Questions — SPS SPS SM Pure (200 questions)

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SPS SPS SM Pure 2022 November Q2
2.
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
SPS SPS SM Pure 2022 November Q3
3. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where k is a constant, has 2 different real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0$$
  2. Hence find the set of possible values of \(k\).
SPS SPS SM Pure 2022 November Q4
4. Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
  1. \(\quad \log _ { 2 } ( 16 x )\),
  2. \(\quad \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\)
    (3)
  3. Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
    (Total 9 marks)
SPS SPS SM Pure 2022 November Q5
5. An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 29 terms is 1102.
  1. Show that \(a + 14 d = 38\).
  2. The sum of the second term and the seventh term is 13 . Find the value of \(a\) and the value of \(d\).
SPS SPS SM Pure 2022 November Q6
6. A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(\quad u _ { n } = 2 n + 5\), for \(n \geq 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. State what type of sequence it is.
  3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).
SPS SPS SM Pure 2022 November Q7
7. There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012-2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10 ^ { b t }\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log _ { 10 } y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\).
  2. Complete the values of \(\log _ { 10 } y\) in the table, draw the graph of \(\log _ { 10 } y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model.
    t12345678910
    \(\log _ { 10 } y\)1.511.581.982.68
SPS SPS SM Pure 2022 November Q8
8.
  1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\).
    \(A B\) is the diameter of a circle.
  2. Find the radius of the circle.
  3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).
SPS SPS SM Pure 2023 June Q1
1. Find $$\int \left( x ^ { 4 } - 6 x ^ { 2 } + 7 \right) \mathrm { d } x$$ giving your answer in simplest form. Curve C has equation $$y = x ^ { 3 } - 7 x ^ { 2 } + 5 x + 4$$ The point \(P ( 2 , - 6 )\) lies on \(C\) Find the equation of the tangent to \(C\) at \(P\)
Give your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
SPS SPS SM Pure 2023 June Q3
3. Express in partial fractions, $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) }$$
SPS SPS SM Pure 2023 June Q4
4. Relative to a fixed origin \(O\),
  • the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
  • the point \(B\) has position vector \(7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\)
  • the point \(C\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }\)
    1. Find \(| \overrightarrow { A B } |\) giving your answer as a simplified surd.
Given that \(A B C D\) is a parallelogram,
  • find the position vector of the point \(D\). The point \(E\) is positioned such that
    • \(A C E\) is a straight line
    • \(A C : C E = 2 : 1\)
    • Find the coordinates of the point \(E\).
    •
    SPS SPS SM Pure 2023 June Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-10_684_689_260_639} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the curve with equation \(y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }\) for \(x \geq 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis, and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }\)
    \(x\)00.511.52
    \(y\)1\(e ^ { 0.05 }\)\(e ^ { 0.2 }\)\(e ^ { 0.45 }\)\(e ^ { 0.8 }\)
    1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
    2. Use your answer to part (a) to deduce an estimate for
      1. \(\quad \int _ { 0 } ^ { 2 } \left( 4 + e ^ { \frac { 1 } { 5 } x ^ { 2 } } \right) d x\)
      2. \(\quad \int _ { 1 } ^ { 3 } e ^ { \frac { 1 } { 5 } ( x - 1 ) ^ { 2 } } d x\)
        giving your answers to 2 decimal places.
    SPS SPS SM Pure 2023 June Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-12_622_1196_251_495} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The shape \(A O C B A\), shown in Figure 2, consists of a sector \(A O B\) of a circle centre \(O\) joined to a triangle \(B O C\). The points \(A , O\) and \(C\) lie on a straight line with \(A O = 7.5 \mathrm {~cm}\) and \(O C = 8.5 \mathrm {~cm}\).
    The size of angle \(A O B\) is 1.2 radians.
    Find, in cm, the perimeter of the shape \(A O C B A\), giving your answer to one decimal place.
    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 SM Pure 2023 October Q1
    1. technology.
      1. Differentiate with respect to \(x\)
        1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
        2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
          [0pt] [BLANK PAGE]
      (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
      (ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).
      [0pt] [BLANK PAGE]