| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find stationary points from derivative |
| Difficulty | Moderate -0.3 This is a straightforward multi-part calculus question requiring standard integration (including x^(-2)), solving f'(x)=0 for stationary points, and using the second derivative test. All techniques are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(x) = \frac{2}{3}x^3 - 7x + 4x^{-1}\ [+c]\) | B2, 1, 0 | Allow terms on different lines; allow unsimplified. |
| \(-\frac{1}{3} = \frac{2}{3} - 7 + 4 + c\) leading to \(c = [2]\) | M1 | Substitute \(f(1) = -\frac{1}{3}\) into an integrated expression and evaluate \(c\). |
| \(f(x) = \frac{2}{3}x^3 - 7x + 4x^{-1} + 2\) | A1 | OE. |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2x^4 - 7x^2 - 4\ [=0]\) | M1 | Forms 3-term quadratic in \(x^2\) with all terms on one side. Accept use of substitution e.g. \(2y^2 - 7y - 4 [=0]\). |
| \((2x^2+1)(x^2-4)\ [=0]\) | M1 | Attempt factors or use formula or complete the square. Allow \(\pm\) sign errors. Factors must expand to give *their* coefficient of \(x^2\) or e.g. \(y\). Must be quartic equation. Accept use of substitution e.g. \((2y+1)(y-4)\). |
| \(x = [\pm]\ 2\) | A1 | If M0 for solving quadratic, SC B1 can be awarded for \([\pm]2\). |
| \(\left[\frac{2}{3}(2)^3 - 7(2) + \frac{4}{2} + 2\right]\) leading to \(\left(2,\ -\frac{14}{3}\right)\) | B1 B1 | B1 B1 for correct coordinates clearly paired; B1 for each correct point; B1 B0 if additional point. |
| \(\left[\frac{2}{3}(-2)^3 - 7(-2) + \frac{4}{-2} + 2\right]\) leading to \(\left(-2,\ \frac{26}{3}\right)\) | ||
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f''(x) = 4x + 8x^{-3}\) | B1 | OE |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f''(2) = 9 > 0\), MINIMUM at \(x = 2\) | B1 FT | FT on \(x = [\pm]2\) provided \(f''(x)\) is correct. Must have correct value of \(f''(x)\) if \(x = 2\) |
| \(f''(-2) = -9 < 0\), MAXIMUM at \(x = -2\) | B1 FT | FT on \(x = [\pm]2\) provided \(f''(x)\) is correct. Must have correct value of \(f''(x)\) if \(x = -2\). Special case: If values not shown and B0B0 scored, SC B1 for \(f''(2) > 0\) MIN and \(f''(-2) < 0\) MAX |
| Alternative: Evaluate \(f'(x)\) for \(x\)-values either side of \(2\) and \(-2\) | M1 | FT on \(x = [\pm]2\) |
| MINIMUM at \(x = 2\), MAXIMUM at \(x = -2\) | A1 FT | FT on \(x = [\pm]2\). Must have correct values of \(f'(x)\) if shown. Special case: If values not shown and M0A0 scored, SC B1 for \(f'(2)\ -/0/+\) MIN and \(f'(-2)\ +/0/-\) MAX |
| Alternative: Justify using correct sketch graph | B1 B1 | Need correct coordinates in (b) for this method |
| Total | 2 |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(x) = \frac{2}{3}x^3 - 7x + 4x^{-1}\ [+c]$ | B2, 1, 0 | Allow terms on different lines; allow unsimplified. |
| $-\frac{1}{3} = \frac{2}{3} - 7 + 4 + c$ leading to $c = [2]$ | M1 | Substitute $f(1) = -\frac{1}{3}$ into an integrated expression and evaluate $c$. |
| $f(x) = \frac{2}{3}x^3 - 7x + 4x^{-1} + 2$ | A1 | OE. |
| **Total: 4** | | |
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x^4 - 7x^2 - 4\ [=0]$ | M1 | Forms 3-term quadratic in $x^2$ with all terms on one side. Accept use of substitution e.g. $2y^2 - 7y - 4 [=0]$. |
| $(2x^2+1)(x^2-4)\ [=0]$ | M1 | Attempt factors or use formula or complete the square. Allow $\pm$ sign errors. Factors must expand to give *their* coefficient of $x^2$ or e.g. $y$. Must be quartic equation. Accept use of substitution e.g. $(2y+1)(y-4)$. |
| $x = [\pm]\ 2$ | A1 | If M0 for solving quadratic, **SC B1** can be awarded for $[\pm]2$. |
| $\left[\frac{2}{3}(2)^3 - 7(2) + \frac{4}{2} + 2\right]$ leading to $\left(2,\ -\frac{14}{3}\right)$ | B1 B1 | B1 B1 for correct coordinates clearly paired; B1 for each correct point; B1 B0 if additional point. |
| $\left[\frac{2}{3}(-2)^3 - 7(-2) + \frac{4}{-2} + 2\right]$ leading to $\left(-2,\ \frac{26}{3}\right)$ | | |
| **Total: 5** | | |
## Question 9(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f''(x) = 4x + 8x^{-3}$ | B1 | OE |
| **Total: 1** | | |
## Question 9(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f''(2) = 9 > 0$, MINIMUM at $x = 2$ | B1 FT | FT on $x = [\pm]2$ provided $f''(x)$ is correct. Must have correct value of $f''(x)$ if $x = 2$ |
| $f''(-2) = -9 < 0$, MAXIMUM at $x = -2$ | B1 FT | FT on $x = [\pm]2$ provided $f''(x)$ is correct. Must have correct value of $f''(x)$ if $x = -2$. **Special case:** If values not shown and B0B0 scored, SC B1 for $f''(2) > 0$ MIN and $f''(-2) < 0$ MAX |
| **Alternative:** Evaluate $f'(x)$ for $x$-values either side of $2$ and $-2$ | M1 | FT on $x = [\pm]2$ |
| MINIMUM at $x = 2$, MAXIMUM at $x = -2$ | A1 FT | FT on $x = [\pm]2$. Must have correct values of $f'(x)$ if shown. **Special case:** If values not shown and M0A0 scored, SC B1 for $f'(2)\ -/0/+$ MIN and $f'(-2)\ +/0/-$ MAX |
| **Alternative:** Justify using correct sketch graph | B1 B1 | Need correct coordinates in (b) for this method |
| **Total** | **2** | |
---9 A curve has equation $y = \mathrm { f } ( x )$, and it is given that $\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }$.
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }$, find $\mathrm { f } ( x )$.
\item Find the coordinates of the stationary points on the curve.
\item Find $\mathrm { f } ^ { \prime \prime } ( x )$.
\item Hence, or otherwise, determine the nature of each of the stationary points.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q9 [12]}}