Questions — SPS SPS SM Pure (200 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 SM Pure 2023 September Q4
4.
    1. Express as a single logarithm $$\log _ { a } 36 - \frac { 1 } { 2 } \log _ { a } 81 + 2 \log _ { a } 4 - 3 \log _ { a } 2$$
  2. (ii) Hence find the value of \(a\), given $$\log _ { a } 36 - \frac { 1 } { 2 } \log _ { a } 81 + 2 \log _ { a } 4 - 3 \log _ { a } 2 = \frac { 3 } { 2 }$$
SPS SPS SM Pure 2023 September Q5
7 marks
5. The curve with equation \(y = x ^ { 3 } - 7 x + 6\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{a1b449df-1096-4b3a-8306-fca410a7e530-10_428_627_342_810} The curve intersects the \(x\)-axis at the points \(A ( - 3,0 ) , B ( 1,0 )\) and \(C\).
  1. Find the coordinates of \(C\).
    [0pt] [1 mark]
  2. Find \(\int \left( x ^ { 3 } - 7 x + 6 \right) \mathrm { d } x\)
    [0pt] [2 marks]
  3. Find the total area of the shaded regions enclosed by the curve and the \(x\)-axis.
    [0pt] [4 marks]
SPS SPS SM Pure 2023 September Q6
6. A curve has equation \(x ^ { 2 } + y ^ { 2 } + 12 x = 64\)
A line has equation \(y = m x + 10\)
    1. In the case that the line intersects the curve at two distinct points, show that $$( 20 m + 12 ) ^ { 2 } - 144 \left( m ^ { 2 } + 1 \right) > 0$$
  2. (ii) Hence find the possible values of \(m\).
    1. On the same diagram, sketch the curve and the line in the case when \(m = 0\)
  4. (ii) State the relationship between the curve and the line.
SPS SPS SM Pure 2023 September Q7
7.
\(( x - 3 )\) is a common factor of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) where: $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 3 } - 11 x ^ { 2 } + ( p - 15 ) x + q
& \mathrm {~g} ( x ) = 2 x ^ { 3 } - 17 x ^ { 2 } + p x + 2 q \end{aligned}$$
    1. Show that \(3 p + q = 90\) and \(3 p + 2 q = 99\) Fully justify your answer.
  2. (ii) Hence find the values of \(p\) and \(q\).
  3. \(\quad \mathrm { h } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\) Using your values of \(p\) and \(q\), fully factorise \(\mathrm { h } ( x )\)
SPS SPS SM Pure 2023 September Q8
4 marks
8. Martin tried to find all the solutions of \(4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
His working is shown below: $$\begin{aligned} & 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0
& \Rightarrow 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = \cos ^ { 2 } \theta
& \Rightarrow 4 \sin ^ { 2 } \theta = 1
& \Rightarrow \sin ^ { 2 } \theta = \frac { 1 } { 4 }
& \Rightarrow \sin \theta = \frac { 1 } { 2 }
& \Rightarrow \theta = 30 ^ { \circ } , 150 ^ { \circ } \end{aligned}$$ Martin did not find all the correct solutions because he made two errors.
  1. Identify the two errors and explain the consequence of each error.
    [0pt] [4 marks]
  2. Find all the solutions that Martin did not find.
SPS SPS SM Pure 2023 September Q9
3 marks
9. Two models are proposed for the value of a car.
  1. The first model suggests that the value of the car, \(V\) pounds, is given by \(V = 18000 - 6000 \sqrt { t }\), where \(t\) is the time in years after the car was first purchased.
    1. State the value of the car when it was first purchased.
  3. (ii) Find \(V\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t }\) when \(t = 4\)
  4. (iii) Interpret your answers to (a)(ii) in the context of the model.
  5. The second model that is proposed suggests that the value of the car, \(V\) pounds, is given by \(V = a b ^ { - t }\), where \(t\) is the time in years after the car was first purchased. When \(t = 0\), both models give the same value for \(V\).
