Questions — SPS SPS FM Pure (237 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
SPS SPS FM Pure 2023 September Q9
9. The equations of two lines are $$\mathbf { r } = \left( \begin{array} { l } 3
0
2 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
3 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { r } - 1
8
2 \end{array} \right) + \mu \left( \begin{array} { c } - 3
1
- 5 \end{array} \right)$$ Find the coordinates of the point where these lines intersect.
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SPS SPS FM Pure 2023 September Q10
10. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geq 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2
0 & 1 \end{array} \right) .$$ [BLANK PAGE]
SPS SPS FM Pure 2023 September Q11
11. A curve has parametric equations \(x = \frac { 1 } { t } - 1\) and \(y = 2 t + \frac { 1 } { t ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the coordinates of the stationary point and, by considering the gradient of the curve on either side of this point, determine its nature.
  3. Find a cartesian equation of the curve.
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SPS SPS FM Pure 2023 September Q12
12.
  1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
  2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A e ^ { n }\).
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SPS SPS FM Pure 2023 September Q13
13. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 }
\frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)$$
  1. The diagram below shows the unit square \(O A B C\). The image of the unit square under the transformation represented by M is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
    \includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-28_778_1095_513_557}
  2. Find the equation of the line of invariant points of this transformation.
    1. Find the determinant of M .
    2. Describe briefly how this value relates to the transformation represented by M .
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SPS SPS FM Pure 2023 September Q14
14. Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).
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SPS SPS FM Pure 2023 February Q1
4 marks
  1. Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
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  2. Solve the equation \(2 z - 5 i z ^ { * } = 12\).
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\section*{3. In this question you must show detailed reasoning.} Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-06_538_723_296_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
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4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3
SPS SPS FM Pure 2023 February Q4
4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \mu \left( \begin{array} { l } 2
0
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) is perpendicular to \(\Pi\).
  2. Hence find a Cartesian equation of \(\Pi\). The line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { r } 4
    - 5
    2 \end{array} \right) + t \left( \begin{array} { r } 1
    6
    - 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
    The point \(A\) lies on \(l\).
    Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
  3. determine the possible coordinates of \(A\).
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SPS SPS FM Pure 2023 February Q6
6 marks
6
- 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
The point \(A\) lies on \(l\).
Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
(c) determine the possible coordinates of \(A\).
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5. Prove by induction that for all positive integers \(n\) $$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
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6. In this question you must show detailed reasoning. Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
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SPS SPS FM Pure 2023 February Q7
7. (a) Prove that $$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$ stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation $$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$ [BLANK PAGE]
SPS SPS FM Pure 2023 February Q8
8. The cubic equation $$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha , \beta\) and \(\gamma\)
The cubic equation $$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$ where \(c\) is a constant, has roots \(( 4 \alpha - 1 ) , ( 4 \beta - 1 )\) and \(( 4 \gamma - 1 )\)
Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
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SPS SPS FM Pure 2023 February Q9
9. (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-18_1173_1209_301_516}
(b) There is a unique complex number \(w\) that satisfies both $$| w - 3 | = 2 \text { and } \arg ( w + 1 ) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
  1. Find the value of \(\alpha\).
  2. Express \(w\) in the form \(r ( \cos \theta + i \sin \theta )\). Give each of \(r\) and \(\theta\) to two significant figures.
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SPS SPS FM Pure 2023 February Q10
10. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ (b) Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f ( x )\).
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SPS SPS FM Pure 2023 February Q11
11. In an Argand diagram, the points \(A , B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2 \mathrm { i }\).
  1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. The points \(D , E\) and \(F\) are the midpoints of the sides of triangle \(A B C\).
  2. Find the exact area of triangle \(D E F\).
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SPS SPS FM Pure 2023 February Q12
2 marks
12. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1
3 & k & 4
3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
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  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p
    & 3 x - 6 y + 4 z = 1
    & 3 x + 2 y - z = 0 \end{aligned}$$
    1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1
      & 3 x - 5 y + 4 z = q
      & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.
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SPS SPS FM Pure 2023 February Q13
13. In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt { \sin \theta } \mathrm { e } ^ { \frac { 1 } { 3 } \cos \theta }\) for \(0 \leqslant \theta \leqslant \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-26_686_1061_317_539}
  1. Find the exact area enclosed by the curve.
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).
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SPS SPS FM Pure 2023 February Q14
14. (a) Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
(b) By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).
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SPS SPS FM Pure 2024 January Q1
  1. Fig. 6 shows the region enclosed by part of the curve \(y = 2 x ^ { 2 }\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(\mathrm { P } ( 1,2 )\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f4c02a9b-802e-4e51-94c2-d1c5d69855b5-04_552_806_287_625} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed.
[0pt] [BLANK PAGE] \section*{2. a)} Find, in terms of \(k\), the set of values of \(x\) for which $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
b) Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ [BLANK PAGE]
SPS SPS FM Pure 2024 January Q3
3. Find the value of the integral:
\(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).
(4 marks)
(Total 4 marks)
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SPS SPS FM Pure 2024 January Q4
4. The points \(A\) and \(B\) have position vectors \(5 \mathbf { j } + 11 \mathbf { k }\) and \(c \mathbf { i } + d \mathbf { j } + 21 \mathbf { k }\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf { r } = 5 \mathbf { j } + 11 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } )\), where \(\lambda\) is a parameter.
  1. Find the value of \(c\) and the value of \(d\).
    (3) The point \(P\) lies on the line \(l\), and \(\overrightarrow { O P }\) is perpendicular to \(l\), where \(O\) is the origin.
  2. Find the position vector of \(P\).
    (6)
  3. Find the area of triangle \(O A B\), giving your answer to 3 significant figures.
    (4)
    (Total 13 marks)
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SPS SPS FM Pure 2024 January Q5
5. Let $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) }$$
  1. Express \(f ( x )\) in terms of partial fractions
  2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
  3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac { 1 } { 2 }\).
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SPS SPS FM Pure 2024 January Q6
6. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5
6 & k \end{array} \right)$$ where \(k\) is a constant.
Given that $$\mathbf { M } ^ { 2 } + 11 \mathbf { M } = a \mathbf { I }$$ where \(a\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
    1. determine the value of \(a\)
    2. show that \(k = - 9\)
  2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf { M }\).
  3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.
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SPS SPS FM Pure 2024 January Q7
7.
  • The point \(P\) represents a complex number \(z\) on an Argand diagram such that
$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle.
    (6) The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$$
  2. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.
  3. Find the complex number for which both \(| z - 6 \mathrm { i } | = 2 | z - 3 |\) and \(\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }\)
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SPS SPS FM Pure 2024 June Q1
1. The matrix \(\mathbf { M }\) is such that \(\mathbf { M } \left( \begin{array} { r r r } 1 & 0 & k
2 & - 1 & 1 \end{array} \right) = \left( \begin{array} { l l l } 1 & - 2 & 0 \end{array} \right)\).
Find
  • the matrix \(\mathbf { M }\),
  • the value of the constant \(k\).
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SPS SPS FM Pure 2024 June Q2
  1. Relative to a fixed origin \(O\),
    the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
    the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
    and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
    Given that \(A B C D\) is a parallelogram,
    1. find the position vector of point \(D\).
    The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
    Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
  2. find the position vector of \(X\).
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