SPS SPS SM Pure (SPS SM Pure) 2020 February

1

1. Given that $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ show that $$x ^ { 2 } - 4 x - 12 = 0$$ [4]
2. 1. Write down the roots of the equation \(x ^ { 2 } - 4 x - 12 = 0\).
   2. State which of these two roots is not a solution of $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ giving a reason for your answer.

2

1. Using integration by parts, find the indefinite integral, with respect to \(x\), of $$x \cos x$$
2. Using the substitution \(u ^ { 2 } = 2 x + 1\), find the indefinite integral, with respect to $$x , \text { of } \frac { 6 x } { \sqrt { 2 x + 1 } } .$$

3 In this question you must show detailed reasoning.

1. Write down the first 5 terms of the geometric series $$\sum _ { r = 0 } ^ { n } 20 \times 0.5 ^ { r }$$
2. Find the smallest value of \(n\) for which the series $$\sum _ { r = 0 } ^ { n } 20 \times 0.5 ^ { r }$$ is greater than \(99.8 \%\) of its sum to infinity.
   [0pt] [5]

4

1. Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\tan ^ { 2 } \theta + 1 \equiv \sec ^ { 2 } \theta\).
   A curve is given parametrically by $$x = a \sec \theta , \quad y = a \tan \theta$$ where \(a\) is a constant.
2. Find a Cartesian equation of the curve.
3. Determine an equation of the tangent to the curve at the point \(\theta = \frac { \pi } { 3 }\), giving your answer in exact form.
   [0pt] [5]

5

1. Find the first three non-zero terms of the expansion, in ascending powers of \(x\), of \(( 4 + x ) ^ { \frac { 1 } { 2 } }\).
2. State the range of values of \(x\) for which your expansion in part (a) is valid. [1]
3. Use your expansion to determine an approximation to \(\sqrt { } 36.9\), showing all the figures on your calculator.

6 In this question you must show detailed reasoning.

1. Given that \(y = \mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } - 16 x ^ { 2 } + 7 x - 3\), determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } < 0$$
2. State the geometrical interpretation of your answer to part (a), in terms of the shape of the graph of \(y = \mathrm { f } ( x )\) (not the gradient of the graph).

7 A scientist is studying the flight of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 140 } { A + 45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } } , \quad 0 \leq t \leq T ,$$ where \(t\) seconds is the time after the bird leaves its nest, and \(A\) and \(T\) are constants. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing the graph of \(H\) against \(t\).
It is given that this seabird's nest is \(\mathbf { 2 0 } \mathbf { ~ m }\) above sea level.

1. Show that \(A = 35\).
   It is also given that $$45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } \equiv 53 \sin ( 2 t - \alpha ) ^ { \circ }$$ where \(\alpha\) is a constant in the range \(0 < \alpha < 90\).
2. Find the value of \(\alpha\) to one decimal place.
   Find, according to this model,
3. the minimum height of the sea bird above sea level, giving your answer to the nearest cm,
   [0pt] [2]
4. the limitation on the value of \(T\).

8 Differentiate from first principles $$y = \frac { 1 } { x }$$

9

1. Describe fully a sequence of transformations that map the line \(y = x\) onto the line $$y = 10 - 2 x$$ The function f is defined as \(\mathrm { f } : x \rightarrow 10 - 2 x , x \in R , x \geq 0\).
   The function ff is denoted by g .
2. Find \(\mathrm { g } ( x )\), giving your answer in a form without brackets.
3. Determine the domain of g .
4. Explain whether \(\mathrm { fg } = \mathrm { gf }\).
5. Find \(\mathrm { g } ^ { - 1 }\) in the form \(\mathrm { g } ^ { - 1 } : x \rightarrow \ldots\)

10 The diagram shows part of the graphs of \(y = \cos ^ { 2 } x\) and \(y = 5 \sin 2 x\) for small positive values of \(x\). The graphs meet at the point \(A\) with \(x\)-coordinate \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{022274c9-7ed2-4436-ae97-d410d7d566fc-14_652_561_402_735}

1. Find the exact area contained between the two graphs (between \(x = 0\) and \(x = \alpha\) ) and the \(y\)-axis. Give your answer in terms of \(\alpha , \cos 2 \alpha\) and/or \(\sin 2 \alpha\).
2. Using the fact that \(\alpha\) is a small positive solution to the equation \(\cos ^ { 2 } x = 5 \sin 2 x\), show that \(\alpha\) satisfies approximately the equation \(\alpha ^ { 2 } + 10 \alpha - 1 = 0\), if terms in \(\alpha ^ { 3 }\) and higher are ignored.
3. Use the equation in part (b) to find an approximate value of \(\alpha\), correct to 3 significant figures.

11 Two circles have equations
\(x ^ { 2 } + y ^ { 2 } - 4 x = 0 \quad\) and
\(x ^ { 2 } + y ^ { 2 } - 6 x - 12 y + 36 = 0\).

1. Find the centre and radius of each circle and hence show that the \(y\)-axis is a tangent to both circles.
2. Find the equation of the line through the centres of both circles.
3. Determine the gradient of a line other than the \(y\)-axis which is a tangent to both circles.
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