7. (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\).
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
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Question 7:
Part (a)
Answer Marks
Guidance
Answer Mark
Guidance
\((3 - \sqrt{x})^2 = 9 - 6\sqrt{x} + x\) M1
Attempt to expand; must have 3 or 4 terms, allow one sign error
\(\div by\ \sqrt{x} \to 9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\) A1 c.s.o. (2)
Fully correct; penalise wrong working
Part (b)
Answer Marks
Guidance
Answer Mark
Guidance
\(\int\!\left(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\right)dx = \frac{9x^{\frac{1}{2}}}{\frac{1}{2}} - 6x + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} (+c)\) M1 A2/1/0
1st M1 some correct integration \(x^n \to x^{n+1}\); A1 at least 2 correct unsimplified terms; A2 all 3 terms correct (unsimplified); ignore \(+c\)
Use \(y = \frac{2}{3}\) and \(x = 1\): \(\frac{2}{3} = 18 - 6 + \frac{2}{3} + c\) M1
2nd M1: use of \(y = \frac{2}{3}\) and \(x = 1\) to find \(c\); No \(+c\) is M0
\(c = -12\) A1 c.s.o.
Award if "\(c = -12\)" stated, even not part of expression for \(y\)
\(y = 18x^{\frac{1}{2}} - 6x + \frac{2}{3}x^{\frac{3}{2}} - 12\) A1 f.t. (6)
f.t. for 3 simplified \(x\) terms with \(y = \ldots\) and numerical \(c\)
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## Question 7:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3 - \sqrt{x})^2 = 9 - 6\sqrt{x} + x$ | M1 | Attempt to expand; must have 3 or 4 terms, allow one sign error |
| $\div by\ \sqrt{x} \to 9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}$ | A1 c.s.o. (2) | Fully correct; penalise wrong working |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int\!\left(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\right)dx = \frac{9x^{\frac{1}{2}}}{\frac{1}{2}} - 6x + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} (+c)$ | M1 A2/1/0 | 1st M1 some correct integration $x^n \to x^{n+1}$; A1 at least 2 correct unsimplified terms; A2 all 3 terms correct (unsimplified); ignore $+c$ |
| Use $y = \frac{2}{3}$ and $x = 1$: $\frac{2}{3} = 18 - 6 + \frac{2}{3} + c$ | M1 | 2nd M1: use of $y = \frac{2}{3}$ and $x = 1$ to find $c$; No $+c$ is M0 |
| $c = -12$ | A1 c.s.o. | Award if "$c = -12$" stated, even not part of expression for $y$ |
| $y = 18x^{\frac{1}{2}} - 6x + \frac{2}{3}x^{\frac{3}{2}} - 12$ | A1 f.t. (6) | f.t. for 3 simplified $x$ terms with $y = \ldots$ and numerical $c$ |
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7. (a) Show that $\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }$ can be written as $9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }$.
Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0$, and that $y = \frac { 2 } { 3 }$ at $x = 1$,\\
(b) find $y$ in terms of $x$.\\
\hfill \mbox{\textit{Edexcel C1 2005 Q7 [8]}}