SPS SPS SM (SPS SM) 2020 June

1. A curve has equation $$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.

2. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows the plan view of a design for a stage at a concert.
The stage is modelled as a sector \(B C D F\), of a circle centre \(F\), joined to two congruent triangles \(A B F\) and \(E D F\). Given that
\(A F E\) is a straight line $$\begin{aligned} & A F = F E = 10.7 \mathrm {~m}
& B F = F D = 9.2 \mathrm {~m} \end{aligned}$$ angle \(B F D = 1.82\) radians
find

1. the perimeter of the stage, in metres, to one decimal place,
2. the area of the stage, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

3. \begin{figure}[h] \end{figure} Figure 2 is a sketch showing the line \(l _ { 1 }\) with equation \(y = 2 x - 1\) and the point \(A\) with coordinates \(( - 2,3 )\). The line \(l _ { 2 }\) passes through \(A\) and is perpendicular to \(l _ { 1 }\)

1. Find the equation of \(l _ { 2 }\) writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The point \(B\) and the point \(C\) lie on \(l _ { 1 }\) such that \(A B C\) is an isosceles triangle with \(A B = A C = 2 \sqrt { 13 }\)
2. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5 x ^ { 2 } - 12 x - 32 = 0$$ Given that \(B\) lies in the 3rd quadrant
3. find, using algebra and showing your working, the coordinates of \(B\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { g } ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).

1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
2. State the largest root of the equation $$\mathrm { g } ( x + 1 ) = 0$$
3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
4. state the range of possible values for \(k\). Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of $$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.

6. A company which makes batteries for electric cars has a 10 -year plan for growth.

• In year 1 the company will make 2600 batteries
• In year 10 the company aims to make 12000 batteries

In order to calculate the number of batteries it will need to make each year, from year 2 to year 9 , the company considers the following model: \section*{the number of batteries made will increase by the same percentage each year} Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10.

7.

1. Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation, $$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
2. (a) A student's attempt at the question
   "Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x
   7 \times \frac { \sin x } { \cos x } = 8 \sin x
   7 \sin x = 8 \sin x \cos x
   7 = 8 \cos x
   \cos x = \frac { 7 } { 8 }
   x = 29.0 ^ { \circ } \text { (to } 3 \mathrm { sf } \text { ) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
   (b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$

8. Prove by contradiction that there are no positive integers \(a\) and \(b\) with \(a\) odd such that $$a + 2 b = \sqrt { 8 a b }$$

9. \begin{figure}[h] \end{figure} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = a b ^ { t }$$ where \(a\) and \(b\) are constants.
The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log _ { 10 } P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model,
find a complete equation for the model giving the value of b to 4 significant figures.

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\).
   (5)
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-10_656_814_1786_662} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).

11.

1. Sketch the curve with equation $$y = k - \frac { 1 } { 2 x } \quad \text { where } k \text { is a positive constant }$$ State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. The straight line \(l\) has equation \(y = 2 x + 3\)
   Given that \(l\) cuts the curve in two distinct places,
2. find the range of values of \(k\), writing your answer in set notation.

12. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.} Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2 ^ { 2 x }$$ The point \(P ( a , 96 \sqrt { 2 } )\) lies on the curve.

1. Find the exact value of \(a\). The curve with equation \(y = 3 \times 2 ^ { 2 x }\) meets the curve with equation \(y = 6 ^ { 3 - x }\) at the point \(Q\).
2. Show that the \(x\) coordinate of \(Q\) is $$\frac { 3 + 2 \log _ { 2 } 3 } { 3 + \log _ { 2 } 3 }$$

13. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\)
The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)