Questions — SPS (1106 questions)

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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
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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 2021 November Q2
4 marks
2. You must show detailed working in this question Determine whether the line with equation \(2 x + 3 y + 4 = 0\) is parallel to the line through the points with coordinates \(( 9,4 )\) and \(( 3,8 )\).
[0pt] [4 marks]
SPS SPS SM 2021 November Q3
4 marks
3. An arithmetic sequence has first term \(a\) and common difference \(d\).
The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. Show that \(4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }\)
[0pt] [4 marks]
SPS SPS SM 2021 November Q4
2 marks
4. Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
[0pt] [2 marks]
SPS SPS SM 2021 November Q5
5 marks
5.
\(\mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)
Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\)
[0pt] [2 marks]
L
L
L
L
L
L Factorise \(\mathrm { p } ( x )\) completely.
[0pt] [3 marks]
L
L
L
L
L
SPS SPS SM 2021 November Q6
3 marks
6. You are not allowed to use a calculator for this question. Show detailed reasoning. Show that \(\frac { 5 \sqrt { 2 } + 2 } { 3 \sqrt { 2 } + 4 }\) can be expressed in the form \(m + n \sqrt { 2 }\), where \(m\) and \(n\) are integers.
[0pt] [3 marks]
SPS SPS SM 2021 November Q7
4 marks
7. The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
Find the possible values of the constant \(k\)
Fully justify your answer.
[0pt] [4 marks]
SPS SPS SM 2021 November Q8
10 marks
8. The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\).
    1. Write down the radius of the circle.
      [0pt] [1 mark]
  2. (ii) Write down the coordinates of \(C\).
    [0pt] [1 mark]
  3. The point \(P ( 5 , - 1 )\) lies on the circle. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = m x + c\)
    [0pt] [4 marks]
  4. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
    [0pt] [4 marks]
SPS SPS SM 2021 November Q9
4 marks
9. David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.
  1. Using David's model:
    1. state the population of rabbits on the island on 1 January 2016;
  3. (ii) predict the population of rabbits on 1 January 2021.
  4. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
  5. Give one reason why David's model may not be appropriate.
    [0pt] [1 mark]
  6. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where \(t\) is the time in years after 1 January 2016.
    Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
    [0pt] [3 marks]
SPS SPS FM 2022 November Q2
  1. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
    (b) Find the value of \(x\) such that
$$\frac { 1 + x } { x } = \sqrt { 3 }$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
[0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8940254-0663-413e-a802-71519742cfcc-06_597_977_130_351} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
      [0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8940254-0663-413e-a802-71519742cfcc-08_721_982_114_347} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
  2. Find an equation for \(l\).
  3. Find the coordinates of the point where \(l\) intersects \(C\) again.
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q5
5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$ [BLANK PAGE]
SPS SPS FM 2022 November Q6
6. A sequence of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(\left\{ \begin{array} { c } u _ { 1 } = 1
u _ { n + 1 } = 3 u _ { n } + 2 \end{array} ( n \geq 1 ) \right.\)
Prove by induction that \(u _ { n } = 2 \left( 3 ^ { n - 1 } \right) - 1\).
(4)
[0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q7
7. (a) Sketch on the same diagram in the space provided the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$ [BLANK PAGE]
SPS SPS FM 2022 November Q8
8. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.
  1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on circle \(C\),
  2. find the coordinates of the centre of \(C\),
  3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y = k$$ where \(a , b\) and \(k\) are integers to be found.
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SPS SPS FM 2022 January Q1
1. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathbf { A } ^ { - 1 }\).
    [0pt] [BLANK PAGE]
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) \(\quad \arg ( z - 3 ) = \arg a\).
    [0pt] [BLANK PAGE]
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 \(A B\).
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 January Q4
4. Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
[0pt] [BLANK PAGE]
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 \(b\).
    [0pt] [BLANK PAGE]
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 \(\pi\).
[0pt] [BLANK PAGE]
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 SM 2022 January Q1
1.
  1. Express \(\frac { 21 } { \sqrt { 7 } }\) in the form \(k \sqrt { 7 }\).
  2. Express \(8 ^ { - \frac { 1 } { 3 } }\) as an exact fraction in its simplest form.