| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Area under curve using integration |
| Difficulty | Moderate -0.3 This is a straightforward multi-part calculus question testing standard techniques: finding x-intercepts by solving an equation, differentiation for tangent and stationary points, and integration for area. All parts follow routine procedures with fractional powers, making it slightly easier than average but still requiring competent execution of multiple techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9\left(x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\right) = 0\) leading to \(9x^{-\frac{3}{2}}(x-4) = 0\) | M1 | OE. Set \(y\) to zero and attempt to solve |
| \(x = 4\) only | A1 | From use of a correct method |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{\mathrm{d}y}{\mathrm{d}x} = 9\left(-\frac{1}{2}x^{-\frac{3}{2}} + 6x^{-\frac{5}{2}}\right)\) | B2, 1, 0 | B2: all 3 terms correct: \(9\), \(-\frac{1}{2}x^{-\frac{3}{2}}\) and \(6x^{-\frac{5}{2}}\); B1: 2 of the 3 terms correct |
| At \(x=4\) gradient \(= 9\left(-\frac{1}{16}+\frac{6}{32}\right) = \frac{9}{8}\) | M1 | Using their \(x=4\) in their differentiated expression and attempt to find equation of the tangent |
| Equation is \(y = \frac{9}{8}(x-4)\) | A1 | or \(y = \frac{9x}{8} - \frac{9}{2}\) OE |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9x^{-\frac{5}{2}}\left(-\frac{1}{2}x + 6\right) = 0\) | M1 | Set their \(\frac{\mathrm{d}y}{\mathrm{d}x}\) to zero and attempt to solve |
| \(x = 12\) | A1 | Condone \((\pm)12\) from use of a correct method |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int 9\left(x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\right)\mathrm{d}x = 9\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}} - \frac{4x^{-\frac{1}{2}}}{-\frac{1}{2}}\right)\) | B2, 1, 0 | B2: all 3 terms correct: \(9\), \(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\), \(\frac{-4x^{-\frac{1}{2}}}{-\frac{1}{2}}\); B1: 2 of the 3 terms correct |
| \(9\left[\left(6 + \frac{8}{3}\right) - (4+4)\right]\) | M1 | Apply limits their \(4 \to 9\) to an integrated expression with no consideration of other areas |
| \(6\) | A1 | Use of \(\pi\) scores A0 |
| Total: 4 |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9\left(x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\right) = 0$ leading to $9x^{-\frac{3}{2}}(x-4) = 0$ | M1 | OE. Set $y$ to zero and attempt to solve |
| $x = 4$ **only** | A1 | From use of a correct method |
| **Total: 2** | | |
## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{\mathrm{d}y}{\mathrm{d}x} = 9\left(-\frac{1}{2}x^{-\frac{3}{2}} + 6x^{-\frac{5}{2}}\right)$ | B2, 1, 0 | B2: all 3 terms correct: $9$, $-\frac{1}{2}x^{-\frac{3}{2}}$ and $6x^{-\frac{5}{2}}$; B1: 2 of the 3 terms correct |
| At $x=4$ gradient $= 9\left(-\frac{1}{16}+\frac{6}{32}\right) = \frac{9}{8}$ | M1 | Using their $x=4$ in their differentiated expression and attempt to find equation of the tangent |
| Equation is $y = \frac{9}{8}(x-4)$ | A1 | or $y = \frac{9x}{8} - \frac{9}{2}$ OE |
| **Total: 4** | | |
## Question 11(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9x^{-\frac{5}{2}}\left(-\frac{1}{2}x + 6\right) = 0$ | M1 | Set their $\frac{\mathrm{d}y}{\mathrm{d}x}$ to zero and attempt to solve |
| $x = 12$ | A1 | Condone $(\pm)12$ from use of a correct method |
| **Total: 2** | | |
## Question 11(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int 9\left(x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\right)\mathrm{d}x = 9\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}} - \frac{4x^{-\frac{1}{2}}}{-\frac{1}{2}}\right)$ | B2, 1, 0 | B2: all 3 terms correct: $9$, $\frac{x^{\frac{1}{2}}}{\frac{1}{2}}$, $\frac{-4x^{-\frac{1}{2}}}{-\frac{1}{2}}$; B1: 2 of the 3 terms correct |
| $9\left[\left(6 + \frac{8}{3}\right) - (4+4)\right]$ | M1 | Apply limits their $4 \to 9$ to an integrated expression with no consideration of other areas |
| $6$ | A1 | Use of $\pi$ scores A0 |
| **Total: 4** | | |11\\
\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548}
The diagram shows the curve with equation $y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)$. The curve crosses the $x$-axis at the point $A$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of $A$.
\item Find the equation of the tangent to the curve at $A$.
\item Find the $x$-coordinate of the maximum point of the curve.
\item Find the area of the region bounded by the curve, the $x$-axis and the line $x = 9$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q11 [12]}}