| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find stationary points from derivative |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring standard power rule integration, finding a constant using a given point, then solving dy/dx = 0 for stationary points. All techniques are routine P1/C1 material with no problem-solving insight needed, making it easier than average but not trivial due to fractional powers. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([y=]\left\{\frac{3x^{\frac{3}{2}}}{\frac{3}{2}}\right\}+\left\{-\frac{3x^{\frac{1}{2}}}{\frac{1}{2}}\right\}[+c]=2x^{\frac{3}{2}}-6x^{\frac{1}{2}}\) | B1 B1 | Marks for correct unsimplified expressions, 1 mark each for contents of \(\{\}\). ISW |
| \(5=2\times3^{\frac{3}{2}}-6\times3^{\frac{1}{2}}+c\) | M1 | Correct use of \((3,5)\) in integrated expression (at least one correct power) including \(+c\) |
| \(y=2x^{\frac{3}{2}}-6x^{\frac{1}{2}}+5\) | A1 | Condone \(c=5\) as final line if \(y=\) or \(f(x)=\) seen elsewhere; coefficients must not contain unresolved double fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3x^{\frac{1}{2}}-3x^{-\frac{1}{2}}=0\) | M1 | Setting given differential to 0 |
| \([x=]\ 1\) | A1 | CAO WWW. Condone extra solution of \(-1\) only if it is rejected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x>1\) or \(x>\) "their 8(b)" | B1FT | Allow \(\geq\) |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[y=]\left\{\frac{3x^{\frac{3}{2}}}{\frac{3}{2}}\right\}+\left\{-\frac{3x^{\frac{1}{2}}}{\frac{1}{2}}\right\}[+c]=2x^{\frac{3}{2}}-6x^{\frac{1}{2}}$ | B1 B1 | Marks for correct unsimplified expressions, 1 mark each for contents of $\{\}$. ISW |
| $5=2\times3^{\frac{3}{2}}-6\times3^{\frac{1}{2}}+c$ | M1 | Correct use of $(3,5)$ in integrated expression (at least one correct power) including $+c$ |
| $y=2x^{\frac{3}{2}}-6x^{\frac{1}{2}}+5$ | A1 | Condone $c=5$ as final line if $y=$ or $f(x)=$ seen elsewhere; coefficients must not contain unresolved double fractions |
---
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $3x^{\frac{1}{2}}-3x^{-\frac{1}{2}}=0$ | M1 | Setting given differential to 0 |
| $[x=]\ 1$ | A1 | CAO WWW. Condone extra solution of $-1$ only if it is rejected |
---
## Question 8(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x>1$ or $x>$ "their 8(b)" | B1FT | Allow $\geq$ |
---8 The equation of a curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }$. The curve passes through the point $( 3,5 )$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the curve.
\item Find the $x$-coordinate of the stationary point.
\item State the set of values of $x$ for which $y$ increases as $x$ increases.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q8 [7]}}