a. Vertical asymptote of \( y = g(x) \):

Solution:

The function \( g(x) = 2\ln x – \ln d \) has a vertical asymptote where the argument of the logarithm approaches zero.

As \( x \to 0^+ \), \( \ln x \to -\infty \), creating a vertical asymptote at \( x = 0 \).

Graph of \( y = g(x) \) showing vertical asymptote at x=0

b(i). Points of intersection equation:

Solution:

Set \( f(x) = g(x) \):

\( \ln(2x – 9) = 2\ln x – \ln d \)

Exponentiate both sides:

\( 2x – 9 = \frac{x^2}{d} \)

Multiply through by \( d \):

\( x^2 – 2dx + 9d = 0 \)

b(ii). Show \( d^2 – 9d > 0 \):

Solution:

For real intersection points, the discriminant must be positive:

\( D = (2d)^2 – 4(1)(9d) = 4d^2 – 36d > 0 \)

Divide by 4:

\( d^2 – 9d > 0 \)

Factor:

\( d(d – 9) > 0 \)

b(iii). Range of possible \( d \) values:

Solution:

From \( d(d – 9) > 0 \), the solution is \( d < 0 \) or \( d > 9 \)

Since \( d > 0 \), the valid range is \( d > 9 \)

c. Value of \( q – p \) when \( d = 10 \):

Solution:

With \( d = 10 \), solve \( x^2 – 20x + 90 = 0 \):

\( x = \frac{20 \pm \sqrt{400 – 360}}{2} = 10 \pm \sqrt{10} \)

Let \( p = 10 – \sqrt{10} \), \( q = 10 + \sqrt{10} \)

Then:

\( q – p = 2\sqrt{10} \)