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 2022 January Q2
2. The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).
  1. Find \(| a |\) and \(\arg a\).
  2. Sketch on a single Argand diagram the loci given by
    (a) \(| z - a | = | a |\),
    (b) \(\quad \arg ( z - 3 ) = \arg a\).
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SPS SPS FM 2022 January Q3
3. Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find a vector equation of the line passing through \(A\) and \(B\).
  2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).
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SPS SPS FM 2022 January Q4
4. Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
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SPS SPS FM 2022 January Q5
5.
  1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).
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SPS SPS FM 2022 January Q6
6. The figure below shows part of the curve \(y = 1 + x ^ { 2 }\), together with the line \(y = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{143eb5e6-a5f5-4c7b-b357-dea3fabec794-14_572_734_258_685} The region enclosed by the curve, the \(y\)-axis, and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the volume of the solid generated, giving your answer in terms of \(\pi\).
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SPS SPS FM 2022 January Q7
7.
  1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
  2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 January Q8
8.
  1. A group of four different letters is chosen from the alphabet of 26 letters, regardless of order.
    1. How many different groups can be chosen?
    2. Find the probability that a randomly chosen group includes the letter P .
  2. A three-digit number greater than 100 is formed using three different digits from the ten digits \(0,1,2,3,4,5,6,7,8,9\).
    1. Show that 648 different numbers can be formed. One of these 648 numbers is chosen at random.
    2. Find the probability that all three digits in the number are even. (You are reminded that 0 is an even number.)
    3. Find the probability that the number is even.
      [0pt] [BLANK PAGE]
SPS SPS FM 2022 January Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{143eb5e6-a5f5-4c7b-b357-dea3fabec794-20_719_969_207_525} The diagram shows the unit square \(O A B C\) and its image \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\) under a transformation T .
  1. Write down the matrix that represents T . The transformation T is equivalent to a transformation P followed by a transformation Q . The matrix that represents \(P\) is \(\left( \begin{array} { r r } 0 & - 1
    1 & 0 \end{array} \right)\).
  2. Give a geometrical description of transformation P .
  3. Find the matrix that represents transformation Q and give a geometrical description of transformation Q .
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SPS SPS FM 2022 February Q1
1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
0 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 1 & 1
0 & - 1 \end{array} \right)\).
  1. Find \(\mathbf { A } + 3 \mathbf { B }\).
  2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated.
SPS SPS FM 2022 February Q2
2. The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,
  1. \(2 z - 3 w\),
  2. \(( \mathrm { i } z ) ^ { 2 }\),
  3. \(\frac { z } { w }\).
SPS SPS FM 2022 February Q3
3. The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{bcba22d8-5f22-4576-b57c-7fdd05d128ad-1_344_349_993_1372}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
  2. Use integration to find the area of the shaded region bounded by the line and the curve.
SPS SPS FM 2022 February Q4
4. The transformation S is a shear parallel to the \(x\)-axis in which the image of the point ( 1,1 ) is the point \(( 0,1 )\).
  1. Draw a diagram showing the image of the unit square under S .
  2. Write down the matrix that represents S .
SPS SPS FM 2022 February Q5
5.
  1. Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
  2. Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
  3. The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$ \section*{6.} In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.
  4. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  5. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
SPS SPS FM 2022 February Q7
7. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { 2 }\) and \(\mathbf { A } ^ { 3 }\).
  2. Hence suggest a suitable form for the matrix \(\mathbf { A } ^ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
SPS SPS FM 2022 February Q8
8.
  1. Expand \(( 1 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }\) in ascending powers of \(x\).
SPS SPS FM 2022 February Q9
9. The position vectors of three points \(A , B\) and \(C\) relative to an origin \(O\) are given respectively by $$\text { and } \quad \begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } ,
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } . \end{aligned}$$
  1. Find the angle between \(A B\) and \(A C\).
  2. Find the area of triangle \(A B C\).
SPS SPS FM 2021 November Q1
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are \(\alpha , \beta\) and \(\gamma\). Without solving the equation, find the value of $$( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )$$
SPS SPS FM 2021 November Q2
3 marks
  1. The equation of a curve in polar coordinates is
$$r = 11 + 9 \sec \theta$$ Show that a cartesian equation of the curve is $$( x - 9 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = 11 x$$ [3 marks]
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SPS SPS FM 2021 November Q3
3. The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$| z - 6 i | = 2 | z - 3 |$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle.
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SPS SPS FM 2021 November Q4
4 marks
4. Prove that $$\sum _ { r = 1 } ^ { n } 18 \left( r ^ { 2 } - 4 \right) = n \left( 6 n ^ { 2 } + 9 n - 69 \right) .$$ [4 marks]
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SPS SPS FM 2021 November Q5
4 marks
5. Use a trigonometrical substitution to show that $$\int _ { 0 } ^ { 2 } \frac { 1 } { \left( 16 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x = \frac { 1 } { 16 \sqrt { 3 } }$$ [4 marks]
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SPS SPS FM 2021 November Q6
6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int _ { 1 } ^ { \infty } \frac { 1 } { \cosh \mathrm { u } } \mathrm { du }$$ giving your answer in an exact form.
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SPS SPS FM 2021 November Q7
11 marks
7. The curve with equation $$y = - x + \tanh ( 36 x ) , \quad x \geq 0$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\).
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac { \sqrt { 35 } } { 6 } - \frac { 1 } { 36 } \ln ( 6 + \sqrt { 35 } )$$ [BLANK PAGE] The function \(f\) is defined by \(f ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).
  3. Find \(\mathrm { f } ^ { \prime \prime \prime } ( \mathrm { x } )\) (i.e. the third derivative of \(f\) ) showing all your intermediate steps. Hence, find the Maclaurin series for \(f ( x )\) up to and including the \(x ^ { 3 }\) term.
    [0pt] [8 marks]
  4. Use the expansion of \(e ^ { x }\) together with the result in part (a) to show that, up to and including the \(x ^ { 3 }\) term, $$e ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } = 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
    [0pt] [3 marks]
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SPS SPS FM 2021 November Q9
9. (a) Show that $$\frac { 1 } { 9 r - 4 } - \frac { 1 } { 9 r + 5 } = \frac { 9 } { ( 9 r - 4 ) ( 9 r + 5 ) }$$ (b) Hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 9 r - 4 ) ( 9 r + 5 ) }$$ [BLANK PAGE]
SPS SPS FM 2021 November Q10
4 marks
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6278666a-d95f-461c-ab81-742c8faae1d5-24_517_1596_331_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 \sqrt { \cos ( 2 \theta ) } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leq \frac { \pi } { 4 } , \quad \frac { 3 \pi } { 4 } < \theta \leq \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1.
    [0pt] [4 marks]
  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.
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