SPS SPS SM Pure (SPS SM Pure) 2023 September

1. The graph of \(y = \mathrm { f } ( x )\) is shown below for \(0 \leq x \leq 6\)
\includegraphics[max width=\textwidth, alt={}, center]{a1b449df-1096-4b3a-8306-fca410a7e530-04_499_551_331_877}

1. Evaluate \(\int _ { 0 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x\)
   [0pt] [2 marks]
2. Deduce values for each of the following, giving reasons for your answers.
3. 1. \(\int _ { 1 } ^ { 7 } \mathrm { f } ( x - 1 ) \mathrm { d } x\)
4. (ii) \(\int _ { 0 } ^ { 6 } ( \mathrm { f } ( x ) - 1 ) \mathrm { d } x\)

2.
\(f ( x ) = \frac { 1 - 2 x ^ { 9 } } { x ^ { 5 } } \quad\) for \(x > 0\) Prove that \(f ( x )\) is a decreasing function.
[0pt] [6 marks]

3.

1. Find the first three terms, in ascending powers of \(x\), of the expansion of $$\left( 3 - \frac { x } { 2 } \right) ^ { 8 }$$ [3 marks]
2. Use your expansion to estimate the value of \(2.995 ^ { 8 }\).
   [0pt] [2 marks]

4.

1. 1. Express as a single logarithm $$\log _ { a } 36 - \frac { 1 } { 2 } \log _ { a } 81 + 2 \log _ { a } 4 - 3 \log _ { a } 2$$
2. (ii) Hence find the value of \(a\), given $$\log _ { a } 36 - \frac { 1 } { 2 } \log _ { a } 81 + 2 \log _ { a } 4 - 3 \log _ { a } 2 = \frac { 3 } { 2 }$$

5. The curve with equation \(y = x ^ { 3 } - 7 x + 6\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{a1b449df-1096-4b3a-8306-fca410a7e530-10_428_627_342_810} The curve intersects the \(x\)-axis at the points \(A ( - 3,0 ) , B ( 1,0 )\) and \(C\).

1. Find the coordinates of \(C\).
   [0pt] [1 mark]
2. Find \(\int \left( x ^ { 3 } - 7 x + 6 \right) \mathrm { d } x\)
   [0pt] [2 marks]
3. Find the total area of the shaded regions enclosed by the curve and the \(x\)-axis.
   [0pt] [4 marks]

6. A curve has equation \(x ^ { 2 } + y ^ { 2 } + 12 x = 64\)
A line has equation \(y = m x + 10\)

1. 1. In the case that the line intersects the curve at two distinct points, show that $$( 20 m + 12 ) ^ { 2 } - 144 \left( m ^ { 2 } + 1 \right) > 0$$
2. (ii) Hence find the possible values of \(m\).
3. 1. On the same diagram, sketch the curve and the line in the case when \(m = 0\)
4. (ii) State the relationship between the curve and the line.

7.
\(( x - 3 )\) is a common factor of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) where: $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 3 } - 11 x ^ { 2 } + ( p - 15 ) x + q
& \mathrm {~g} ( x ) = 2 x ^ { 3 } - 17 x ^ { 2 } + p x + 2 q \end{aligned}$$

1. 1. Show that \(3 p + q = 90\) and \(3 p + 2 q = 99\) Fully justify your answer.
2. (ii) Hence find the values of \(p\) and \(q\).
3. \(\quad \mathrm { h } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\) Using your values of \(p\) and \(q\), fully factorise \(\mathrm { h } ( x )\)

8. Martin tried to find all the solutions of \(4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
His working is shown below: $$\begin{aligned} & 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0
& \Rightarrow 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = \cos ^ { 2 } \theta
& \Rightarrow 4 \sin ^ { 2 } \theta = 1
& \Rightarrow \sin ^ { 2 } \theta = \frac { 1 } { 4 }
& \Rightarrow \sin \theta = \frac { 1 } { 2 }
& \Rightarrow \theta = 30 ^ { \circ } , 150 ^ { \circ } \end{aligned}$$ Martin did not find all the correct solutions because he made two errors.

1. Identify the two errors and explain the consequence of each error.
   [0pt] [4 marks]
2. Find all the solutions that Martin did not find.

9. Two models are proposed for the value of a car.

1. The first model suggests that the value of the car, \(V\) pounds, is given by \(V = 18000 - 6000 \sqrt { t }\), where \(t\) is the time in years after the car was first purchased.
2. 1. State the value of the car when it was first purchased.
3. (ii) Find \(V\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t }\) when \(t = 4\)
4. (iii) Interpret your answers to (a)(ii) in the context of the model.
5. The second model that is proposed suggests that the value of the car, \(V\) pounds, is given by \(V = a b ^ { - t }\), where \(t\) is the time in years after the car was first purchased. When \(t = 0\), both models give the same value for \(V\).
   When \(t = 4\), both models give the same value for \(V\). Find the value of \(a\) and the value of \(b\).
   [0pt] [3 marks]
6. Explain, with a reason, which model is likely to be the better model over time.

10. The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 ^ { x } , x \in \mathbb { R }
& \mathrm {~g} ( x ) = \sqrt { 1 - x } , x \in \mathbb { R } , x \leq a \end{aligned}$$

1. State the maximum possible value of \(a\).
2. The function h is defined by \(\mathrm { h } ( x ) = \mathrm { gf } ( x )\)
3. 1. Write down an expression for \(\mathrm { h } ( x )\)
4. (ii) Using set notation, state the greatest possible domain of h .
5. (iii) State the range of h .

11. A geometric sequence, \(S _ { 1 }\), has first term \(a\) and common ratio \(r\) where \(a \neq 0\) and \(r \in ( - 1,1 )\) A new sequence, \(S _ { 2 }\), is formed by squaring each term of \(S _ { 1 }\)

1. Given that the sum to infinity of \(S _ { 2 }\) is twice the sum to infinity of \(S _ { 1 }\), show that \(a = 2 ( 1 + r )\) Fully justify your answer.
2. Determine the set of possible values for \(a\). \section*{Additional Answer Space } \section*{Additional Answer Space }