Questions SPS FM Pure (237 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2024 June Q3
3. (a) Sketch on the Argand diagram below the locus of points satisfying the equation \(| z - 2 | = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-08_1260_1303_260_468}
(b) Given that \(| z - 2 | = 2\) and \(\arg ( z - 2 ) = - \frac { \pi } { 3 }\), express \(z\) in the form \(a + b i\) where \(a , b \in \mathbb { R }\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q4
4. Prove by induction that the sum of the first \(n\) cube numbers is \(\frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q5
5. (a) The diagram shows the graph of \(y = a \sec ( b x ) + 1\) for \(x \in [ 0 , \pi )\). Find the values of \(a\) and \(b\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-12_818_556_201_897}
(b) The diagram shows the graph of \(y = \arccos ( x + c )\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-12_511_766_1667_790}
  1. State the value of c .
  2. State the coordinates of the point \(P\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q6
6. In this question you must show detailed reasoning. Given that $$( 1 + a x ) ^ { n } = 1 + 6 x - 6 x ^ { 2 } + \ldots$$ where \(a\) and \(n\) are constants, find the values of \(a\) and \(n\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q7
7. In the quartic equation \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d = 0\), the coefficients \(a , b , c\) and \(d\) are real. Two of the roots of the equation are i and \(2 - \mathrm { i }\). Find the value of \(a , b , c\) and \(d\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q8
8. Using the substitution \(x = \mathrm { e } ^ { u }\), find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace492d8-1dd0-401e-af74-505ca19d5e9c-20_679_1136_132_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 2.
  1. Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
  2. Show that the \(x\) coordinate of \(Q\) satisfies $$x = \frac { 8 } { 1 + \ln x }$$
  3. Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with \(x _ { 1 } = 3.5\) to
    1. find the value of \(x _ { 5 }\) to 4 decimal places,
    2. find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.
SPS SPS FM Pure 2024 June Q10
10. $$\begin{aligned} & \boldsymbol { v } _ { \mathbf { 1 } } = \left( \begin{array} { c } \sqrt { 17 }
\cos 2 \theta
- 4 \end{array} \right)
& \boldsymbol { v } _ { \mathbf { 2 } } = \left( \begin{array} { c } - \sin 2 \theta
2 \sqrt { 2 }
1 \end{array} \right) \end{aligned}$$ Given that \(\boldsymbol { v } _ { \mathbf { 1 } }\) and \(\boldsymbol { v } _ { \mathbf { 2 } }\) are perpendicular and that \(0 \leq \theta \leq \pi\), find all possible values of \(\theta\). Give your answers to 3 significant figures.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace492d8-1dd0-401e-af74-505ca19d5e9c-24_387_752_137_749} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl.
When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)
Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\)
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q12
12. A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace492d8-1dd0-401e-af74-505ca19d5e9c-28_583_917_155_676} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$
  1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
  2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q14
14. A theme park ride lasts for 70 seconds. The height above ground, \(H\) metres, of a passenger on the theme park ride is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = \frac { t \sin \left( \frac { \pi t } { 5 } \right) } { 10 H } \quad 0 \leqslant t \leqslant 70$$ where \(t\) seconds is the time from the start of the ride.
Given that the passenger is 5 m above ground at the start of the ride find the height above ground of the passenger 52 seconds after the start of the ride.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q15
15. Two angles, \(x\) and \(y\), are acute. $$\begin{aligned} \sin x \cos y & = \frac { 1 + \sqrt { 3 } } { 4 }
\cos x \sin y & = \frac { - 1 + \sqrt { 3 } } { 4 } \end{aligned}$$
  1. Find the exact value of \(\sin ( x + y )\).
  2. Find all possible pairs of values of \(x\) and \(y\), giving your answers in terms of \(\pi\). Fully justify your answer.
