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SPS SPS SM Pure 2023 June Q7
7. A ball is released from rest from a height of 5 m and bounces repeatedly on horizontal ground. After hitting the ground for the first time, the ball rises to a maximum height of 3 m . In a model for the motion of the ball
  • the maximum height after each bounce is \(60 \%\) of the previous maximum height
  • the motion takes place in a vertical line
    1. Using the model
      1. show that the maximum height after the 3rd bounce is 1.08 m ,
      2. find the total distance the ball travels from release to when the ball hits the ground for the 5th time.
According to the model, after the ball is released, there is a limit, \(D\) metres, to the total distance the ball will travel.
  • Find the value of \(D\) With reference to the model,
  • give a reason why, in reality, the ball will not travel \(D\) metres in total.
    •
    SPS SPS SM Pure 2023 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-16_801_1049_255_555} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A circle with centre \(( 9 , - 6 )\) touches the \(x\)-axis as shown in Figure 4.
    1. Write down an equation for the circle. A line \(l\) is parallel to the \(x\)-axis.
      The line \(l\) cuts the circle at points \(P\) and \(Q\).
      Given that the distance \(P Q\) is 8
    2. find the two possible equations for \(l\).
    SPS SPS SM Pure 2023 June Q9
    9. A curve has equation $$y = 4 x ^ { 2 } - 5 x$$ The curve passes through the point \(P ( 2,6 )\)
    Use differentiation from first principles to find the value of the gradient of the curve at \(P\).
    (4)
    SPS SPS SM Pure 2023 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-20_424_1241_246_299} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-20_385_1205_792_379} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} A suspension bridge cable \(P Q R\) hangs between the tops of two vertical towers, \(A P\) and \(B R\), as shown in Figure 5. A walkway \(A O B\) runs between the bases of the towers, directly under the cable.
    The towers are 100 m apart and each tower is 24 m high.
    At the point \(O\), midway between the towers, the cable is 4 m above the walkway.
    The points \(P , Q , R , A , O\) and \(B\) are assumed to lie in the same vertical plane and \(A O B\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(P Q R\) used to model this cable.
    Given that \(O\) is the origin,
    1. find an equation for the curve \(P Q R\). Lee can safely inspect the cable up to a height of 12 m above the walkway.
      A defect is reported on the cable at a location 19 m horizontally from one of the towers.
    2. Determine whether, according to the model, Lee can safely inspect this defect.
    SPS SPS SM Pure 2023 June Q11
    11. The function f is defined by $$\mathrm { f } ( x ) = \frac { 12 x } { 3 x + 4 } \quad x \in \mathbb { R } , x \geqslant 0$$
    1. Find the range of f.
    2. Find \(f ^ { - 1 }\).
    3. Show, for \(x \in \mathbb { R } , x \geqslant 0\), that $$\mathrm { ff } ( x ) = \frac { 9 x } { 3 x + 1 }$$
    4. Show that \(\mathrm { ff } ( x ) = \frac { 7 } { 2 }\) has no solutions.
    SPS SPS SM Pure 2023 June Q12
    12.
    1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation $$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$ giving your answers to 2 decimal places.
    2. Hence find the smallest positive solution of the equation $$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$ giving your answer to 2 decimal places.
    SPS SPS SM Pure 2023 June Q13
    13. A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N \mathrm { ppm }\) (parts per million), is modelled by the equation $$N = 65 - 3 \mathrm { e } ^ { 0.1 t } \quad t \in \mathbb { R } t \geq 0$$ where \(t\) hours is the time after the treatment was applied. \section*{Use the equation of the model to answer parts (a) and (b).}
    1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm .
    2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place.
    SPS SPS SM Pure 2023 June Q14
    14.
    1. Prove that the sum of the squares of 2 consecutive odd integers is always 2 more than a multiple of 8
    2. Use proof by contradiction to show that \(\log _ { 2 } 5\) is irrational.
    SPS SPS SM Pure 2023 June Q15
    15. The resting metabolic rate, \(R \mathrm { ml }\) of oxygen consumed per hour, of a particular species of mammal is modelled by the formula, $$R = a M ^ { b }$$ where
    • \(M\) grams is the mass of the mammal
    • \(\quad a\) and \(b\) are constants
      1. Show that this relationship can be written in the form
    $$\log _ { 10 } R = b \log _ { 10 } M + \log _ { 10 } a$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-30_700_901_1005_667} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A student gathers data for \(R\) and \(M\) and plots a graph of \(\log _ { 10 } R\) against \(\log _ { 10 } M\)
    The graph is a straight line passing through points \(( 0.7,1.2 )\) and \(( 1.8,1.9 )\) as shown in Figure 3.
  • Using this information, find a complete equation for the model. Write your answer in the form $$R = a M ^ { b }$$ giving the value of each of \(a\) and \(b\) to 3 significant figures.
  • With reference to the model, interpret the value of the constant \(a\)
    •
    SPS SPS SM Pure 2023 June Q16
    16. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-32_538_672_219_733} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H \mathrm {~m}\), is modelled by the equation $$H = 0.8 + k \cos ( 30 t - 70 ) ^ { \circ }$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).
    1. Find the time of day at which the height of the sea is at its maximum. Given that the maximum height of the sea relative to the path is 2 m ,
      1. find a complete equation for the model,
      2. state the minimum height of the sea relative to the path. It is safe to use the path when the sea is 10 centimetres or more below the path.