    When \(t = 4\), both models give the same value for \(V\). Find the value of \(a\) and the value of \(b\).
    [0pt] [3 marks]
  6. Explain, with a reason, which model is likely to be the better model over time.
SPS SPS SM Pure 2023 September Q10
10. The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 ^ { x } , x \in \mathbb { R }
& \mathrm {~g} ( x ) = \sqrt { 1 - x } , x \in \mathbb { R } , x \leq a \end{aligned}$$
  1. State the maximum possible value of \(a\).
  2. The function h is defined by \(\mathrm { h } ( x ) = \mathrm { gf } ( x )\)
    1. Write down an expression for \(\mathrm { h } ( x )\)
  4. (ii) Using set notation, state the greatest possible domain of h .
  5. (iii) State the range of h .
SPS SPS SM Pure 2023 September Q11
11. A geometric sequence, \(S _ { 1 }\), has first term \(a\) and common ratio \(r\) where \(a \neq 0\) and \(r \in ( - 1,1 )\) A new sequence, \(S _ { 2 }\), is formed by squaring each term of \(S _ { 1 }\)
  1. Given that the sum to infinity of \(S _ { 2 }\) is twice the sum to infinity of \(S _ { 1 }\), show that \(a = 2 ( 1 + r )\) Fully justify your answer.
  2. Determine the set of possible values for \(a\). \section*{Additional Answer Space } \section*{Additional Answer Space }
SPS SPS SM Pure 2024 February Q1
1. Find \(\int \left( 2 x ^ { 4 } - x \sqrt { x } \right) \mathrm { d } x\).
SPS SPS SM Pure 2024 February Q2
2. The coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 x + k ) ^ { 12 }\), where \(k\) is a positive integer, is 79200000.
Determine the value of \(k\).
SPS SPS SM Pure 2024 February Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-06_709_1052_287_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\)
The table below shows corresponding values of \(x\) and \(y\) for this curve between \(x = 0.5\) and \(x = 0.9\) The values of \(y\) are given to 4 significant figures.
\(x\)0.50.60.70.80.9
\(y\)1.6321.7111.7861.8591.930
  1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for $$\int _ { 0.5 } ^ { 0.9 } \mathrm { f } ( x ) \mathrm { d } x$$ Give your answer to 3 significant figures.
  2. Using your answer to part (a), deduce an estimate for $$\int _ { 0.5 } ^ { 0.9 } ( 3 \mathrm { f } ( x ) + 2 ) \mathrm { d } x$$
SPS SPS SM Pure 2024 February Q4
4. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\),
and the point \(C\) has position vector \(( a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k } )\), where \(a\) is a constant and \(a < 0\)
\(D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).
  1. Find the position vector of \(D\).
    (2) Given \(| \overrightarrow { A C } | = 4\)
  2. find the value of \(a\).
    (3)
SPS SPS SM Pure 2024 February Q5
5. The diagram shows the graph of \(y = 1.5 + \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-10_513_1266_349_210}
  1. Show that the equation of the graph can be written in the form \(y = a - b \cos 2 x\) where \(a\) and \(b\) are constants to be determined.
  2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
  3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(y = 1 + \cos 2 x\) in the interval \(0 \leqslant x \leqslant 2 \pi\).
SPS SPS SM Pure 2024 February Q6
6. Curve \(C\) has equation $$y = \left( x ^ { 2 } - 5 x + 8 \right) \mathrm { e } ^ { x ^ { 2 } } \quad x \in \mathbb { R }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( 2 x ^ { 3 } - 10 x ^ { 2 } + 18 x - 5 \right) \mathrm { e } ^ { x ^ { 2 } }$$ Given that
    • \(C\) has only one stationary point
    • the stationary point has \(x\) coordinate \(\alpha\)
    • \(\frac { \mathrm { d } y } { \mathrm {~d} x } \approx - 0.5\) at \(x = 0.3\)
    • \(\frac { \mathrm { d } y } { \mathrm {~d} x } \approx 0.9\) at \(x = 0.4\)
    • explain why \(0.3 < \alpha < 0.4\)
    • Show that \(\alpha\) is a solution of the equation
    $$x = \frac { 5 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 9 \right) }$$
  2. Using the iteration formula $$x _ { n + 1 } = \frac { 5 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 9 \right) } \quad \text { with } x _ { 1 } = 0.3$$ find
    1. the value of \(x _ { 3 }\) to 4 decimal places,
    2. the value of \(\alpha\) to 4 decimal places.
