Questions — SPS SPS SM Pure (200 questions)

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SPS SPS SM Pure 2021 May Q2
2. (a) Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
(b) Explain how the trapezium rule could be used to obtain a more accurate estimate.
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SPS SPS SM Pure 2021 May Q3
3. Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
  2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
  3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).
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SPS SPS SM Pure 2021 May Q4
4. Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
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SPS SPS SM Pure 2021 May Q5
5.
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
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SPS SPS SM Pure 2021 May Q6
6. Shona makes the following claim.
" \(n\) is an even positive integer greater than \(2 \Rightarrow 2 ^ { n } - 1\) is not prime"
Prove that Shona's claim is true.
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SPS SPS SM Pure 2021 May Q7
7. A curve has parametric equations $$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$ for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
  2. Find the cartesian equation of the curve.
  3. State the set of values that \(x\) can take and hence sketch the curve.
    [0pt] [BLANK PAGE] \section*{8. In this question you must show detailed reasoning.} The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
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SPS SPS SM Pure 2021 May Q9
9. In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{5795db3d-2fcb-444e-a878-79e83c846334-20_747_481_233_826} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
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SPS SPS SM Pure 2021 May Q1
1.
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta\).
  2. Hence determine an approximate solution to \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta\).
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SPS SPS SM Pure 2021 May Q2
3 marks
2. Solve the equation \(| 2 x - 1 | = | x + 3 |\).
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SPS SPS SM Pure 2021 May Q3
3. Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).
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SPS SPS SM Pure 2021 May Q4
4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
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SPS SPS SM Pure 2021 May Q5
5. A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
  2. Find the equation of the normal to the curve at the point \(( 1,2 )\).
    [0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} A circle touches the lines \(y = \frac { 1 } { 2 } x\) and \(y = 2 x\) at \(( 6,3 )\) and \(( 3,6 )\) respectively.
    \includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-14_515_524_338_790} Find the equation of the circle.
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SPS SPS SM Pure 2021 May Q7
7. It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3 \tan 2 x - 8 \tan x = 4$$
  1. Show that \(t = \sqrt [ 3 ] { \frac { 1 } { 2 } + \frac { 1 } { 4 } t - \frac { 1 } { 2 } t ^ { 2 } }\), where \(t = \tan x\).
  2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8 .
  3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration.
  4. Solve the equation \(3 \tan 4 y - 8 \tan 2 y = 4\) for \(0 < y < \frac { 1 } { 2 } \pi\).
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SPS SPS SM Pure 2021 May Q8
8. Find the general solution of the differential equation $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 20 x - 35 )$$ Give your answer in the form \(y = \mathrm { f } ( x )\).
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SPS SPS SM Pure 2021 May Q9
9. (i) Show that the two non-stationary points of inflection on the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\) are at \(x = \pm \frac { 1 } { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-20_492_1064_237_513} The diagram shows the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac { 1 } { 2 }\) and \(x = - \frac { 1 } { 2 }\).
(ii) Show that the area of the shaded region is given by $$\int _ { 0 } ^ { \ln 2 } \sqrt { \mathrm { e } ^ { y } - 1 } \mathrm {~d} y$$ (iii) Show that the substitution \(\mathrm { e } ^ { y } = \sec ^ { 2 } \theta\) transforms the integral in part (ii) to \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } 2 \tan ^ { 2 } \theta \mathrm {~d} \theta\).
(iv) Hence find the exact area of the shaded region.
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SPS SPS SM Pure 2020 October Q1
  1. a) Find
$$\int \frac { x } { x ^ { 2 } + 1 } d x$$ b) Find. $$\int 2 \pi ( 4 x + 3 ) ^ { 10 } d x$$ c) Find. $$\int \frac { 2 } { e ^ { 4 x } } d x$$
SPS SPS SM Pure 2020 October Q2
  1. a) Find \(\frac { d y } { d x }\) if \(y = 4 \ln ( 3 x )\)
    b) Differentiate \(\frac { 2 x } { \sqrt { 3 x + 1 } }\) giving your answer in the form \(\frac { 3 x + c } { \sqrt { ( 3 x + 1 ) ^ { p } } }\), where \(c , p \in \mathbb { N }\)
  2. Expand \(( x - 2 y ) ^ { 5 }\).
  3. What transformations could be used, and in which order, to transform the curve \(y = \sin x\) into the curve \(y = 2 \sin ( 3 x + 30 )\) ?
Find the equation of the tangent to the curve $$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$ at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).
SPS SPS SM Pure 2020 October Q6
6.
  1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
  2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).
SPS SPS SM Pure 2020 October Q7
7.
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
SPS SPS SM Pure 2020 October Q8
8.
    1. Sketch the graph of \(y = \operatorname { cosec } x\) for \(0 < x < 4 \pi\).
    2. It is given that \(\operatorname { cosec } \alpha = \operatorname { cosec } \beta\), where \(\frac { 1 } { 2 } \pi < \alpha < \pi\) and \(2 \pi < \beta < \frac { 5 } { 2 } \pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\).
    1. Write down the identity giving \(\tan 2 \theta\) in terms of \(\tan \theta\).
    2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2 \phi \tan 4 \phi\), showing all your working.
SPS SPS SM Pure 2020 October Q9
9. Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$
  1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\).
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures.
SPS SPS SM Pure 2020 October Q10
10.
  1. Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
  2. Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
SPS SPS SM Pure 2022 June Q1
6 marks
1.
  1. The expression \(\left( 2 + x ^ { 2 } \right) ^ { 3 }\) can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Demonstrate clearly, using the binomial expansion, that \(p = 12\) and find the value of \(q\).
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  2. Hence find \(\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
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SPS SPS SM Pure 2022 June Q2
2. The trapezium \(A B C D\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-06_296_586_237_778} The line \(A B\) has equation \(2 x + 3 y = 14\) and \(D C\) is parallel to \(A B\). The point D has coordinates ( 3,7 ).
  1. Find an equation of the line DC
    (2 marks)
  2. The angle \(B A D\) is a right angle. Find an equation of the line \(A D\), giving your answer in the form \(m x + n y + p = 0\), where \(m , n\) and \(p\) are integers.
    (3 marks)
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SPS SPS SM Pure 2022 June Q3
3. A circle has centre \(C ( 3 , - 8 )\) and radius 10.
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
  3. The line with equation \(y = 2 x + 1\) intersects the circle.
    1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
    2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
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