correct order, or use $F(x=1) - F(x=0)$. Detail of use of limits needed as **A.G.**
- $= 16 - 12\ln 3$ **A.G.** **A1** Obtain $16 - 12\ln 3$ **A.G.**
**(ii)**
- $\pi \times 4^2 \times 1$ **M1** Attempt vol of cylinder
- $16\pi - \pi(16-12\ln 3)$ **M1** Dep on first M mark. Attempt vol of cylinder $- \pi(16-12\ln 3)$
- $= 12\pi\ln 3$ **A1** Obtain $12\pi\ln 3$10 (i) By using the substitution $u = 3 - 2 x$, or otherwise, show that $\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3$.\\
(ii)\\
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The diagram shows the region $R$, which is bounded by the curve $y = \frac { 4 x } { 3 - 2 x }$, the $y$-axis and the line $y = 4$. Find the exact volume generated when the region $R$ is rotated completely around the $x$-axis.
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q10 [10]}}