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 June Q16
16. $$\begin{gathered} M _ { 1 } = \left( \begin{array} { c c } 2 k - 9 & 5 - k
- k & k - 2 \end{array} \right)
M _ { 2 } = \left( \begin{array} { c c } 5 & 1
2 k - 3 & k - 3 \end{array} \right)
k \in \mathbb { R } \end{gathered}$$ Matrices \(M _ { 1 }\) and \(M _ { 2 }\) represent transformations \(T _ { 1 }\) and \(T _ { 2 }\) respectively.
\(\Delta\) is a triangle in the \(x y\)-plane with vertices at \(( 0,0 ) , ( 4,0 )\) and \(( 3,2 )\).
The image of \(\Delta\) under \(T _ { 1 }\) is \(\Delta _ { 1 }\) and the image of \(\Delta\) under \(T _ { 2 }\) is \(\Delta _ { 2 }\).
The area of \(\Delta _ { 2 }\) is greater than the area of \(\Delta _ { 1 }\).
Find the range of possible values of \(k\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q1
1. $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 3
k & 1 \end{array} \right]$$
  1. Find \(\mathbf { A } ^ { - 1 }\)
  2. The determinant of \(\mathbf { A } ^ { 2 }\) is equal to 4 . Find the possible values of \(k\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q2
2. A sequence \(u _ { n }\) is defined by \(u _ { n + 1 } = 2 u _ { n } + 3\) and \(u _ { 1 } = 1\). Prove by induction that \(u _ { n } = 4 \times 2 ^ { n - 1 } - 3\) for all positive integers \(n\).
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q3
3. A finite region is bounded by the curve with equation \(y = x + x ^ { - \frac { 3 } { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi ( a \sqrt { 2 } + b )\), where \(a\) and \(b\) are rational numbers to be determined.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q4
6 marks
4. The curve \(C\) has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
    The line \(l\) is the normal to \(C\) at \(P\).
  2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.
    [0pt] [BLANK PAGE]
  4. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg ( z + \mathrm { i } ) = \frac { \pi } { 6 }$$ [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-12_1307_1351_516_463}
  5. \(\quad z _ { 1 }\) is a point on \(L\) such that \(| z |\) is a minimum. Find the exact value of \(z _ { 1 }\) in the form \(a + b \mathrm { i }\)
    [0pt] [4 marks]
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q6
8 marks
6. A curve has equation \(y = x \mathrm { e } ^ { \frac { x } { 2 } }\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection.
[0pt] [8 marks]
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q7
8 marks
7. (a) Prove the identity \(\frac { \cos x } { \sec x + 1 } + \frac { \cos x } { \sec x - 1 } \equiv 2 \cot ^ { 2 } x\)
[0pt] [3 marks]
(b) Hence, solve the equation $$\frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) + 1 } = \cot \left( 2 \theta + \frac { \pi } { 3 } \right) - \frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) - 1 }$$ in the interval \(0 \leq \theta \leq 2 \pi\), giving your values of \(\theta\) to three significant figures where appropriate.
[0pt] [5 marks]
[0pt] [BLANK PAGE] \section*{8. A population of meerkats is being studied.} The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q9
6 marks
9. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
  2. The equation \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer.
    2. Show that the roots of \(\mathrm { f } ( x ) = 0\) satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
    3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to three decimal places.
      [0pt] [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e ^ { - \frac { x } { 2 } }\) and \(y = x\), and the position of \(x _ { 1 }\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      [0pt] [2 marks] \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1d67c98c-e81c-4967-8a0b-a78afd95a0aa-22_1236_1566_1519_360}
      \end{figure} [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q1
  1. The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
  2. Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
  3. It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
$$z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80 = 0$$ where \(p\) and \(r\) are real numbers.
  1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
  2. Find the value of \(p\) and the value of \(r\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q4
4. Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series $$( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots$$ where \(n\) is a positive integer. Give your answer in fully factorised form.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q5
6 marks
5. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers \(n \geq 1\), $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$ [6 marks]
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2024 February Q6
6.
  1. Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x\) is an improper integral.
  2. Prove that $$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.
    [0pt] [BLANK PAGE]