A curve $C$ has equation $y = px + \sqrt{qx} - \ln(rx)$, where $p$, $q$ and $r$ are constants. Given that $C$ crosses the $x$-axis at the points where $x = 1$, $x = 2$ and $x = 5$, find the values of $p$, $q$ and $r$, giving your answers correct to 3 decimal places.

1. Substitute the given $x$-intercepts into the equation to form three equations.

Since the curve crosses the $x$-axis at $x = 1, 2$ and $5$, we set $y = 0$ for each coordinate:
$p(1) + \sqrt{q(1)} - \ln(r \cdot 1) = 0$
$p(2) + \sqrt{q(2)} - \ln(r \cdot 2) = 0$
$p(5) + \sqrt{q(5)} - \ln(r \cdot 5) = 0$

2. Manipulate the equations using properties of surds and logarithms to form a system of linear equations.

Using the properties $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\ln(ab) = \ln a + \ln b$:
$p + \sqrt{q} - \ln r = 0 \quad \text{--- (1)}$
$2p + \sqrt{2}\sqrt{q} - (\ln 2 + \ln r) = 0 \implies 2p + \sqrt{2}\sqrt{q} - \ln r = \ln 2 \quad \text{--- (2)}$
$5p + \sqrt{5}\sqrt{q} - (\ln 5 + \ln r) = 0 \implies 5p + \sqrt{5}\sqrt{q} - \ln r = \ln 5 \quad \text{--- (3)}$
We now have a system of linear equations with variables $p$, $\sqrt{q}$, and $\ln r$.

3. Solve the linear system using a graphing calculator.

Entering the coefficients into the simultaneous equation solver on a graphing calculator:
$p \approx -0.451835$
$\sqrt{q} \approx 2.764232$
$\ln r \approx 2.312397$

4. Calculate the final values of $q$ and $r$, and round all answers to 3 decimal places.

$p \approx -0.452$
$q = (\sqrt{q})^2 \approx (2.764232)^2 \approx 7.64098 \approx 7.641$
$r = e^{\ln r} \approx e^{2.312397} \approx 10.0986 \approx 10.099$