| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Area under curve using substitution |
| Difficulty | Standard +0.3 This is a straightforward integration by substitution question with a linear substitution (u = 3 - 2x), which is one of the most basic types. The substitution is given explicitly, and finding area under a curve using a provided substitution is a standard P3 exercise requiring routine application of technique rather than problem-solving insight. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule correctly to obtain \(p(x+5)(3-2x)^n + q(3-2x)^{\frac{1}{2}}\) | *M1 | Allow with incorrect chain rule. BOD over sign errors unless incorrect rule quoted |
| Obtain correct derivative in any form | A1 | e.g. \(-(x+5)(3-2x)^{-\frac{1}{2}} + (3-2x)^{\frac{1}{2}}\) |
| Equate derivative to zero and obtain a linear equation | DM1 | Allow with surd factor e.g. \((3-2x)^{-\frac{1}{2}}(-(x+5)+(3-2x))=0\) |
| Obtain a correct linear equation | A1 | e.g. \(-(x+5)+3-2x=0\) |
| Obtain answer \(\left(-\frac{2}{3}, \frac{13\sqrt{39}}{9}\right)\) | A1 | Or exact equivalent e.g. \(\left(-\frac{2}{3}, \frac{13\sqrt{13}}{3\sqrt{3}}\right)\) or \(\left(-\frac{2}{3}, \frac{\sqrt{2197}}{\sqrt{27}}\right)\). Accept with \(x\), \(y\) stated separately. ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(y^2\) and differentiate | *M1 | Ignore their LHS i.e. their \(\frac{d}{dx}y^2\) |
| Obtain correct derivative in any form | A1 | e.g. \(-6x^2 - 34x - 20\) |
| Equate derivative to zero and solve for \(x\) | DM1 | |
| Obtain \(-\frac{2}{3}\) | A1 | Ignore \(-5\) if seen |
| Obtain answer \(\left(-\frac{2}{3}, \frac{13\sqrt{39}}{9}\right)\) only | A1 | Or exact equivalent e.g. \(\left(-\frac{2}{3}, \frac{13\sqrt{13}}{3\sqrt{3}}\right)\) or \(\left(-\frac{2}{3}, \frac{\sqrt{2197}}{\sqrt{27}}\right)\). ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use substitution and reach \(a\int\left(\frac{13}{2} - \frac{u}{2}\right)u^{\frac{1}{2}}\,du\) | *M1 | OE. Need to see \(-2\) or \(-\frac{1}{2}\) used. Condone if \(du\) missing or integral sign missing. Allow M1A0 for complete substitution into \(\int x\sqrt{3-2x}\,dx\) to obtain first term of line below |
| Obtain correct integral \(-\frac{1}{2}\int\left(\frac{13}{2} - \frac{u}{2}\right)u^{\frac{1}{2}}\,du\) | A1 | OE e.g. \(-\frac{1}{2}\left[\int\frac{3-u}{2}\sqrt{u}\,du + 5\int\sqrt{u}\,du\right]\). Ignore limits at this stage. Condone if \(du\) missing |
| \(x = -5\) and \(\frac{3}{2}\) | B1 | SOI e.g. by \(u=13\) and \(0\). In any order and at any stage |
| Use correct limits the right way round in integral of form \(a\left(\frac{26}{3}u^{\frac{3}{2}} - \frac{2}{5}u^{\frac{5}{2}}\right)\) | DM1 | |
| Obtain answer \(\frac{169}{15}\sqrt{13}\) or \(a = \frac{169}{15}\) | A1 | Or exact equivalents |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule correctly to obtain $p(x+5)(3-2x)^n + q(3-2x)^{\frac{1}{2}}$ | *M1 | Allow with incorrect chain rule. BOD over sign errors unless incorrect rule quoted |
| Obtain correct derivative in any form | A1 | e.g. $-(x+5)(3-2x)^{-\frac{1}{2}} + (3-2x)^{\frac{1}{2}}$ |
| Equate derivative to zero and obtain a linear equation | DM1 | Allow with surd factor e.g. $(3-2x)^{-\frac{1}{2}}(-(x+5)+(3-2x))=0$ |
| Obtain a correct linear equation | A1 | e.g. $-(x+5)+3-2x=0$ |
| Obtain answer $\left(-\frac{2}{3}, \frac{13\sqrt{39}}{9}\right)$ | A1 | Or exact equivalent e.g. $\left(-\frac{2}{3}, \frac{13\sqrt{13}}{3\sqrt{3}}\right)$ or $\left(-\frac{2}{3}, \frac{\sqrt{2197}}{\sqrt{27}}\right)$. Accept with $x$, $y$ stated separately. ISW |
**Alternative Method for 10(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $y^2$ and differentiate | *M1 | Ignore their LHS i.e. their $\frac{d}{dx}y^2$ |
| Obtain correct derivative in any form | A1 | e.g. $-6x^2 - 34x - 20$ |
| Equate derivative to zero and solve for $x$ | DM1 | |
| Obtain $-\frac{2}{3}$ | A1 | Ignore $-5$ if seen |
| Obtain answer $\left(-\frac{2}{3}, \frac{13\sqrt{39}}{9}\right)$ only | A1 | Or exact equivalent e.g. $\left(-\frac{2}{3}, \frac{13\sqrt{13}}{3\sqrt{3}}\right)$ or $\left(-\frac{2}{3}, \frac{\sqrt{2197}}{\sqrt{27}}\right)$. ISW |
---
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use substitution and reach $a\int\left(\frac{13}{2} - \frac{u}{2}\right)u^{\frac{1}{2}}\,du$ | *M1 | OE. Need to see $-2$ or $-\frac{1}{2}$ used. Condone if $du$ missing or integral sign missing. Allow M1A0 for complete substitution into $\int x\sqrt{3-2x}\,dx$ to obtain first term of line below |
| Obtain correct integral $-\frac{1}{2}\int\left(\frac{13}{2} - \frac{u}{2}\right)u^{\frac{1}{2}}\,du$ | A1 | OE e.g. $-\frac{1}{2}\left[\int\frac{3-u}{2}\sqrt{u}\,du + 5\int\sqrt{u}\,du\right]$. Ignore limits at this stage. Condone if $du$ missing |
| $x = -5$ and $\frac{3}{2}$ | B1 | SOI e.g. by $u=13$ and $0$. In any order and at any stage |
| Use correct limits the right way round in integral of form $a\left(\frac{26}{3}u^{\frac{3}{2}} - \frac{2}{5}u^{\frac{5}{2}}\right)$ | DM1 | |
| Obtain answer $\frac{169}{15}\sqrt{13}$ or $a = \frac{169}{15}$ | A1 | Or exact equivalents |
---\begin{enumerate}[label=(\alph*)]
\item Find the exact coordinates of $M$.
\item Using the substitution $u = 3 - 2 x$, find by integration the area of the shaded region bounded by the curve and the $x$-axis. Give your answer in the form $a \sqrt { 13 }$, where $a$ is a rational number. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q10 [10]}}