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(a) Show that \( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} \) can be written as \( 9x^{1/2} - 6x + x^{1/3} \) - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 1
Question 8
(a) Show that \( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} \) can be written as \( 9x^{1/2} - 6x + x^{1/3} \). (b) Given that \( \frac{dy}{dx} = \frac{(3 - \sqrt{x})^3}{\sq... show full transcript
Worked Solution & Example Answer:(a) Show that \( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} \) can be written as \( 9x^{1/2} - 6x + x^{1/3} \) - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 1
Step 1
Show that \( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} \) can be written as \( 9x^{1/2} - 6x + x^{1/3} \)
Answer
To show the validity of the statement, we begin by expanding the numerator ( (3 - \sqrt{x})^3 ).
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Expand the cubic expression:
[ (3 - \sqrt{x})^3 = 3^3 - 3 \cdot 3^2 \cdot \sqrt{x} + 3 \cdot 3 \cdot x - \sqrt{x}^3 = 27 - 27\sqrt{x} + 9x - x^{3/2} ]
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Divide by ( \sqrt{x} ):
[ \frac{(3 - \sqrt{x})^3}{\sqrt{x}} = \frac{27 - 27\sqrt{x} + 9x - x^{3/2}}{\sqrt{x}} = \frac{27}{\sqrt{x}} - 27 + 9\sqrt{x} - x ]
This simplifies to:
[ 9x^{1/2} - 6x + x^{1/3} ]
Therefore, we have shown that ( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} = 9x^{1/2} - 6x + x^{1/3} ).
Step 2
find \( y \) in terms of \( x \)
Answer
We are given ( \frac{dy}{dx} = \frac{(3 - \sqrt{x})^3}{\sqrt{x}} ). To find ( y ) in terms of ( x ), we will integrate the right side.
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Rewrite the differential equation:
[ dy = \frac{(3 - \sqrt{x})^3}{\sqrt{x}} dx ]
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Integrate both sides:
[ y = \int \frac{(3 - \sqrt{x})^3}{\sqrt{x}} dx + C ]
We already derived ( \frac{(3 - \sqrt{x})^3}{\sqrt{x}} ) in part (a). Now substituting it we get:
[ y = \int (9\sqrt{x} - 6x + x^{1/3}) dx ]
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Integrate term by term:
- ( \int 9\sqrt{x} dx = 9 \cdot \frac{2}{3} x^{3/2} = 6 x^{3/2} )
- ( \int -6x dx = -3x^2 )
- ( \int x^{1/3} dx = \frac{3}{4} x^{4/3} )
Thus:
[ y = 6 x^{3/2} - 3x^2 + \frac{3}{4} x^{4/3} + C ]
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Use the initial condition: Given that ( y = \frac{1}{3} ) when ( x = 1 ), we substitute these values to find ( C ):
[ \frac{1}{3} = 6(1)^{3/2} - 3(1)^{2} + \frac{3}{4}(1)^{4/3} + C ]
Simplifying gives:
[ \frac{1}{3} = 6 - 3 + \frac{3}{4} + C ]
[ \frac{1}{3} = 3 + \frac{3}{4} + C ]
Collect terms on the right:
[ \frac{1}{3} - \frac{3}{4} = C \implies C = -\frac{5}{12} ]
So, the final expression for ( y ) is:
[ y = 6 x^{3/2} - 3x^2 + \frac{3}{4} x^{4/3} - \frac{5}{12} ]