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 SM 2024 October Q3
3. The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where p is a constant. Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).
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SPS SPS SM 2024 October Q4
4. The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
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SPS SPS SM 2024 October Q6
6. The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).
\(t\)051025
\(m\)200160
  1. Find the values missing from the table.
  2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.
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SPS SPS SM 2024 October Q7
7. A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0
& \log _ { 3 } x - 2 = 0
& \log _ { 3 } x = 2
& x = 8 \end{aligned}$$
  1. Identify the two mistakes that the student has made.
  2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
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SPS SPS SM 2024 October Q8
8. In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac { 5 } { 3 x - 4 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 2 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 3 } + \ldots$$ is convergent.
  1. Find the set of possible values of \(x\), giving your answer in set notation.
  2. Given that the sum to infinity of the series is \(\frac { 2 } { 3 }\), find the value of \(x\).
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SPS SPS FM Pure 2025 January Q1
1.
\includegraphics[max width=\textwidth, alt={}]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-04_400_513_169_774}
The diagram shows the curve \(y = 6 x - x ^ { 2 }\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning.
(Total 4 marks)
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SPS SPS FM Pure 2025 January Q2
2.
  1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).
    (Total 8 marks)
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SPS SPS FM Pure 2025 January Q3
3. $$\mathbf { A } = \left( \begin{array} { c r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is constant. }$$ A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
  1. Find the value of \(k\) for which the line \(y = 2 x\) is mapped onto itself under \(T\).
  2. Show that \(\mathbf { A }\) is non-singular for all values of \(k\).
  3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
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SPS SPS FM Pure 2025 January Q4
4. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0
0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
  2. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3
    3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
  3. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
  4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).
    (3)
    [0pt] [BLANK PAGE] With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
    & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  5. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
  6. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\), with position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\), lies on \(l _ { 1 }\)
    The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\)
  7. Find the position vector of \(B\).
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SPS SPS FM Pure 2025 January Q6
9 marks
6. You are given the complex number \(w = 2 + 2 \sqrt { 3 } i\).
  1. Express \(w\) in modulus-argument form.
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
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    (Total 12 marks)
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SPS SPS FM Pure 2025 January Q7
7. 7 A candlestick has base diamater 8 cm and height 28 cm , as shown in Figure 9. A model of the candlestick is shown in Figure 10, together with the equations that were used to create the model. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 9} \includegraphics[alt={},max width=\textwidth]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-16_835_428_456_276}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 10} \includegraphics[alt={},max width=\textwidth]{1b1cfccc-20f3-42f8-a1e1-d9405e7afcb9-16_846_762_447_934}
\end{figure} a Show that the volume generated by rotating the shaded region (in Figure 10) \(2 \pi\) radians about the \(y\)-axis is \(\frac { 112 } { 15 } \pi\)
b Hence find the volume of metal needed to create the candlestick.
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SPS SPS FM 2025 February Q1
1.
\includegraphics[max width=\textwidth, alt={}, center]{b073ed4d-319a-4b97-8ff1-59d66aa22f24-04_680_942_118_651} The diagram shows the curve with equation \(y = 5 x ^ { 4 } + a x ^ { 3 } + b x\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units \({ } ^ { 2 }\), determine the \(y\)-coordinate of \(P\).
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SPS SPS FM 2025 February Q2
2. (i) Find the first three terms in the expansion of \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), where \(| x | < \frac { 1 } { 2 }\).
(ii) Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).
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SPS SPS FM 2025 February Q3
4 marks
3. Express \(\frac { 9 x ^ { 2 } + 43 x + 8 } { ( 3 + x ) ( 1 - x ) ( 2 x + 1 ) }\) in partial fractions.
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[0pt] [BLANK PAGE] The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)\), where \(a\) is a constant.
  1. Find \(\mathrm { A } ^ { - 1 }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { l l } 2 & a
    4 & 1 \end{array} \right)\).
  2. Given that \(\mathrm { PA } = \mathrm { B }\), find the matrix P .
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SPS SPS FM 2025 February Q5
5. (a) \(\mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P . Write down the matrix A .
(b)
\(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). B is the matrix that represents Q . Find the matrix \(B\).
(c) T is P followed by Q. C is the matrix that represents T. Determine the matrix \(\mathbf { C }\).
(d) \(L\) is the line whose equation is \(y = x\). Explain whether or not \(L\) is a line of invariant points under \(T\).
(e) An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\). Explain what the value of the determinant of \(\mathbf { C }\) means about
  • the area of \(N\) compared to the area of \(M\),
  • the orientation of \(N\) compared to the orientation of \(M\).
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SPS SPS FM 2025 February Q6
  1. The equations of two lines are
$$r = i + 2 j + \lambda ( 2 i + j + 3 k ) \text { and } r = 6 i + 8 j + k + \mu ( i + 4 j - 5 k )$$
  1. Show that these lines meet, and find the coordinates of the point of intersection.
  2. Find the acute angle between these lines.
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SPS SPS FM 2025 February Q7
7. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geq 9\).
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SPS SPS FM 2025 February Q8
8. A locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \{ z : | z + \mathrm { i } | \leq | z - 2 | \}\).
  1. Indicate by shading on the Argand diagram below the region representing \(C _ { 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{b073ed4d-319a-4b97-8ff1-59d66aa22f24-18_883_940_408_317}
  2. Find the cartesian equation of the boundary line of the region representing \(C _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\).
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SPS SPS FM 2025 February Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b073ed4d-319a-4b97-8ff1-59d66aa22f24-20_880_501_139_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b073ed4d-319a-4b97-8ff1-59d66aa22f24-20_775_583_242_1279} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(A B C D E F G H A\) of the bottle with the measurements taken by the student. The horizontal cross-section between \(C F\) and \(D E\) is a circle of diameter 8 cm and the horizontal cross-section between \(B G\) and \(A H\) is a circle of diameter 2 cm . The student thinks that the curve \(G F\) could be modelled as a curve with equation $$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Find the value of \(a\) and the value of \(b\) according to the model.
  2. Use the model to find the volume of water that the bottle can contain.
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SPS SPS SM 2025 February Q1
  1. Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).
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  2. (a)
    \includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-06_412_919_187_694}
The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the cross-section \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\). Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(\alpha\) and \(b\) are integers to be determined.
(b) Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
(d) Explain why, for this roof, \(x\) must be less than a certain value, which you should state.
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SPS SPS SM 2025 February Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-08_469_471_130_877} The diagram shows a sector \(A O B\) of a circle with centre \(O\). The length of the \(\operatorname { arc } A B\) is 6 cm and the area of the sector \(A O B\) is \(24 \mathrm {~cm} ^ { 2 }\). Find the area of the shaded segment enclosed by the \(\operatorname { arc } A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.
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SPS SPS SM 2025 February Q4
4. (a) The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 . Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
(b) Prove that \(K\) has at least two distinct factors other than 1 and \(K\).
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SPS SPS SM 2025 February Q5
5. A curve has the following properties:
  • The gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 x\).
  • The curve passes through the point \(( 4 , - 13 )\).
Determine the coordinates of the points where the curve meets the line \(y = 2 x\).
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SPS SPS SM 2025 February Q6
6. For all real values of \(x\), the functions f and g are defined by \(\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }\) and \(\mathrm { g } ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.
  1. Find \(\mathrm { fg } ( x )\). Determine the range of \(\mathrm { fg } ( x )\) in terms of \(a\).
  2. If \(f g ( 2 ) = 144\), find the value of \(a\).
  3. Determine whether the function fg has an inverse.
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SPS SPS SM 2025 February Q7
7. (a) Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$ (b) Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.
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