SPS SPS SM Pure (SPS SM Pure) 2024 September

1. Express

$$f ( x ) = \frac { x ^ { 2 } + x - 5 } { ( x - 2 ) ( x - 1 ) ^ { 2 } }$$ in partial fractions. \section*{(Total for Question 1 is 3 marks)}

1. The curve \(C\) has equation

$$y = \frac { 5 x ^ { 3 } - 8 } { 2 x ^ { 2 } } \quad x > 0$$ Find an equation for the tangent to \(C\) at \(x = 2\) writing your answer in the form
\(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(5) \section*{(Total for Question 2 is 5 marks)}

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { x + 3 } { x - 4 } \quad x \in \mathbb { R } , x \neq 4$$

1. Find ff (6)
   (2)
2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain
   (3) \section*{(Total for Question 3 is 5 marks)}

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. Solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ (Total for Question 4 is 4 marks)

5. Find the coefficient of the term in \(x ^ { 7 }\) of the binomial expansion of $$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.
(3) \section*{(Total for Question 5 is 3 marks)}

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\)
Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\)
Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)
(3)

7. The circle \(C\)

• has centre \(A ( 3,5 )\)
• passes through the point \(B ( 8 , - 7 )\)

The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
Given that \(M N\)

• lies above the \(x\)-axis
• is parallel to the \(x\)-axis
• has length \(4 \sqrt { 22 }\)

Find an equation for the line passing through points \(M\) and \(N\).
(5)
(Total for Question 7 is 5 marks)

8. (a) Sketch the curve with equation $$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\)
On your sketch show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of any asymptotes to the curve.
  (3)
  (b) Use the trapezium rule with 5 trapeziums to find an approximate value for

$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(3)
(c) Using the answer to part (b), find an approximate value for

1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)

9. The sum to infinity of the geometric series $$a + a r + a r ^ { 2 } + \ldots$$ is 10 .
The sum to infinity of the series formed by the squares of the terms is 100/9.
a) Show that \(r = 4 / 5\) and find \(a\).
b) Find the sum to infinity of the series formed by the cubes of the terms. \section*{(Total for Question 9 is 5 marks)}

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
2. Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ \section*{(Total for Question 10 is 6 marks)}

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-24_641_732_778_680} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
   The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
   Given that the \(y\) coordinate of \(P\) is 7
   (a) state the value of \(A\),
   (b) find the exact coordinates of \(Q\),
   (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.

12. \begin{figure}[h] \end{figure} The diagram shows a quadrilateral \(O A C B\) in which $$\overrightarrow { O A } = 4 \mathbf { a } \quad \overrightarrow { O B } = 3 \mathbf { b } \quad \overrightarrow { B C } = 2 \mathbf { a } + \mathbf { b }$$ The point \(P\) lies on \(A C\) such that \(A P : P C = 3 : 2\)
The point \(Q\) is such that \(O P Q\) and \(B C Q\) are straight lines.
Using a vector method, find \(\overrightarrow { O Q }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\)
Give your answer in its simplest form.
Show your working clearly.

13. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.
The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\) Use algebraic integration to find the exact area of \(R\).
(8) \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET }