Questions — SPS SPS FM Pure (237 questions)

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SPS SPS FM Pure 2025 June Q14
14. The three dimensional non-zero vector \(\boldsymbol { u }\) has the following properties:
  • The angle \(\theta\) between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 1
    5
    9 \end{array} \right)\) is acute.
  • The (non-reflex) angle between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 9
    5
    1 \end{array} \right)\) is \(2 \theta\).
  • \(\boldsymbol { u }\) is perpendicular to the vector \(\left( \begin{array} { l } 1
    1
    1 \end{array} \right)\).
Find the angle \(\theta\).
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SPS SPS FM Pure 2024 September Q1
  1. (a) Sketch the graph with equation
$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.
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SPS SPS FM Pure 2024 September Q2
2. $$\mathbf { P } = \frac { 1 } { 2 } \left( \begin{array} { r r } 1 & \sqrt { 3 }
- \sqrt { 3 } & 1 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)$$ The matrices \(\mathbf { P }\) and \(\mathbf { Q }\) represent linear transformations, \(P\) and \(Q\) respectively, of the plane.
The linear transformation \(M\) is formed by first applying \(P\) and then applying \(Q\).
  1. Find the matrix \(\mathbf { M }\) that represents the linear transformation \(M\).
  2. Show that the invariant points of the linear transformation \(M\) form a line in the plane, stating the equation of this line.
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SPS SPS FM Pure 2024 September Q3
3. (a) Sketch, on an Argand diagram, the set of points $$X = \{ z \in \mathbb { C } : | z - 4 - 2 i | < 3 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg ( z ) \leqslant \frac { \pi } { 4 } \right\}$$ On your diagram
  • shade the part of the diagram that is included in the set
  • use solid lines to show the parts of the boundary that are included in the set, and use dashed lines to show the parts of the boundary that are not included in the set
    (b) Show that the complex number \(z = 5 + 4 \mathrm { i }\) is in the set \(X\).
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SPS SPS FM Pure 2024 September Q4
  1. (a) Prove by induction that, for all \(n \in \mathbb { Z } ^ { + }\)
$$\mathrm { f } ( n ) = n ^ { 5 } + 4 n$$ is divisible by 5
(b) Show that \(\mathrm { f } ( - x ) = - \mathrm { f } ( x )\) for all \(x \in \mathbb { R }\)
(c) Hence prove that \(\mathrm { f } ( n )\) is divisible by 5 for all \(n \in \mathbb { Z }\)
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SPS SPS FM Pure 2024 September Q5
5. (a) Show that the binomial expansion of $$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.
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SPS SPS FM Pure 2024 September Q6
6. $$\mathrm { f } ( z ) = 8 z ^ { 3 } + 12 z ^ { 2 } + 6 z + 65$$ Given that \(\frac { 1 } { 2 } - \mathrm { i } \sqrt { 3 }\) is a root of the equation \(\mathrm { f } ( z ) = 0\)
  1. write down the other complex root of the equation,
  2. use algebra to solve the equation \(\mathrm { f } ( z ) = 0\) completely.
  3. Show the roots of \(\mathrm { f } ( z )\) on a single Argand diagram.
  4. Show that the roots of \(\mathrm { f } ( z )\) form the vertices of an equilateral triangle in the complex plane.
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SPS SPS FM Pure 2024 September Q7
5 marks
7. The region bounded by the curve with equation \(y = 3 + \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use integration to show that the volume generated is \(\frac { 125 \pi } { 2 }\)
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SPS SPS FM Pure 2024 September Q8
8. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017.
Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).
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SPS SPS FM Pure 2024 September Q9
9. In a chemical reaction, compound B is formed from compound A and other compounds. The mass of B at time \(t\) minutes is \(x \mathrm {~kg}\). The total mass of A and B is always 1 kg . Sadiq formulates a simple model for the reaction in which the rate at which the mass of \(B\) increases is proportional to the product of the masses of \(A\) and \(B\).
  1. Show that the model can be written as \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 1 - x )\), where \(k\) is a constant. Initially, the mass of B is 0.2 kg .
  2. Solve the differential equation, expressing \(x\) in terms of \(k\) and \(t\). After 15 minutes, the mass of B is measured to be 0.9 kg .
  3. Find the value of \(k\), correct to 3 significant figures.
  4. Find the mass of B after 30 minutes.
  5. Explain what the model predicts for the mass of A remaining for large values of \(t\).
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SPS SPS FM Pure 2024 September Q10
10.
  1. Find \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 2 \tan x \mathrm {~d} x\) giving your answer in the form \(\ln p\).
  2. Show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan x \mathrm {~d} x\) is undefined explaining your reasoning.
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SPS SPS FM Pure 2025 February Q1
  1. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows:
$$\mathbf { A } = \left( \begin{array} { l } 1
SPS SPS FM Pure 2025 February Q3
3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices.
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2. The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\). Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
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3. Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
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SPS SPS FM Pure 2025 February Q4
4. The cubic equation $$2 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 12 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(\alpha + 3\) ), ( \(\beta + 3\) ) and \(( \gamma + 3 )\), giving your answer in the form \(p w ^ { 3 } + q w ^ { 2 } + r w + s = 0\), where \(p , q , r\) and \(s\) are integers to be found.
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SPS SPS FM Pure 2025 February Q5
5. In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
  1. Find the equation of \(C\), giving your answer in the form $$| z - a | = b \quad a \in \mathbb { C } , b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 \mathrm { i } | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
  2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
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SPS SPS FM Pure 2025 February Q6
6. $$\mathrm { f } ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. show all the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram,
  2. find the value of \(p\) and the value of \(q\).
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SPS SPS FM Pure 2025 February Q7
2 marks
7. Line \(l _ { 1 }\) has Cartesian equation $$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$
  1. Write the equation of line \(l _ { 1 }\) in the form $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$$ where \(\lambda\) is a parameter and \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
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  2. Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
    1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
  4. (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.
  5. (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that $$\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }$$ where \(p\) and \(q\) are constants to be found.
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SPS SPS FM Pure 2025 February Q8
8. Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0
4 & - 1 \end{array} \right)\).
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SPS SPS FM Pure 2025 February Q9
9. $$\mathrm { f } ( z ) = z ^ { 3 } - 8 z ^ { 2 } + p z - 24$$ where \(p\) is a real constant.
Given that the equation \(\mathrm { f } ( z ) = 0\) has distinct roots $$\alpha , \beta \text { and } \left( \alpha + \frac { 12 } { \alpha } - \beta \right)$$
  1. solve completely the equation \(\mathrm { f } ( z ) = 0\)
  2. Hence find the value of \(p\).
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SPS SPS FM Pure 2025 February Q1
  1. The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).
Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
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SPS SPS FM Pure 2025 February Q2
2. Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
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SPS SPS FM Pure 2025 February Q3
3. The curve \(C\) has equation $$y = 31 \sinh x - 2 \sinh 2 x \quad x \in \mathbb { R }$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\).
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SPS SPS FM Pure 2025 February Q4
4. The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
  2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } \cdot ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
  3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
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SPS SPS FM Pure 2025 February Q7
7. The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$
  1. Sketch the curve.
  2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
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SPS SPS FM Pure 2025 February Q8
5 marks
8. (a) Solve the equation \(z ^ { 3 } = \sqrt { 2 } - \sqrt { 6 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leq \theta < 2 \pi\)
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(b) The transformation represented by the matrix \(\mathbf { M } = \left[ \begin{array} { l l } 5 & 1
1 & 3 \end{array} \right]\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points.
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