SPS SPS SM Pure (SPS SM Pure) 2024 February

1. Find \(\int \left( 2 x ^ { 4 } - x \sqrt { x } \right) \mathrm { d } x\).

2. The coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 x + k ) ^ { 12 }\), where \(k\) is a positive integer, is 79200000.
Determine the value of \(k\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\)
The table below shows corresponding values of \(x\) and \(y\) for this curve between \(x = 0.5\) and \(x = 0.9\) The values of \(y\) are given to 4 significant figures.

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for $$\int _ { 0.5 } ^ { 0.9 } \mathrm { f } ( x ) \mathrm { d } x$$ Give your answer to 3 significant figures.
2. Using your answer to part (a), deduce an estimate for $$\int _ { 0.5 } ^ { 0.9 } ( 3 \mathrm { f } ( x ) + 2 ) \mathrm { d } x$$

4. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\),
and the point \(C\) has position vector \(( a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k } )\), where \(a\) is a constant and \(a < 0\)
\(D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).

1. Find the position vector of \(D\).
   (2) Given \(| \overrightarrow { A C } | = 4\)
2. find the value of \(a\).
   (3)

5. The diagram shows the graph of \(y = 1.5 + \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-10_513_1266_349_210}

1. Show that the equation of the graph can be written in the form \(y = a - b \cos 2 x\) where \(a\) and \(b\) are constants to be determined.
2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(y = 1 + \cos 2 x\) in the interval \(0 \leqslant x \leqslant 2 \pi\).

6. Curve \(C\) has equation $$y = \left( x ^ { 2 } - 5 x + 8 \right) \mathrm { e } ^ { x ^ { 2 } } \quad x \in \mathbb { R }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( 2 x ^ { 3 } - 10 x ^ { 2 } + 18 x - 5 \right) \mathrm { e } ^ { x ^ { 2 } }$$ Given that
   • \(C\) has only one stationary point
   • the stationary point has \(x\) coordinate \(\alpha\)
   • \(\frac { \mathrm { d } y } { \mathrm {~d} x } \approx - 0.5\) at \(x = 0.3\)
   • \(\frac { \mathrm { d } y } { \mathrm {~d} x } \approx 0.9\) at \(x = 0.4\)
   • explain why \(0.3 < \alpha < 0.4\)
   • Show that \(\alpha\) is a solution of the equation
   $$x = \frac { 5 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 9 \right) }$$
2. Using the iteration formula $$x _ { n + 1 } = \frac { 5 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 9 \right) } \quad \text { with } x _ { 1 } = 0.3$$ find
   1. the value of \(x _ { 3 }\) to 4 decimal places,
   2. the value of \(\alpha\) to 4 decimal places.

7. The function f is defined by $$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.

1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point,
2. find the range of possible values of \(k\).

8.

1. Evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$
2. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$ [4 marks]

9.

1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-18_456_1150_488_584} \captionsetup{labelformat=empty} \caption{Figure 6}
   \end{figure} Figure 6 shows the cross-section of a water wheel.
   The wheel is free to rotate about a fixed axis through the point \(C\).
   The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
   The water level is assumed to be horizontal and of constant height.
   The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
   Using the model, find
2. 1. the maximum height of \(P\) above the water level,
   2. the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
3. find the value of \(T\) giving your answer to 3 significant figures.
   (Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
4. Explain how the equation of the model should be refined to take this into account.

10. Prove by contradiction that there are infinitely many prime numbers.

11. The curves \(y = \mathrm { h } ( x )\) and \(y = \mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(y = \mathrm { h } ( x )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(y = \mathrm { h } ^ { - 1 } ( x )\) crosses the \(x\)-axis at D and the \(y\)-axis at C .
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-22_789_798_568_242}

1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
2. Determine the coordinates of A, B, C and D.
3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
4. Points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a , b\) and \(r ^ { 2 }\) are constants to be determined.

12.

1. Sketch the graph with equation $$y = | 3 x - 2 a |$$ where \(a\) is a positive constant.
   State the coordinates of each point where the graph cuts or meets the coordinate axes.
2. Solve, in terms of \(a\), the inequality $$| 3 x - 2 a | \leqslant x + a$$ Given that \(| 3 x - 2 a | \leqslant x + a\)
3. find, in terms of \(a\), the range of possible values of \(\mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = 5 a - \left| \frac { 1 } { 2 } a - x \right|$$

13. A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-28_485_917_445_210}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Hence show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at P is - 1 .
3. Find the time at which the particle is travelling in the direction opposite to that at P .
4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).

14.

1. Use the substitution \(x = u ^ { 2 } + 1\) to show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$ where \(p\) and \(q\) are positive constants to be found.
2. Hence, using algebraic integration, show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$ where \(a\) is a rational constant to be found.
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