SPS SPS SM Pure (SPS SM Pure) 2024 June

1. The functions \(f\) and \(g\) are defined by

$$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geq 0
\mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geq 0 \end{array}$$

1. Write down the range of f
2. Find the value of fg (1.5)
3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 } ( x )\)

2. Differentiate \(f ( x ) = a x ^ { 2 } + b x\) from first principles
(Total for Question 2 is 4 marks)

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$

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

Solutions based entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } - 4 x ^ { 2 } - 7 x - 2$$ a) Use the factor theorem to show that ( \(2 x + 1\) ) is a factor of \(f ( x )\).
b) Write \(\frac { 3 x + 4 } { f ( x ) }\) in partial fraction form.

5. In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\), length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Hence find the value of \(x\) for which \(S\) is stationary and justify that this value of \(x\) gives the minimum value of \(S\).
3. Hence find the minimum surface area of the brick.

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$f ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence, using set notation, find the set of values of \(x\) for which $$f ( - x ) \geq 6 \text { and } x < 0$$

1. (i) Using the laws of logarithms, solve

$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$ (ii) Given that $$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } ( y ) \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$ find \(y\) in terms of \(a\), giving your answer in simplest form.
(3)

8. ABCD is a parallelogram and ADM is a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{b063f4ea-372b-4193-b8fe-a9f8017d7349-16_497_1102_278_294} \section*{Diagram NOT accurately drawn} \(\overrightarrow { A B } = a \quad \overrightarrow { B C } = b \quad \overrightarrow { D M } = \frac { 1 } { 2 } b\)
K is the point on AB such that \(\mathrm { AK } : \mathrm { AB } = \lambda : 1\)
L is the point on CD such that \(\mathrm { CL } : \mathrm { CD } = \mu : 1\)
KLM is a straight line.
Give that \(\lambda : \mu = 1 : 2\) use a vector method to find the value of \(\lambda\) and the value of \(\mu\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of

• the curve with equation \(y = \tan x\)
• the straight line \(l\) with equation \(y = \pi x\)
  in the interval \(- \pi < x < \pi\)
  1. State the period of \(\tan x\)
     (1)
  2. Write down the number of roots of the equation
     1. \(\tan x = ( \pi + 2 ) x\) in the interval \(- \pi < x < \pi\)
        (1)
     2. \(\tan x = \pi x\) in the interval \(- 2 \pi < x < 2 \pi\)
        (1)
     3. \(\tan x = \pi x\) in the interval \(- 100 \pi < x < 100 \pi\)
        (1)

1. a) Use proof by contradiction to prove that there are no positive integers, \(x\) and \(y\), such that

$$x ^ { 2 } - y ^ { 2 } = 1$$ b) Prove, by counter-example, that the statement \section*{" if \(a\) is rational and \(b\) is irrational then \(\log _ { a } b\) is irrational"} is false.
c) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 "$$ \section*{(Total for Question 10 is 6 marks)}

1. In this question you must show detailed reasoning.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve for \(- \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology or numerical methods are not acceptable. A geometric sequence has first term \(a\) and common ratio \(r\), where \(r > 0\)
Given that

• the 3rd term is 20
• the 5th term is 12.8
  1. show that \(r = 0.8\)
  2. Hence find the value of \(a\).

Given that the sum of the first \(n\) terms of this sequence is greater than 156

find the smallest possible value of \(n\).

13. The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a circle, \(C _ { 2 }\) Given that

• \(p\) is a positive constant
• \(Q R\) is a diameter of \(C _ { 2 }\)
  find the exact value of \(p\).
  (4)
  (Total for Question 13 is 4 marks)

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = \left( 2 + \frac { k x } { 8 } \right) ^ { 7 } \quad \text { where } k \text { is a non-zero constant }$$

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(\mathrm { f } ( x )\). Give each term in simplest form. Given that, in the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 3 }\) are the first 3 terms of an arithmetic progression,
2. find, using algebra, the possible values of \(k\).
   (Solutions relying entirely on calculator technology are not acceptable.)

15. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).
A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below, with the \(y\) values rounded to 4 decimal places where appropriate.

1. Use the trapezium rule with all the values of \(y\) in the table to find an approximation for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$ giving your answer to 3 decimal places.
   (2) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_627_581_1142_404} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_524_442_1238_1183} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The region \(R\), shown shaded in Figure 1, is bounded by
   • the curve \(C _ { 1 }\)
   • the curve \(C _ { 2 }\) with equation \(y = 2 - \frac { 1 } { 4 } x ^ { 2 }\)
   • the line with equation \(x = 2\)
   • the \(y\)-axis
   The region \(R\) forms part of the design for a logo shown in Figure 2.
   The design consists of the shaded region \(R\) inside a rectangle of width 2 and height 3
   Using calculus and the answer to part (a),
2. calculate an estimate for the percentage of the logo which is shaded.

16. An area of sea floor is being monitored. The area of the sea floor, \(S \mathrm {~km} ^ { 2 }\), covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$

1. find, according to the model, the area of sea floor covered by coral reefs when \(t = 2\)
2. find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(q\)

17. \begin{figure}[h] \end{figure} \section*{In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geq 0$$

1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
   The line \(l\) is the tangent to \(C\) at the point \(P\).
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
2. Find, using calculus, the exact area of \(R\).
   (6) ADDITIONAL SHEET ADDITIONAL SHEET ADDITIONAL SHEET