Moderate -0.8 This is a straightforward integration question requiring standard application of the power rule to two terms with fractional indices, followed by using a given point to find the constant of integration. It's slightly easier than average as it involves routine technique with no substitution complexity or problem-solving insight needed.
3 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 ( 4 x - 7 ) ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(\left( 4 , \frac { 5 } { 2 } \right)\).
Find the equation of the curve.
Condone \(c = 5\) as their final line if either \(y =\) or \(f(x) =\) seen elsewhere. Coefficients must not contain unresolved double fractions
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{(4x-7)^{\frac{3}{2}}}{\frac{3}{2}\times 4}\right] + \left[-\frac{4}{1}x^{\frac{1}{2}}\right] \Rightarrow \frac{1}{2}(4x-7)^{\frac{3}{2}} - 8x^{\frac{1}{2}}$ | **B1 B1** | Marks can be awarded for correct unsimplified expressions ISW |
| $\frac{5}{2} = \frac{1}{2}(9)^{\frac{3}{2}} - 8\times 4^{\frac{1}{2}} + c \quad [\Rightarrow c = 5]$ | **M1** | Using $(4, \frac{5}{2})$ in an integrated expression (defined by at least one correct power) including $+ c$ |
| $y = \frac{3}{6}(4x-7)^{\frac{3}{2}} - 8x^{\frac{1}{2}} + 5$ | **A1** | Condone $c = 5$ as their final line if either $y =$ or $f(x) =$ seen elsewhere. Coefficients must not contain unresolved double fractions |
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3 The equation of a curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 ( 4 x - 7 ) ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } }$. It is given that the curve passes through the point $\left( 4 , \frac { 5 } { 2 } \right)$.
Find the equation of the curve.\\
\hfill \mbox{\textit{CAIE P1 2022 Q3 [4]}}