Questions — SPS (1106 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 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 \(A\).
[0pt] [BLANK PAGE]
5. Prove by induction that for all positive integers \(n\) $$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
[0pt] [6]
[0pt] [BLANK PAGE]
6. In this question you must show detailed reasoning. Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
[0pt] [BLANK PAGE]
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 \(c\).
[0pt] [BLANK PAGE]
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.
    [0pt] [BLANK PAGE]
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 \(y = f ( x )\).
[0pt] [BLANK PAGE]
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 \(D E F\).
    [0pt] [BLANK PAGE]
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.
    [0pt] [2]
  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.
      [0pt] [BLANK PAGE]
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 \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).
    [0pt] [BLANK PAGE]
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 \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 February Q1
1. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$
  1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(f ( x )\).
    (2)
  2. Factorise \(f ( x )\) to a linear and quadratic factor.
    (2)
  3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)
    (3)
SPS SPS SM Pure 2023 February Q2
2.
\(f ( x ) = 3 x ^ { 2 } + 2 x . \quad\) Find \(f ^ { \prime } ( x )\) from first principles.
(4)
SPS SPS SM Pure 2023 February Q3
3.
a) Show that when \(x\) is small, \(2 \cos x - 3 \sin x\) can be written as \(a + b x + c x ^ { 2 }\), where \(a , b\) and \(c\) are integers to be found.
b) Hence find a small positive value of \(x\) that is an approximate solution to \(2 \cos x - 3 \sin x = 7 x\)
SPS SPS SM Pure 2023 February Q4
4. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the coordinates of the stationary point on \(C\).
  3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.
SPS SPS SM Pure 2023 February Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-12_832_931_260_502} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2 The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the arcs \(P T\) and \(S Q\) of the curve. Use integration to find the exact area of the shaded region \(R\).
SPS SPS SM Pure 2023 February Q6
6. $$f ( x ) = ( 3 - 2 x ) ^ { - 4 }$$ a) Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.
(3)
b) For what values of \(x\) is the expansion valid?
SPS SPS SM Pure 2023 February Q7
2 marks
7. The function f is defined by $$\mathrm { f } ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$
  1. \(\quad \mathrm { f } ( x ) = 0\) has a single solution at the point \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 1
    [0pt] [2 marks]
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$
  3. (ii) Use the Newton-Raphson method with \(x _ { 1 } = 1\) to find \(x _ { 3 }\), an approximation for \(\alpha\). Give your answer to five decimal places.
  4. (iii) Explain why the Newton-Raphson method fails to find \(\alpha\) with \(x _ { 1 } = 0\)
SPS SPS SM Pure 2023 February Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-18_563_853_274_566} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(C\) with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Using your answer to (a), find the exact coordinates of the stationary point on the curve \(C\) for which \(x > 0\). Write each coordinate in its simplest form.
    (4) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 3\)
  3. Complete the table below with the value of \(y\) corresponding to \(x = 1\)
    \(x\)0123
    \(y\)0\(\frac { 3 } { 5 } \ln 5\)\(\frac { 3 } { 10 } \ln 10\)
    (1)
  4. Use the trapezium rule with all the \(y\) values in the completed table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
    (2)
SPS SPS SM Pure 2023 February Q9
9. The function g is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$
  1. Sketch the graph with equation \(y = \mathrm { g } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
    (2)
  2. Solve the equation $$| 8 - 2 x | = x + 5$$ The function \(f\) is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
  3. Find fg(5).
  4. Find the range of f . You must make your method clear.
SPS SPS SM Pure 2023 February Q10
10.
a) Show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$ where \(a\) is a rational constant to be found.
b) By using a suitable substitution, find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$
SPS SPS SM Pure 2023 February Q11
11.
  1. Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (4)
  2. Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.
    (3)
SPS SPS SM Pure 2023 February Q12
12. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$
  1. find the values of the constants \(A , B , C\) and \(D\).
    (3)
  2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
SPS SPS SM Pure 2023 February Q13
13. The curve \(C\) has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point \(A\) with coordinates \(( 5,3 )\) lies on \(C\).
  1. Find the value of \(t\) at the point \(A\).
  2. Show that an equation of the normal to \(C\) at \(A\) is $$3 y = 10 x - 41$$ The normal to \(C\) at \(A\) cuts \(C\) again at the point \(B\).
  3. Find the exact coordinates of \(B\).
SPS SPS SM Pure 2023 February Q14
7 marks
14. The sum to infinity of a geometric series is 96
The first term of the series is less than 30
The second term of the series is 18
  1. Find the first term and common ratio of the series.
    [0pt] [4 marks]
    1. Show that the \(n\)th term of the series, \(u _ { n }\), can be written as $$u _ { n } = \frac { 3 ^ { n } } { 2 ^ { 2 n - 5 } }$$ [3 marks]
  3. (ii) Hence show that $$\log _ { 3 } u _ { n } = n \left( 1 - 2 \log _ { 3 } 2 \right) + 5 \log _ { 3 } 2$$
SPS SPS FM Mechanics 2023 January Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15f4500a-8eb8-4b5f-896c-de730272a35b-06_312_979_157_568} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hockey stick is modelled as a uniform rod \(O A\) of length \(14 r\) joined to a uniform semicircular arc \(A B\) of diameter \(2 r\), as shown in Figure 1. The rod and the arc lie in the same plane and are made of the same material.
  1. Find, in terms of \(\pi\) and \(r\), the distance of the centre of mass of the hockey stick from the line \(A B\). The hockey stick is freely suspended from \(O\) and hangs in equilibrium.
    Given that the centre of mass of the hockey stick is a distance \(\frac { \pi r } { ( 14 + \pi ) }\) from \(O A\),
  2. find, in degrees to 3 significant figures, the angle between \(O A\) and the vertical.
    [0pt] [Question 2 Continued]
SPS SPS FM Mechanics 2023 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15f4500a-8eb8-4b5f-896c-de730272a35b-08_396_860_178_641} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question use \(\boldsymbol { g } = \mathbf { 1 0 m s } \boldsymbol { s } ^ { \mathbf { - 2 } }\).} A light elastic string has natural length \(a\) metres and modulus of elasticity \(\lambda\) newtons. A particle \(P\) of mass 2 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\) on a rough inclined plane. The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\) The point \(B\) on the plane lies below \(A\) on the line of greatest slope of the plane through \(A\) and \(A B = 3 a\) metres, as shown in Figure 3. The particle \(P\) is held at \(B\) and then released from rest. The particle first comes to instantaneous rest at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\)
  1. Show that \(\lambda = 24\)
  2. Find the magnitude of the acceleration of \(P\) at the instant it is released from \(B\).
  3. Explain why the answer to part (b) is the greatest value of the magnitude of the acceleration of \(P\) as \(P\) moves from \(B\) to \(A\).
    [0pt] [Question 3 Continued] \section*{4.}