Questions — SPS SPS FM (245 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 2024 October Q1
  1. The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(f ( x )\).
Use differentiation from first principles to determine the value of \(f ^ { \prime } ( 2 )\).
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SPS SPS FM 2024 October Q2
2. 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 FM 2024 October Q3
3. (i) Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
(ii) In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
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SPS SPS FM 2024 October Q4
4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
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SPS SPS FM 2024 October Q5
5. A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
    1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    2. Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
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SPS SPS FM 2024 October Q6
6. 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 FM 2024 October Q7
7. In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).
\(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
Calculate the possible values of the angle \(A D B\).
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SPS SPS FM 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 2024 October Q9
9. Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
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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 FM 2025 October Q1
  1. Determine the equation of the line that passes through the point \(( 1,3 )\) and is perpendicular to the line with equation \(3 x + 6 y - 5 = 0\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be determined.
  2. In a triangle \(A B C , A B = 9 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(A C = 4 \mathrm {~cm}\).
    1. Show that \(\cos C A B = \frac { 2 } { 3 }\).
    2. Hence find the exact value of \(\sin C A B\).
    3. Find the exact area of triangle \(A B C\).
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    4. Given the function \(f ( x ) = 3 x ^ { 3 } - 7 x - 1\), defined for all real values of \(x\), prove from first principles that \(f ^ { \prime } ( x ) = 9 x ^ { 2 } - 7\).
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    The cubic polynomial \(2 x ^ { 3 } - k x ^ { 2 } + 4 x + k\), where \(k\) is a constant, is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( 2 ) = 16\).
  3. Show that \(k = 3\). For the remainder of the question, you should use this value of \(k\).
  4. Use the factor theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
  5. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
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SPS SPS FM 2025 October Q5
5. In this question you must show detailed reasoning. Consider the expansion of \(\left( \frac { x ^ { 2 } } { 2 } + \frac { a } { x } \right) ^ { 6 }\). The constant term is 960 .
Find the possible values of \(a\).
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SPS SPS FM 2025 October Q6
6. The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac { x } { 2 } - \frac { 1 } { 3 \sqrt { x } }$$ a) Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a ^ { b }\) where a and b are rational numbers.
b) Find the nature of the stationary point, justifying your answer.
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SPS SPS FM 2025 October Q7
7. The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point C . The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.
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SPS SPS FM 2025 October Q8
8. A sequence of terms \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4
a _ { n + 1 } & = k a _ { n } + 3 \end{aligned}$$ where \(k\) is a constant.
Given that
  • \(\sum _ { n = 1 } ^ { 3 } a _ { n } = 12\)
  • all terms of the sequence are different
    find the value of \(k\)
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SPS SPS FM 2025 October Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa025dee-b19f-4743-b212-2fff9a868eaf-18_689_830_127_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a quartic expression in \(x\). The curve
  • has maximum turning points at \(( - 1,0 )\) and \(( 5,0 )\)
  • crosses the \(y\)-axis at \(( 0 , - 75 )\)
  • has a minimum turning point at \(x = 2\)
    1. Find the set of values of \(x\) for which
$$\mathrm { f } ^ { \prime } ( x ) \geqslant 0$$ writing your answer in set notation.
  • Find the equation of \(C\). You may leave your answer in factorised form. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant.
    Given that the graph of \(C _ { 1 }\) intersects the \(x\)-axis at exactly four places,
  • find the range of possible values for \(k\).
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    SPS SPS FM 2025 October Q10
    10. The graph of \(y = \mathrm { e } ^ { x }\) can be transformed to the graph of \(y = \mathrm { e } ^ { 2 x - 1 }\) by a stretch parallel to the \(x\)-axis followed by a translation.
      1. State the scale factor of the stretch.
      2. Give full details of the translation. Alternatively the graph of \(y = \mathrm { e } ^ { x }\) can be transformed to the graph of \(y = \mathrm { e } ^ { 2 x - 1 }\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.
    2. State the scale factor of the stretch parallel to the \(y\)-axis.
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    SPS SPS FM 2025 October Q11
    11. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 3 } { 2 } \ln x & x > 0
    \mathrm {~g} ( x ) = \frac { 4 x + 3 } { 2 x + 1 } & x > 0 \end{array}$$
    1. Find \(\operatorname { gf } \left( e ^ { 2 } \right)\) writing your answer in simplest form.
    2. Find the range of the function fg .
    3. Given that \(\mathrm { f } ( 8 )\) and \(\mathrm { f } ( 2 )\) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational.
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