1. Apply a vertical stretch by a factor of 2. This transformation changes the graph of \(y = f(x)\) to \(y = 2f(x)\), effectively doubling the y-values of all points on the graph.

2. Translate the graph vertically by \(\begin{pmatrix} 0 \\ -14 \end{pmatrix}\). This moves the graph down by 14 units, resulting in the graph of \(y = g(x)\).

(a)(i) For an arithmetic progression, the difference between consecutive terms is constant. Thus, \(k^2 - 4k = 8k - k^2\). Solving gives:

\(2k^2 = 12k\)

\(k(k - 6) = 0\)

Since \(k\) is non-zero, \(k = 6\).

(a)(ii) The first term \(a = 4k = 24\) and the common difference \(d = k^2 - 4k = 12\). The sum of the first 20 terms is:

\(S_{20} = \frac{20}{2} (2 \times 24 + 19 \times 12) = 2760\)

(b) Let the first term be \(a\) and the common ratio be \(r\). Then \(ar^3 = 36\) and \(ar^5 = 6\). Dividing gives:

\(r^2 = \frac{1}{6}\)

\(r = \frac{\sqrt{6}}{6}\)

\(a = \frac{36}{r^3} = \frac{1296}{\sqrt{6}}\)

The sum to infinity is:

\(S_\infty = \frac{a}{1 - r} = \frac{1296}{\sqrt{6} - 1}\)

(a) To express \(x^2 + 4x + 2\) in the form \((x+a)^2 + b\), complete the square:

\(x^2 + 4x + 2 = (x+2)^2 - 4 + 2 = (x+2)^2 - 2\).

Thus, \(a = 2\) and \(b = -2\).

(b)(i) To find \(f^{-1}(x)\), start with \(y = (x+2)^2 - 2\).

Rearrange to solve for \(x\):

\(y + 2 = (x+2)^2\)

\(x = \pm \sqrt{y+2} - 2\)

Since \(x \leq -2\), choose the negative root:

\(f^{-1}(x) = -\sqrt{x+2} - 2\).

(b)(ii) To find \((gf)^{-1}(x)\), first find \(gf(x)\):

\(gf(x) = g(f(x)) = g((x+2)^2 - 2) = -(x+2)^2 - 2 - 4 = -(x+2)^2 - 6\).

Now, solve \(y = -(x+2)^2 - 6\) for \(x\):

\(y + 6 = -(x+2)^2\)

\(-(x+2)^2 = y + 6\)

\(x+2 = \pm \sqrt{-y-6}\)

Since \(x \leq -2\), choose the negative root:

\((gf)^{-1}(x) = -\sqrt{-x-2} - 2\).

First, solve the linear equation \(2x + y + 4 = 0\) for \(y\):

\(y = -2x - 4\).

Substitute \(y = -2x - 4\) into the curve equation \(2xy + 5y^2 = 24\):

\(2x(-2x - 4) + 5(-2x - 4)^2 = 24\).

Simplify and expand:

\(-4x^2 - 8x + 5(4x^2 + 16x + 16) = 24\).

\(-4x^2 - 8x + 20x^2 + 80x + 80 = 24\).

Combine like terms:

\(16x^2 + 72x + 56 = 0\).

Divide the entire equation by 8:

\(2x^2 + 9x + 7 = 0\).

Factor the quadratic equation:

\((2x + 7)(x + 1) = 0\).

Set each factor to zero and solve for \(x\):

\(2x + 7 = 0 \Rightarrow x = -\frac{7}{2}\).

\(x + 1 = 0 \Rightarrow x = -1\).

Substitute \(x = -\frac{7}{2}\) into \(y = -2x - 4\):

\(y = -2\left(-\frac{7}{2}\right) - 4 = 3\).

Substitute \(x = -1\) into \(y = -2x - 4\):

\(y = -2(-1) - 4 = -2\).

Thus, the points of intersection are \((-1, -2)\) and \(\left(-\frac{7}{2}, 3\right)\).

To find the coefficient of \(x^7\) in the expansion of \(\left( px^2 + \frac{4}{p}x \right)^5\), we use the binomial theorem. The general term in the expansion is given by:

\(\binom{5}{r} (px^2)^{5-r} \left( \frac{4}{p}x \right)^r\)

We need the power of \(x\) to be 7, so:

\(2(5-r) + r = 7\)

\(10 - 2r + r = 7\)

\(10 - r = 7\)

\(r = 3\)

Substitute \(r = 3\) into the general term:

\(\binom{5}{3} (px^2)^{5-3} \left( \frac{4}{p}x \right)^3\)

\(= \binom{5}{3} (px^2)^2 \left( \frac{4}{p}x \right)^3\)

\(= \binom{5}{3} p^2 x^4 \cdot \frac{64}{p^3} x^3\)

\(= \binom{5}{3} \cdot \frac{64p^2}{p^3} x^7\)

\(= 10 \cdot \frac{64}{p} x^7\)

The coefficient of \(x^7\) is \(\frac{640}{p}\), and we know this equals 1280:

\(\frac{640}{p} = 1280\)

\(640 = 1280p\)

\(p = \frac{640}{1280} = \frac{1}{2}\)