SPS SPS SM Pure (SPS SM Pure) 2023 June

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.

3. Express in partial fractions, $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) }$$

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 \(E\).

5. \begin{figure}[h] \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 \(y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }\)

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.

6. \begin{figure}[h] \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.

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.

8. \begin{figure}[h] \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 \(l\).

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)

10. \begin{figure}[h] \end{figure} \begin{figure}[h] \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.

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.

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.

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.

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.

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] \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 \(a\)

16. \begin{figure}[h] \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 ,
2. 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.)

17. \begin{figure}[h] \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 \(\frac { 1 } { 3 }\)

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 }