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 2021 March Q1
  1. Differentiate the following with respect to \(x\), simplifying your answers fully
    a) \(y = e ^ { 3 x } + \ln 2 x\)
    b) \(y = \left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\)
    c) \(y = \frac { 2 x } { \left( 5 - 3 x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } }\)
    d) \(y = e ^ { - \frac { 8 } { 3 } x } \ln \left( 1 + x ^ { 3 } \right)\)
  1. Express \(2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }\) in terms of \(\sec \theta\).
  2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$
SPS SPS FM 2021 March Q3
3. $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x - 3 , x \in \mathbb { R } , x \geq 1$$
  1. Write down the domain and range of \(\mathrm { f } ^ { - 1 }\).
  2. Sketch the graph of \(\mathrm { f } ^ { - 1 }\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes.
  3. Find the gradient of \(f ^ { - 1 } ( x )\) when \(f ^ { - 1 } ( x ) = \frac { 5 } { 3 }\)
SPS SPS FM 2021 March Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{ef0c9a48-9d23-48ed-89b8-2e116114d7ed-07_527_718_191_651} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
SPS SPS FM 2021 March Q5
5. (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24$$ where \(t\) hours is the number of hours after 1200 .
(b) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(c) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).
SPS SPS FM 2021 March Q6
6. $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
0 & 1 \end{array} \right) .$$ Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
0 & 1 \end{array} \right)\), for all positive integers \(n\).
SPS SPS FM 2021 March Q7
7. This is the graph of \(y = \frac { 5 } { 4 x - 3 } - \frac { 3 } { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{ef0c9a48-9d23-48ed-89b8-2e116114d7ed-10_513_547_207_817} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a , b , c \in Q\)
SPS SPS FM 2021 March Q8
8. The function f is defined, for any complex number \(z\), by $$\mathrm { f } ( z ) = \frac { \mathrm { i } z - 1 } { \mathrm { i } z + 1 }$$ Suppose throughout that \(x\) is a real number.
  1. Show that $$\operatorname { Re } f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } \quad \text { and } \quad \operatorname { Im } f ( x ) = \frac { 2 x } { x ^ { 2 } + 1 }$$
  2. Show that \(\mathrm { f } ( x ) \mathrm { f } ( x ) ^ { * } = 1\), where \(\mathrm { f } ( x ) ^ { * }\) is the complex conjugate of \(\mathrm { f } ( x )\).
SPS SPS FM 2021 April Q1
  1. i) Differentiate the following with respect to \(x\), simplifying your answers fully
    a) \(y = e ^ { 3 x } + \ln 2 x\)
    b) \(y = \left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\)
    c) \(y = \frac { 2 x } { \left( 5 - 3 x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } }\)
    d) \(y = e ^ { - \frac { 8 } { 3 } x } \ln \left( 1 + x ^ { 3 } \right)\)
    ii) Integrate with respect to \(x\)
\begin{displayquote}
  1. \(\frac { 7 } { ( 2 x - 5 ) ^ { 5 } } - \frac { 3 } { 2 x - 5 }\)
  2. \(\frac { 4 x ^ { 2 } + 5 x - 3 } { 2 x - 5 }\) \end{displayquote}
SPS SPS FM 2021 April Q2
  1. solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation
$$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$
SPS SPS FM 2021 April Q3
3. $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x - 3 , x \in \mathbb { R } , x \geq 1$$
  1. Write down the domain and range of \(\mathrm { f } ^ { - 1 }\).
  2. Sketch the graph of \(\mathrm { f } ^ { - 1 }\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes.
  3. Find the gradient of \(f ^ { - 1 } ( x )\) when \(f ^ { - 1 } ( x ) = - \frac { 5 } { 3 }\)
SPS SPS FM 2021 April Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{11e1fe10-7d93-4a49-9151-83b40391a329-07_520_714_196_653} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
SPS SPS FM 2021 April Q6
6. This is the graph of \(y = \frac { 5 } { 4 x - 3 } - \frac { 3 } { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{11e1fe10-7d93-4a49-9151-83b40391a329-09_753_917_301_630} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a , b , c \in Q\)
SPS SPS FM 2021 April Q7
7. $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
0 & 1 \end{array} \right) .$$ Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
0 & 1 \end{array} \right)\), for all positive integers \(n\).
SPS SPS FM 2021 April Q8
8. The function f is defined, for any complex number \(z\), by $$\mathrm { f } ( z ) = \frac { \mathrm { i } z - 1 } { \mathrm { i } z + 1 }$$ Suppose throughout that \(x\) is a real number.
  1. Show that $$\operatorname { Re } f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } \quad \text { and } \quad \operatorname { Im } f ( x ) = \frac { 2 x } { x ^ { 2 } + 1 }$$
  2. Show that \(\mathrm { f } ( x ) \mathrm { f } ( x ) ^ { * } = 1\), where \(\mathrm { f } ( x ) ^ { * }\) is the complex conjugate of \(\mathrm { f } ( x )\). Spare Paper
SPS SPS FM Mechanics 2021 June Q1
  1. A train is travelling between two stations that are 4.8 km apart on a straight horizontal track.
It accelerates uniformly from rest to a speed of \(40 \mathrm {~ms} ^ { - 1 }\) covering a distance of 400 m .
