Questions — SPS SPS FM Pure (237 questions)

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SPS SPS FM Pure 2024 February Q7
7. In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).
  1. Find the area of \(S\).
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q8
8. A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2
\lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
    [0pt] [BLANK PAGE] \section*{9. In this question you must show detailed reasoning.} The complex number \(- 4 + \mathrm { i } \sqrt { 48 }\) is denoted by \(z\).
  3. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
  4. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
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SPS SPS FM Pure 2024 February Q10
11 marks
10. The diagram shows the polar curve \(C _ { 1 }\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C _ { 2 }\) with equation \(r = 1 + \cos 2 \theta\)
\includegraphics[max width=\textwidth, alt={}, center]{44ff7962-1982-46f0-aa02-be7127485bde-18_405_959_331_995}
  1. On the diagram above, complete the sketch of \(C _ { 2 }\)
    [0pt] [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k \pi + m \alpha - \sin 2 \alpha + q \sin 4 \alpha$$ where \(\alpha = \sin ^ { - 1 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \right)\), and \(k , m\) and \(q\) are rational numbers.
    [0pt] [9 marks]
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SPS SPS FM Pure 2024 February Q11
11. Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3
( k - 1 ) x + ( 3 - k ) y + 2 z & = 1
7 x - 3 y + 4 z & = 2 \end{aligned}$$
  1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q12
7 marks
12. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = \frac { x ^ { 3 } } { \sqrt { 4 - 2 x - x ^ { 2 } } }$$ where \(0 < x < \sqrt { 5 } - 1\)
[0pt] [7 marks]
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SPS SPS FM Pure 2024 February Q13
13. In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh \frac { 1 } { 2 } x ^ { 2 }\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q14
14. Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
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SPS SPS FM Pure 2024 February Q15
15. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer.
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SPS SPS FM Pure 2025 January Q1
1.
\includegraphics[max width=\textwidth, alt={}]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-04_400_513_169_774}
The diagram shows the curve \(y = 6 x - x ^ { 2 }\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning.
(Total 4 marks)
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SPS SPS FM Pure 2025 January Q2
2.
  1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).
    (Total 8 marks)
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2025 January Q3
3. $$\mathbf { A } = \left( \begin{array} { c r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is constant. }$$ A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
  1. Find the value of \(k\) for which the line \(y = 2 x\) is mapped onto itself under \(T\).
  2. Show that \(\mathbf { A }\) is non-singular for all values of \(k\).
  3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2025 January Q4
4. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0
0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
  2. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3
    3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
  3. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
  4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).
    (3)
    [0pt] [BLANK PAGE] With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
    & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  5. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
  6. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\), with position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\), lies on \(l _ { 1 }\)
    The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\)
  7. Find the position vector of \(B\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2025 January Q6
9 marks
6. You are given the complex number \(w = 2 + 2 \sqrt { 3 } i\).
  1. Express \(w\) in modulus-argument form.
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
    [0pt] [9]
    (Total 12 marks)
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SPS SPS FM Pure 2025 January Q7
7. 7 A candlestick has base diamater 8 cm and height 28 cm , as shown in Figure 9. A model of the candlestick is shown in Figure 10, together with the equations that were used to create the model. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 9} \includegraphics[alt={},max width=\textwidth]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-16_835_428_456_276}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 10} \includegraphics[alt={},max width=\textwidth]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-16_846_762_447_934}
\end{figure} a Show that the volume generated by rotating the shaded region (in Figure 10) \(2 \pi\) radians about the \(y\)-axis is \(\frac { 112 } { 15 } \pi\)
b Hence find the volume of metal needed to create the candlestick.
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SPS SPS FM Pure 2025 June Q1
  1. The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).
    [0pt] [BLANK PAGE]
  2. (a) Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\)
A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\)
(b) Give a reason why the student should not use \(x = \frac { 1 } { 2 }\)
(c) Substitute \(x = \frac { 1 } { 11 }\) into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.