    3. Find the times between which it is safe to use the path.
      (Solutions relying entirely on calculator technology are not acceptable.)
    SPS SPS SM Pure 2023 June Q17
    17. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-36_894_899_244_607} \captionsetup{labelformat=empty} \caption{Figure 7}
    \end{figure} Figure 7 shows the curves with equations $$\begin{aligned} & y = k x ^ { 2 } \quad x \geqslant 0
    & y = \sqrt { k x } \quad x \geqslant 0 \end{aligned}$$ where \(k\) is a positive constant.
    The finite region \(R\), shown shaded in Figure 7, is bounded by the two curves.
    Show that, for all values of \(k\), the area of \(R\) is \(\frac { 1 } { 3 }\)
    SPS SPS SM Pure 2023 June Q18
    18. Given that \(p\) is a positive constant,
    1. show that $$\sum _ { n = 1 } ^ { 11 } \ln \left( p ^ { n } \right) = k \ln p$$ where \(k\) is a constant to be found,
    2. show that $$\sum _ { n = 1 } ^ { 11 } \ln \left( 8 p ^ { n } \right) = 33 \ln \left( 2 p ^ { 2 } \right)$$
    3. Hence find the set of values of \(p\) for which $$\sum _ { n = 1 } ^ { 11 } \ln \left( 8 p ^ { n } \right) < 0$$ giving your answer in set notation. \section*{Additional Answer Space } \section*{Additional Answer Space } \section*{Additional Answer Space }
    SPS SPS FM Pure 2023 October Q1
    1. A curve is described by the equation
    $$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). A point \(Q\) lies on the curve.
      The tangent to the curve at \(Q\) is parallel to the \(y\)-axis.
      Given that the \(x\) coordinate of \(Q\) is negative,
    2. use your answer to part (a) to find the coordinates of \(Q\).
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 October Q2
    2. Given that \(x \geqslant 2\), find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ [BLANK PAGE]
    SPS SPS FM Pure 2023 October Q3
    3. Liquid is pouring into a large vertical circular cylinder at a constant rate of \(1600 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is \(4000 \mathrm {~cm} ^ { 2 }\).
    1. Show that at time \(t\) seconds, the height \(h \mathrm {~cm}\) of liquid in the cylinder satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - k \sqrt { } h , \text { where } k \text { is a positive constant. }$$ When \(h = 25\), water is leaking out of the hole at \(400 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
    2. Show that \(k = 0.02\)
    3. Separate the variables of the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - 0.02 \sqrt { } h$$ to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by $$\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { h } } \mathrm {~d} h$$ Using the substitution \(h = ( 20 - x ) ^ { 2 }\), or otherwise,
    4. find the exact value of \(\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { h } } \mathrm {~d} h\).
    5. Hence find the time taken to fill the cylinder from empty to a height of 100 cm , giving your answer in minutes and seconds to the nearest second.
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 October Q4
    4. The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
    1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
    2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 October Q5
    5. (i) Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$ (ii) Show that \(( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1\).
    (iii) Use the results in parts (i) and (ii) and the standard result for \(\sum _ { r = 1 } ^ { n } r\) to show that $$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$ [BLANK PAGE]
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q1
    1. Show that \(\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = 1\).
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q2
    2. (i) Sketch the graph of \(y = | 2 x - 7 a |\), where \(a\) is a positive constant. State the coordinates of the points where the graph meets each axis.
    (ii) Solve the inequality \(| 2 x - 7 a | < 4 a\).
    (iii) Deduce the largest integer \(N\) satisfying the inequality \(| 2 \ln N - 10.5 | < 6\).
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q3
    3. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. \(\sin \frac { 1 } { 2 } x = 0.8\)
    2. \(\sin x = 3 \cos x\)
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q4
    4. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geq 9\).
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q5
    5.
    \includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-12_423_743_123_731} The diagram shows the curve \(y = \mathrm { f } ( x )\), where f is the function defined for all real values of \(x\) by $$f ( x ) = 3 + 4 e ^ { - x }$$
    1. State the range of f.
    2. Find an expression for \(f ^ { - 1 } ( x )\), and state the domain and range of \(f ^ { - 1 }\).
    3. The straight line \(y = x\) meets the curve \(y = \mathrm { f } ( x )\) at the point \(P\). By using an iterative process based on the equation \(x = \mathrm { f } ( x )\), with a starting value of 3 , find the coordinates of the point \(P\). Show all your working and give each coordinate correct to 3 decimal places.
    4. How is the point \(P\) related to the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) ?
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q6
    6. The matrix \(\left( \begin{array} { l l } 1 & 3
    0 & 1 \end{array} \right)\) represents a transformation \(P\).
    1. Describe fully the transformation \(P\). The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1
      - 1 & 0 \end{array} \right)\).
    2. Given that \(M\) represents transformation \(Q\) followed by transformation \(P\), find the matrix that represents the transformation Q and describe fully the transformation Q .
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q7
    7. The complex number \(2 - \mathrm { i }\) is denoted by \(z\).
    1. Find \(| z |\) and arg \(z\).
    2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 September Q8
    8. A curve has equation \(y = 2 + \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The region \(R\) is bounded by the curve and by the straight lines \(x = 0 , x = 4\) and \(y = 0\). Find the exact volume of the solid obtained when \(R\) is rotated completely about the x-axis.
    [0pt] [BLANK PAGE]