Questions — SPS (1106 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 2024 June Q4
4. Prove by induction that the sum of the first \(n\) cube numbers is \(\frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
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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 \(P\).
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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 \(n\).
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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 \(d\).
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SPS SPS FM Pure 2024 June Q8
8. Using the substitution \(x = \mathrm { e } ^ { u }\), find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
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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.
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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 \(h = 10\)
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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.)
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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 \(\ln \left( \frac { 3 } { 2 } \right)\).
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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.
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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.
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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 \(k\).
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SPS SPS SM Pure 2024 June Q1
  1. The functions \(f\) and \(g\) are defined by
$$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geq 0
\mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geq 0 \end{array}$$
  1. Write down the range of f
  2. Find the value of fg (1.5)
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 } ( x )\)
SPS SPS SM Pure 2024 June Q2
2. Differentiate \(f ( x ) = a x ^ { 2 } + b x\) from first principles
(Total for Question 2 is 4 marks)
SPS SPS SM Pure 2024 June Q3
3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
  2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$
SPS SPS SM Pure 2024 June Q4
  1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } - 4 x ^ { 2 } - 7 x - 2$$ a) Use the factor theorem to show that ( \(2 x + 1\) ) is a factor of \(f ( x )\).
b) Write \(\frac { 3 x + 4 } { f ( x ) }\) in partial fraction form.
SPS SPS SM Pure 2024 June Q5
5. In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-10_629_988_370_577} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\), length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)
  1. Show that the surface area of the brick, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
  2. Hence find the value of \(x\) for which \(S\) is stationary and justify that this value of \(x\) gives the minimum value of \(S\).
  3. Hence find the minimum surface area of the brick.
SPS SPS SM Pure 2024 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-12_735_1081_239_500} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.
  1. Using Figure 1, find the range of values of \(x\) for which $$f ( x ) < 6$$
  2. State the largest solution of the equation $$f ( 2 x ) = 6$$
    1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
    2. Hence, using set notation, find the set of values of \(x\) for which $$f ( - x ) \geq 6 \text { and } x < 0$$
SPS SPS SM Pure 2024 June Q7
  1. (i) Using the laws of logarithms, solve
$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$ (ii) Given that $$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } ( y ) \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$ find \(y\) in terms of \(a\), giving your answer in simplest form.
(3)
SPS SPS SM Pure 2024 June Q8
8. ABCD is a parallelogram and ADM is a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{b063f4ea-372b-4193-b8fe-a9f8017d7349-16_497_1102_278_294} \section*{Diagram NOT accurately drawn} \(\overrightarrow { A B } = a \quad \overrightarrow { B C } = b \quad \overrightarrow { D M } = \frac { 1 } { 2 } b\)
K is the point on AB such that \(\mathrm { AK } : \mathrm { AB } = \lambda : 1\)
L is the point on CD such that \(\mathrm { CL } : \mathrm { CD } = \mu : 1\)
KLM is a straight line.
Give that \(\lambda : \mu = 1 : 2\) use a vector method to find the value of \(\lambda\) and the value of \(\mu\).
SPS SPS SM Pure 2024 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-18_718_882_219_596} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of
  • the curve with equation \(y = \tan x\)
  • the straight line \(l\) with equation \(y = \pi x\)
    in the interval \(- \pi < x < \pi\)
    1. State the period of \(\tan x\)
      (1)
    2. Write down the number of roots of the equation
      1. \(\tan x = ( \pi + 2 ) x\) in the interval \(- \pi < x < \pi\)
        (1)
      2. \(\tan x = \pi x\) in the interval \(- 2 \pi < x < 2 \pi\)
        (1)
      3. \(\tan x = \pi x\) in the interval \(- 100 \pi < x < 100 \pi\)
        (1)
SPS SPS SM Pure 2024 June Q10
  1. a) Use proof by contradiction to prove that there are no positive integers, \(x\) and \(y\), such that
$$x ^ { 2 } - y ^ { 2 } = 1$$ b) Prove, by counter-example, that the statement \section*{" if \(a\) is rational and \(b\) is irrational then \(\log _ { a } b\) is irrational"} is false.
c) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 "$$ \section*{(Total for Question 10 is 6 marks)}
SPS SPS SM Pure 2024 June Q11
  1. In this question you must show detailed reasoning.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
  2. Hence solve for \(- \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$
SPS SPS SM Pure 2024 June Q12
  1. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology or numerical methods are not acceptable. A geometric sequence has first term \(a\) and common ratio \(r\), where \(r > 0\)
Given that
  • the 3rd term is 20
  • the 5th term is 12.8
    1. show that \(r = 0.8\)
    2. Hence find the value of \(a\).
Given that the sum of the first \(n\) terms of this sequence is greater than 156
  • find the smallest possible value of \(n\).
    •