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SPS SPS FM Pure 2025 June Q3
3. Describe a sequence of transformations which maps the graph of $$y = | 2 x - 5 |$$ onto the graph of $$y = | x |$$ [BLANK PAGE]
SPS SPS FM Pure 2025 June Q4
4. Given that $$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.
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SPS SPS FM Pure 2025 June Q5
5. Two matrices \(\mathbf { A }\) and \(\mathbf { B }\) satisfy the equation $$\mathbf { A B } = \boldsymbol { I } + 2 \mathbf { A }$$ where \(\boldsymbol { I }\) is the identity matrix and \(\mathbf { B } = \left[ \begin{array} { c c } 3 & - 2
- 4 & 8 \end{array} \right]\) \section*{Find \(\mathbf { A }\).} [BLANK PAGE]
SPS SPS FM Pure 2025 June Q6
6. (a) Prove that $$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.
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SPS SPS FM Pure 2025 June Q7
7. Fig. 10 shows the graph of \(x ^ { 3 } + y ^ { 3 } = x y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-16_538_527_251_785} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. P is the maximum point on the curve. The parabola \(y = k x ^ { 2 }\) intersects the curve at P . Find the value of the constant \(k\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2025 June Q8
8. (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{14f14bf3-88ee-413c-a62d-0914f41a485d-18_1339_1383_370_402}
(b) There is a unique complex number \(w\) that satisfies both $$| w - 3 | = 2 \quad \text { and } \quad \arg ( w + 1 ) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
(b) (i) Find the value of \(\alpha\).
(b) (ii) Express \(w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
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SPS SPS FM Pure 2025 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-20_707_823_130_701} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(A \mathrm { e } ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
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SPS SPS FM Pure 2025 June Q10
10. Prove by induction that \(f ( n ) = 2 ^ { 4 n } + 5 ^ { 2 n } + 7 ^ { n }\) is divisible by 3 for all positive integers \(n\).
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SPS SPS FM Pure 2025 June Q11
11. Fig. 15 shows the graph of \(\mathrm { f } ( x ) = 2 x + \frac { 1 } { x } + \ln x - 4\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-24_1008_771_212_669} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure}
  1. Show that the equation $$2 x + \frac { 1 } { x } + \ln x - 4 = 0$$ has a root, \(\alpha\), such that \(0.1 < \alpha < 0.9\).
  2. Obtain the following Newton-Raphson iteration for the equation in part (i). $$x _ { r + 1 } = x _ { r } - \frac { 2 x _ { r } ^ { 3 } + x _ { r } + x _ { r } ^ { 2 } \left( \ln x _ { r } - 4 \right) } { 2 x _ { r } ^ { 2 } - 1 + x _ { r } }$$
  3. Explain why this iteration fails to find \(\alpha\) using each of the following starting values.
    (A) \(x _ { 0 } = 0.4\)
    (B) \(x _ { 0 } = 0.5\)
    (C) \(x _ { 0 } = 0.6\)
    [0pt] [BLANK PAGE] \section*{12. In this question you must show detailed reasoning.}
    \includegraphics[max width=\textwidth, alt={}]{14f14bf3-88ee-413c-a62d-0914f41a485d-26_819_589_173_826}
    The curve \(C\) has parametric equations $$x = \frac { 1 } { \sqrt { 2 + t } } , \quad y = \ln ( 1 + t ) , \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac { 1 } { 2 }\).
    (a) Find the exact \(y\)-coordinate of \(P\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).
    (b) Determine the exact coordinates of \(Y\). The curve \(C\) and the line segment \(P Y\) are rotated \(2 \pi\) radians about the \(y\)-axis.
    (c) Determine the exact volume of the solid generated. Give your answer in the form \(\pi ( \ln p + q )\), where \(p\) and \(q\) are rational numbers.
    [0pt] [You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ]
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SPS SPS FM Pure 2025 June Q13
13.
  1. Using a suitable substitution, find $$\int \sqrt { 1 - x ^ { 2 } } d x$$
  2. Show that the differential equation $$\frac { d y } { d x } = 2 \sqrt { 1 - x ^ { 2 } - y ^ { 2 } + x ^ { 2 } y ^ { 2 } }$$ given that \(y = 0\) when \(x = 0 , | x | < 1\) and \(| y | < 1\), has the solution $$y = x \cos \left( x \sqrt { 1 - x ^ { 2 } } \right) + \sqrt { 1 - x ^ { 2 } } \sin \left( x \sqrt { 1 - x ^ { 2 } } \right) .$$ [BLANK PAGE]