Given that $$rac{6x+3x^{rac{5}{2}}}{ ext{√}x}$$ can be written in the form $$6x^{p} + 3x^{q}$$, (a) write down the value of p and the value of q - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 1

Given the expression from part (a), we have:

$\frac{dy}{dx} = 6x^{\frac{1}{2}} + 3x^{2}$

To find $y$, we need to integrate this expression with respect to $x$:

$y = \int (6x^{\frac{1}{2}} + 3x^{2})dx$

Integrating the first term:

$\int 6x^{\frac{1}{2}}dx = 6 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = 4x^{\frac{3}{2}}$

For the second term:

$\int 3x^{2}dx = 3 \cdot \frac{x^{3}}{3} = x^{3}$

Combining these gives:

$y = 4x^{\frac{3}{2}} + x^{3} + C$

Now, we can use the condition that $y = 90$ when $x = 4$ to find $C$:

Substituting:

$90 = 4(4)^{\frac{3}{2}} + (4)^{3} + C$

Calculating:

$4(\sqrt{64}) + 64 = 4 \cdot 8 + 64 = 32 + 64 = 96$

Thus:

$90 = 96 + C$

Which leads to:

$C = 90 - 96 = -6$

Finally, substituting back for $C$ in the expression for $y$ gives:

$y = 4x^{\frac{3}{2}} + x^{3} - 6$