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SPS SPS SM Pure 2024 June Q13
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)
SPS SPS SM Pure 2024 June Q14
  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.)
SPS SPS SM Pure 2024 June Q15
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.
\(x\)00.511.52
\(y\)32.68332.42.14661.92
  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.
SPS SPS SM Pure 2024 June Q16
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 \(q\)
SPS SPS SM Pure 2024 June Q17
17. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-34_803_1048_228_529} \captionsetup{labelformat=empty} \caption{Figure 3}
\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
SPS SPS FM Pure 2023 September Q1
1. $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 3
k & 1 \end{array} \right]$$
  1. Find \(\mathbf { A } ^ { - 1 }\)
  2. The determinant of \(\mathbf { A } ^ { 2 }\) is equal to 4 . Find the possible values of \(k\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q2
2. A sequence \(u _ { n }\) is defined by \(u _ { n + 1 } = 2 u _ { n } + 3\) and \(u _ { 1 } = 1\). Prove by induction that \(u _ { n } = 4 \times 2 ^ { n - 1 } - 3\) for all positive integers \(n\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q3
3. A finite region is bounded by the curve with equation \(y = x + x ^ { - \frac { 3 } { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi ( a \sqrt { 2 } + b )\), where \(a\) and \(b\) are rational numbers to be determined.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q4
6 marks
4. The curve \(C\) has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
    The line \(l\) is the normal to \(C\) at \(P\).
  2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.
    [0pt] [BLANK PAGE]
  4. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg ( z + \mathrm { i } ) = \frac { \pi } { 6 }$$ [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-12_1307_1351_516_463}
  5. \(\quad z _ { 1 }\) is a point on \(L\) such that \(| z |\) is a minimum. Find the exact value of \(z _ { 1 }\) in the form \(a + b \mathrm { i }\)
    [0pt] [4 marks]
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q6
8 marks
6. A curve has equation \(y = x \mathrm { e } ^ { \frac { x } { 2 } }\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection.
[0pt] [8 marks]
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q7
8 marks
7. (a) Prove the identity \(\frac { \cos x } { \sec x + 1 } + \frac { \cos x } { \sec x - 1 } \equiv 2 \cot ^ { 2 } x\)
[0pt] [3 marks]
(b) Hence, solve the equation $$\frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) + 1 } = \cot \left( 2 \theta + \frac { \pi } { 3 } \right) - \frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) - 1 }$$ in the interval \(0 \leq \theta \leq 2 \pi\), giving your values of \(\theta\) to three significant figures where appropriate.
[0pt] [5 marks]
[0pt] [BLANK PAGE] \section*{8. A population of meerkats is being studied.} The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q9
6 marks
9. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
  2. The equation \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer.
    2. Show that the roots of \(\mathrm { f } ( x ) = 0\) satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
    3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to three decimal places.
      [0pt] [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e ^ { - \frac { x } { 2 } }\) and \(y = x\), and the position of \(x _ { 1 }\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      [0pt] [2 marks] \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-22_1236_1566_1519_360}
      \end{figure} [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q1
2 marks
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.
    1. \(\int _ { 1 } ^ { 7 } \mathrm { f } ( x - 1 ) \mathrm { d } x\)
  4. (ii) \(\int _ { 0 } ^ { 6 } ( \mathrm { f } ( x ) - 1 ) \mathrm { d } x\)
SPS SPS SM Pure 2023 September Q2
6 marks
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]
SPS SPS SM Pure 2023 September Q3
5 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]
SPS SPS SM Pure 2023 September Q4
4.
    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 $$\log _ { a } 36 - \frac { 1 } { 2 } \log _ { a } 81 + 2 \log _ { a } 4 - 3 \log _ { a } 2 = \frac { 3 } { 2 }$$
SPS SPS SM Pure 2023 September Q5
7 marks
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]
SPS SPS SM Pure 2023 September Q6
6. A curve has equation \(x ^ { 2 } + y ^ { 2 } + 12 x = 64\)
A line has equation \(y = m x + 10\)
    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 \(m\).
    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.
SPS SPS SM Pure 2023 September Q7
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. 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 \(\mathrm { h } ( x )\)
SPS SPS SM Pure 2023 September Q8
4 marks
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.
SPS SPS SM Pure 2023 September Q9
3 marks
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.
    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.
SPS SPS SM Pure 2023 September Q10
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 \(\mathrm { h } ( x ) = \mathrm { gf } ( x )\)
    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 .
SPS SPS SM Pure 2023 September Q11
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 }
SPS SPS FM Mechanics 2024 January Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-06_397_878_159_571} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
  1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
  2. Find the coefficient of restitution between the spheres.
    [0pt] [Question 3 Continued]
SPS SPS FM Mechanics 2024 January Q4
4. A uniform heavy lamina occupies the region shaded in Fig. 3. This region is formed by removing a square of side 1 unit from a square of side \(a\) units (where \(a > 1\) ). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-08_558_594_299_699} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Relative to the axes shown in Fig. 3, the centre of mass of the lamina is at \(( \bar { x } , \bar { y } )\).
  1. Show that \(\bar { x } = \bar { y } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).
    [0pt] [You may need to use the result \(\frac { a ^ { 3 } - 1 } { 2 \left( a ^ { 2 } - 1 \right) } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).]
  2. Show that the centre of mass of the lamina lies on its perimeter if \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
  3. With the value of \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\) the lamina is suspended from A and hangs in equilibrium. Find the angle that the line OA makes with the vertical.
    [0pt] [Question 4 Continued] \section*{5.}