Moderate -0.8 This is a straightforward integration question requiring only basic power rule application and finding a constant using given conditions. The algebra of simplifying x√x to x^(3/2) and integrating terms like x^(-1/2) is routine C1 material with no problem-solving element, making it easier than average.
8.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$
Given that \(y = 37\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.
A1: one term integrated correctly (need not be simplified) e.g. \(\frac{6}{\frac{1}{2}}x^{\frac{1}{2}}\) or \(+\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\). No need for \(+c\). A1: other term integrated correctly; need not simplify nor need \(+c\)
Use \(x=4\), \(y=37\): \(37 = 12\sqrt{4} + \frac{2}{5}(\sqrt{4})^5 + c\)
M1
Substitute \(x=4\), \(y=37\) to produce equation in \(c\)
\(\Rightarrow c = \frac{1}{5}\) or equivalent e.g. \(0.2\)
A1
Correctly calculates \(c=\frac{1}{5}\) or equivalent
cso; allow \(5y=60x^{\frac{1}{2}}+2x^{\frac{5}{2}}+1\) and fully simplified equivalents e.g. \(y=12\sqrt{x}+\frac{2}{5}\sqrt{x^5}+\frac{1}{5}\)
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x\sqrt{x} = x^{\frac{3}{2}}$ | B1 | May be implied by $+\frac{x^{\frac{5}{2}}}{\frac{5}{2}}$ in subsequent work |
| $x^n \rightarrow x^{n+1}$ | M1 | In at least one case; see either $x^{\frac{1}{2}}$ or $x^{\frac{5}{2}}$ or both |
| $y = \frac{6}{\frac{1}{2}}x^{\frac{1}{2}} + \frac{x^{\frac{5}{2}}}{\frac{5}{2}}\ (+c)$ | A1, A1 | A1: one term integrated correctly (need not be simplified) e.g. $\frac{6}{\frac{1}{2}}x^{\frac{1}{2}}$ or $+\frac{x^{\frac{5}{2}}}{\frac{5}{2}}$. No need for $+c$. A1: other term integrated correctly; need not simplify nor need $+c$ |
| Use $x=4$, $y=37$: $37 = 12\sqrt{4} + \frac{2}{5}(\sqrt{4})^5 + c$ | M1 | Substitute $x=4$, $y=37$ to produce equation in $c$ |
| $\Rightarrow c = \frac{1}{5}$ or equivalent e.g. $0.2$ | A1 | Correctly calculates $c=\frac{1}{5}$ or equivalent |
| $(y) = 12x^{\frac{1}{2}} + \frac{2}{5}x^{\frac{5}{2}} + \frac{1}{5}$ | A1 | cso; allow $5y=60x^{\frac{1}{2}}+2x^{\frac{5}{2}}+1$ and fully simplified equivalents e.g. $y=12\sqrt{x}+\frac{2}{5}\sqrt{x^5}+\frac{1}{5}$ |
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8.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$
Given that $y = 37$ at $x = 4$, find $y$ in terms of $x$, giving each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C1 2014 Q8 [7]}}