SPS SPS FM (SPS FM) 2024 October

1. Solve the following simultaneous equations:

$$\begin{aligned} & y = 4 x ^ { 2 } + 2 x - 5
& y = | 4 x + 1 | \end{aligned}$$

1. The graph of \(y = f ( x )\) (where \(- 2 \leq x \leq 6\) ) has the following features:

• A local maximum at \(x = 0\).
• A local minimum at \(x = 2\).
• No other turning points.
• Three stationary points.

Sketch a possible graph of \(y = f ^ { \prime } ( x )\) on the axes provided.
You can ignore the scale for the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-04_1269_1354_826_447}
[0pt] [BLANK PAGE]

3. The function \(g ( x )\) is defined as follows: $$g ( x ) = e ^ { \sin \left( x ^ { \circ } \right) } , - 90 \leq x \leq 90$$

1. Find \(g ^ { - 1 } ( x )\), stating its domain.
2. Sketch \(y = g ^ { - 1 } ( x )\) on the axes provided below, being sure to label all key points.
   \includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-07_1524_1591_459_342}

4. The polynomial \(P ( x )\) is defined as follows:
\(P ( x ) \equiv x ^ { 8 } + 8 x ^ { 7 } + 28 x ^ { 6 } + 56 x ^ { 5 } + 70 x ^ { 4 } + 56 x ^ { 3 } + 28 x ^ { 2 } + 8 x , x \in \mathbb { R }\)
By first factorising \(P ( x )\) find all of its real roots.
[0pt] [BLANK PAGE]

5. While working on a logarithms problem on one of the whiteboards in the Maths corridor, Rehman confidently asserts the following:
"For all positive real numbers \(x\) and \(y\), we know that \(\log _ { 2 } x - \log _ { 2 } y \equiv \frac { \log _ { 2 } x } { \log _ { 2 } y }\)."
Soufiane, who happens to be passing, knows that this is wrong. He attempts to prove this by counterexample, picking two values of \(x\) and \(y\) off the top of his head and plugging both sides of the false identity into his calculator. To his dismay, both sides give exactly the same answer, and Rehman smugly walks off uncorrected. What is the range of possible values of \(x\) that Soufiane picked?
[0pt] [6]
[0pt] [BLANK PAGE]

6. Neil enjoys playing around with sequences in his spare time, and one day he decides to create a new one. He does this by taking a normal arithmetic sequence but repeatedly halving the common difference between the terms. For instance, if the difference between the first two terms is 16 , the difference between the third term and the second term will be 8 and the difference between the fourth term and the third term will be 4 (continuing in this fashion for as long as he likes).

1. If \(u _ { 1 } = a\) and \(u _ { 2 } = a + d\), find a closed form expression for \(u _ { n }\), where \(n \in \mathbb { N }\). "Closed form" means that you can't have a "..." or sigma notation in your final answer.
2. With \(u _ { n }\) defined as above, find a closed form expression for \(S _ { n } = \sum _ { k = 1 } ^ { n } u _ { k }\), where \(n \in \mathbb { N }\).
3. With \(u _ { n }\) and \(S _ { n }\) defined as above, find the values of \(a\) and \(d\) if \(u _ { 7 } = 23\) and \(S _ { 7 } = 41\).
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]