It then travels at \(40 \mathrm {~ms} ^ { - 1 }\) for \(T\) seconds and decelerates uniformly at \(0.8 \mathrm {~ms} ^ { - 2 }\) for the final part of the journey until it arrives at the next station. This is represented in the velocity-time graph below.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-02_595_1394_497_210}
i. Work out the acceleration during the first 400 m of the journey.
ii. Find the value of \(T\).
[0pt] [BLANK PAGE]
SPS SPS FM Mechanics 2021 June Q2
2. A passenger in a lift has a mass of 84 kg . The lift starts to accelerate at \(1.2 \mathrm {~ms} ^ { - 2 }\). Find the difference between the two possible values of the normal reaction between the lift floor and the passenger.
[0pt] [BLANK PAGE]
SPS SPS FM Mechanics 2021 June Q3
3. A particle \(P\) moves along a straight line such that at time \(t\) seconds its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by: $$v ( t ) = t ^ { 2 } - 5 t + 4$$ Find the distance travelled by the particle between \(t = 1\) and \(t = 5.5\).
[0pt] [BLANK PAGE]
SPS SPS FM Mechanics 2021 June Q4
4. A particle of mass \(m \mathrm {~kg}\) is attached to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are 100 cm apart on a horizontal ceiling. The particle hangs in equilibrium as shown in the diagram, which is not drawn to scale.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-08_328_904_301_648} The string \(A C\) has length 80 cm and the string \(B C\) has length 60 cm .
Given that the tension in \(A C\) is 29.4 N , find:
i. the tension in \(B C\)
ii. the value of \(m\).
[0pt] [BLANK PAGE]
SPS SPS FM Mechanics 2021 June Q5
5. Two particles \(P\) and \(Q\) have masses 2 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially both \(P\) and \(Q\) are 2.1 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-10_417_182_358_1032}
i. Show that the acceleration of \(Q\) as it descends is: \(4.2 \mathrm {~ms} ^ { - 2 }\)
ii. Find the tension in the string as \(Q\) descends.
iii. Explain how you have used the information that the string is inextensible and that the pulley is smooth. When \(Q\) hits the ground it does not rebound and the string becomes slack. Particle \(P\) then moves freely under gravity without reaching the pulley.
iv. Find the greatest height above the ground that \(P\) reaches.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS FM Mechanics 2021 June Q6
6. A ski slope is modelled as a rough slope at an angle of \(30 ^ { \circ }\) to the horizontal. A skier of mass 72 kg is being towed up the slope at a constant speed of \(7 \mathrm {~ms} ^ { - 1 }\) by a rope inclined at an angle of \(30 ^ { \circ }\) to the slope. The skier is modelled as a particle \(P\) and the coefficient of friction between the skier and the slope is \(\frac { \sqrt { 3 } } { 23 }\). This situation is represented in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-13_396_625_388_790}
i. Show that the value of the normal reaction between the skier and the slope is \(23 \sqrt { 3 } g\) and find a similar expression in terms of \(g\) for the exact value of the tension in the rope.
ii. The skier lets go of the tow rope. Find the time the skier travels for before coming instantaneously to rest, giving your answer as a rational number of seconds.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 June Q1
  1. Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.
Time taken ( \(t\) minutes)Number of people
\(0 < t \leq 10\)\(x\)
\(10 < t \leq 20\)30
\(20 < t \leq 30\)35
\(30 < t \leq 50\)28
\(50 < t \leq 90\)12
The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 June Q2
2. Events \(A\) and \(B\) are such that \(P ( A \cup B ) = 0.95 , P ( A \cap B ) = 0.6\) and \(P ( A \mid B ) = 0.75\).
i. Find \(P ( B )\).
ii. Find \(P ( A )\).
iii. Show that the events \(A ^ { \prime }\) and \(B\) are independent.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 June Q3
3. The letters of the word CHAFFINCH are written on cards.
i. In how many ways can the letters be rearranged with no restrictions.
ii. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 June Q4
4. The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6 kg and three in five sacks weigh more than 5.5 kg .
i. Find the mean and standard deviation of the weights of potato sacks.
ii. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6 kg ? You must explain your reasoning clearly in this question.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2021 June Q5
5. Eleven students in a class sit a Mathematics exam and their average score is \(67 \%\) with a standard deviation of \(12 \%\). One student from the class is absent and sits the paper later, achieving a score of \(85 \%\).
i. Find the mean score for the whole class and the standard deviation for the whole class.
ii. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier.
[0pt] [BLANK PAGE] \section*{6. Only two airlines fly daily into an airport.} AMP Air has 70 flights per day and Volt Air has 65 flights per day.
Passengers flying with AMP Air have an \(18 \%\) probability of losing their luggage and passengers flying with Volt Air have a \(23 \%\) probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost.
Find the exact probability that she travelled with Volt Air, giving your answer as a rational number.
[0pt] [BLANK PAGE]