SPS SPS FM (SPS FM) 2022 October

1. a) Solve the inequality:

$$\frac { x - 9 } { 2012 } + \frac { x - 8 } { 2013 } + \frac { x - 7 } { 2014 } + \frac { x - 6 } { 2015 } + \frac { x - 5 } { 2016 } \leq \frac { x - 2012 } { 9 } + \frac { x - 2013 } { 8 } + \frac { x - 2014 } { 7 } + \frac { x - 2015 } { 6 } + \frac { x - 2016 } { 5 }$$ b) Find all ( \(x , y , z\) ) such that: $$\frac { 1 } { x } + \frac { 1 } { y + z } = \frac { 1 } { 3 } , \quad \frac { 1 } { y } + \frac { 1 } { z + x } = \frac { 1 } { 5 } , \quad \frac { 1 } { z } + \frac { 1 } { x + y } = \frac { 1 } { 7 }$$ [Question 1 - Continued]
[0pt] [Question 1 - Continued]

2. A function is defined by: $$f ( x ) = \sqrt { \frac { 1 - x } { 1 + x } } , x \in \mathbb { R } , | x | < 1$$ a) P and Q are points on the curve with \(x\)-coordinates \(x\) and \(x + h\) respectively. Find the gradient of the line segment PQ . Simplify your answer to a single fraction.
b) Use differentiation from first principles to show that: $$f ^ { \prime } ( x ) = - \frac { 1 } { ( 1 + x ) \sqrt { 1 - x ^ { 2 } } }$$ c) Sketch the curve on the axes provided over the page, showing clearly the behaviour of the curve near \(x = 0\) and \(x = \pm 1\).
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
\includegraphics[max width=\textwidth, alt={}, center]{5023b2d9-ed3d-4a4a-b0c6-f529550b2e3e-09_1731_1566_913_260}

3. If for some \(x , y \in \mathbb { R }\) we have \(| x + y | + | x - y | = 2\), find the maximal value of \(x ^ { 2 } - 6 x + y ^ { 2 }\).
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]

4. A sequence is defined by \(u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }\) for \(n \geq 1\).
a) In the case where \(r = \frac { 6 } { 5 }\) find the smallest value of \(n\) such that \(u _ { n } > 10 ^ { 50 }\). A convergent sequence is defined by \(v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }\) for \(n \geq 1\).
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of \(r\), giving your answer in exact form.
c) Evaluate the infinite product: $$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$ [Question 4 - Continued]
[0pt] [Question 4 - Continued]
[0pt] [Question 4 - Continued]

5. A cotangent function \(\cot x\) is defined as \(\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }\).
a) If \(- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }\) and \(\cot \alpha = - \frac { 12 } { 5 }\), find the exact value of \(\sin \alpha\) and \(\cos \alpha\).
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle.
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]

6. A function is defined by: $$f ( x ) = \frac { a x + b } { c x + d } , x \in \mathbb { R } , x \neq - \frac { d } { c }$$ a) Find and simplify an expression for \(f ^ { - 1 } ( x )\), stating the domain. A function is defined by: $$g ( x ) = \frac { x - 6 } { x - 4 } , x \in \mathbb { R } , x \neq 4$$ b) Find \(g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\), stating an appropriate domain for each function.
c) Find \(g ^ { - 1 } ( x ) , g ^ { - 2 } ( x )\) and \(g ^ { - 3 } ( x )\), stating an appropriate domain for each function. NB: \(g ^ { - n } ( x ) = g ^ { - 1 } \left( g ^ { - 1 } \left( \cdots \left( g ^ { - 1 } ( x ) \right) \cdots \right) \right)\) with \(n\) copies of \(g ^ { - 1 }\).
d) State the range of \(g ( x ) , g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\). A function is defined (over an appropriate domain) by \(h ( x ) = g ( x ) + g ^ { - 1 } ( x )\).
e) Solve the inequality \(h ( x ) \geq 4\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]