Moderate -0.3 This is a straightforward integration problem requiring a simple substitution (u = 3x + 2) followed by applying a boundary condition to find the constant. The substitution is standard and the algebra is routine, making it slightly easier than average but still requiring proper technique.
4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 3 x + 2 ) ^ { 2 } }\). The curve passes through the point \(\left( 2,5 \frac { 2 } { 3 } \right)\).
Find the equation of the curve.
Substituting \(\left(2,\,5\frac{2}{3}\right)\) into *their* integrated expression — defined by power \(=-1\), or dividing by their power. \(+c\) needed
\(y=-\dfrac{8}{3(3x+2)}+6\)
A1
OE e.g. \(y=-\frac{8}{3}(3x+2)^{-1}+6\)
Total
4
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y=-\dfrac{\frac{8}{3}}{(3x+2)}[+c]$ | *B1 | For $(3x+2)^{-1}$ |
| | DB1 | For $-\frac{8}{3}$ |
| $5\dfrac{2}{3}=-\dfrac{\frac{8}{3}}{(3\times2+2)}+c$ | M1 | Substituting $\left(2,\,5\frac{2}{3}\right)$ into *their* integrated expression — defined by power $=-1$, or dividing by their power. $+c$ needed |
| $y=-\dfrac{8}{3(3x+2)}+6$ | A1 | OE e.g. $y=-\frac{8}{3}(3x+2)^{-1}+6$ |
| **Total** | **4** | |
4 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 3 x + 2 ) ^ { 2 } }$. The curve passes through the point $\left( 2,5 \frac { 2 } { 3 } \right)$.\\
Find the equation of the curve.\\
\hfill \mbox{\textit{CAIE P1 2021 Q4 [4]}}