SPS SPS SM Pure 2024 February Q7
7. The function f is defined by $$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.
  1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point,
  2. find the range of possible values of \(k\).
SPS SPS SM Pure 2024 February Q8
4 marks
8.
  1. Evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$
  2. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$ [4 marks]
SPS SPS SM Pure 2024 February Q9
9.
  1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-18_456_1150_488_584} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} Figure 6 shows the cross-section of a water wheel.
    The wheel is free to rotate about a fixed axis through the point \(C\).
    The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
    The water level is assumed to be horizontal and of constant height.
    The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
    Using the model, find
    1. the maximum height of \(P\) above the water level,
    2. the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
  3. find the value of \(T\) giving your answer to 3 significant figures.
    (Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
  4. Explain how the equation of the model should be refined to take this into account.
SPS SPS SM Pure 2024 February Q10
10. Prove by contradiction that there are infinitely many prime numbers.
SPS SPS SM Pure 2024 February Q11
11. The curves \(y = \mathrm { h } ( x )\) and \(y = \mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(y = \mathrm { h } ( x )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(y = \mathrm { h } ^ { - 1 } ( x )\) crosses the \(x\)-axis at D and the \(y\)-axis at C .
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-22_789_798_568_242}
  1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
  2. Determine the coordinates of A, B, C and D.
  3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
  4. Points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a , b\) and \(r ^ { 2 }\) are constants to be determined.
SPS SPS SM Pure 2024 February Q12
12.
  1. Sketch the graph with equation $$y = | 3 x - 2 a |$$ where \(a\) is a positive constant.
    State the coordinates of each point where the graph cuts or meets the coordinate axes.
  2. Solve, in terms of \(a\), the inequality $$| 3 x - 2 a | \leqslant x + a$$ Given that \(| 3 x - 2 a | \leqslant x + a\)
  3. find, in terms of \(a\), the range of possible values of \(\mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = 5 a - \left| \frac { 1 } { 2 } a - x \right|$$
SPS SPS SM Pure 2024 February Q13
13. A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-28_485_917_445_210}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Hence show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at P is - 1 .
  3. Find the time at which the particle is travelling in the direction opposite to that at P .
  4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).
SPS SPS SM Pure 2024 February Q14
14.
  1. Use the substitution \(x = u ^ { 2 } + 1\) to show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$ where \(p\) and \(q\) are positive constants to be found.
  2. Hence, using algebraic integration, show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$ where \(a\) is a rational constant to be found.
    (6) Use this page for any additional working. Use this page for any additional working. Use this page for any additional working. Use this page for any additional working.
SPS SPS SM Pure 2024 September Q1
  1. Express
$$f ( x ) = \frac { x ^ { 2 } + x - 5 } { ( x - 2 ) ( x - 1 ) ^ { 2 } }$$ in partial fractions. \section*{(Total for Question 1 is 3 marks)}
SPS SPS SM Pure 2024 September Q2
  1. The curve \(C\) has equation
$$y = \frac { 5 x ^ { 3 } - 8 } { 2 x ^ { 2 } } \quad x > 0$$ Find an equation for the tangent to \(C\) at \(x = 2\) writing your answer in the form
\(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(5) \section*{(Total for Question 2 is 5 marks)}
SPS SPS SM Pure 2024 September Q3
  1. The function f is defined by
$$\mathrm { f } ( x ) = \frac { x + 3 } { x - 4 } \quad x \in \mathbb { R } , x \neq 4$$
  1. Find ff (6)
    (2)
  2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain
    (3) \section*{(Total for Question 3 is 5 marks)}