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 2022 October Q4
4. Let \(f ( x )\) be given by: $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$ a) Use the factor theorem to show that ( \(x + 3\) ) is a factor of \(f ( x )\)
b) Factorise \(f ( x )\) into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation \(f ( x ) = 0\), showing detailed reasoning with your working
c) Hence write down the one solution to the equation $$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$ in the form \(\ln ( a + \sqrt { b } )\)
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q5
5. Solve, for \(0 < \theta < 360 ^ { \circ }\),
a) \(5 \cos ( \theta + 30 ) = 3\)
b) \(\cos ^ { 2 } ( x ) + 4 \sin ^ { 2 } ( x ) + 4 \sin ( x ) = 0\) Give each non-exact solution to one decimal place.
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q6
6. The curve \(C\) has the equation \(y = 6 x ^ { 2 } + 2 \sqrt { x }\). Find the equation of the normal of the curve at the point where \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a x + b y = k\) where \(a , b\) and \(k\) are positive integers. For this question, show detailed reasoning with your working
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q7
7. A sequence of positive integers is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = u _ { n } + n ( 3 n + 1 ) , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that $$u _ { n } = n ^ { 2 } ( n - 1 ) + 1 , \quad n \in \mathbb { Z } ^ { + }$$ [BLANK PAGE]
SPS SPS FM 2022 October Q8
8. Given that \(k\) is a positive constant,
a) sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis
b) find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$ [BLANK PAGE]
SPS SPS FM 2022 October Q9
9. a) Write the following as a single logarithm $$3 \log ( x ) - \frac { 1 } { 2 } \log ( y ) + 2$$ b) Solve \(2 ^ { x } e ^ { 3 x + 1 } = 10\) Giving your answer to (b) in the form \(\frac { \ln a + b } { \ln c + d }\), where \(a , b , c\) and \(d\) are integers.
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q10
10. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.
a) Show that \(n k ( n - 1 ) = 252\)
b) Find the value of \(k\) and the value of \(n\).
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q11
11.
a) Without using a calculator, show that \(5 > 3 \sqrt { 2 }\)
b) Two circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 5 y = \frac { 39 } { 4 } \text { and } x ^ { 2 } + y ^ { 2 } + 2 x - y = \frac { 3 } { 4 }$$ respectively. Show that \(C _ { 2 }\) lies completely inside \(C _ { 1 }\)
[0pt] [BLANK PAGE] End of Examination
SPS SPS SM 2022 October Q1
1 Simplify \(\left( \frac { x ^ { 12 } } { 16 } \right) ^ { - \frac { 3 } { 4 } }\)
SPS SPS SM 2022 October Q2
2 A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = - 3 x ^ { 2 } + 12 x + 8$$
  1. Write \(f ( x )\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The curve \(C\) has a maximum turning point at \(M\).
  2. Find the coordinates of \(M\). Solutions relying on calculator technology are not acceptable. Simplify $$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$ giving your answer in the form \(b \sqrt { } 2 + c\), where \(b\) and \(c\) are integers.
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q4
4. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q5
5. (a) Given that $$y = \log _ { 3 } x$$ find expressions in terms of \(y\) for
  1. \(\log _ { 3 } \left( \frac { x } { 9 } \right)\)
  2. \(\log _ { 3 } \sqrt { x }\) Write each answer in its simplest form.
    (b) Hence or otherwise solve $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) - \log _ { 3 } \sqrt { x } = 2$$ [BLANK PAGE]
SPS SPS SM 2022 October Q6
6. An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54
  1. show that $$a + 4 d = 6$$ Given also that the 8th term is half the 7th term,
  2. find the values of \(a\) and \(d\).
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q7
7. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Using algebra, find all solutions of the equation $$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
  2. Hence find all real solutions of $$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$ [BLANK PAGE]
SPS SPS SM 2022 October Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba499f70-f2ee-4eff-b15c-33c3f09297f0-14_622_805_141_660} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = p m ^ { q }$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg .
Figure 2 illustrates the linear relationship between \(\log _ { 10 } h\) and \(\log _ { 10 } m\)
The line meets the vertical \(\log _ { 10 } h\) axis at 2.25 and has a gradient of - 0.235
  1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
  2. Comment on the suitability of the model for this mammal.
  3. With reference to the model, interpret the value of the constant \(p\).
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q9
9. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$ where \(k\) is a constant.
Given that
  • the sequence is a periodic sequence of order 3
  • \(a _ { 1 } = 2\)
    1. show that
$$k ^ { 2 } + k - 2 = 0$$
  • For this sequence explain why \(k \neq 1\)
  • Find the value of $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$ [BLANK PAGE]
    •
    SPS SPS SM 2022 October Q10
    10. A circle \(C\) with radius \(r\)
    • lies only in the 1st quadrant
    • touches the \(x\)-axis and touches the \(y\)-axis
    The line \(l\) has equation \(2 x + y = 12\)
    1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
    2. find the two possible values of \(r\), giving your answers as fully simplified surds.
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q1
    1. (i) Solve the equation
    $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
    (ii) Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$
    SPS SPS FM Pure 2023 November Q2
    2. $$\mathrm { f } ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
    1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
    2. Factorise f(x) completely.
    3. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$ [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q3
    3. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
    3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
      [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q4
    4. A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 2
    u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 . \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1\).
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q5
    5. (a) Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    [0pt] [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q6
    6. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$ [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q7
    7. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
    1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are integers to be found.
    2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
      1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
      2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$ [BLANK PAGE]
    SPS SPS FM Pure 2023 November Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3dcde139-bc6b-412d-8d1f-c45543d67430-16_703_851_150_701} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.
    1. Find the coordinates of \(P\).
    2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
      Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
    3. find the range of possible values of \(a\), writing your answer in set notation.
      [0pt] [BLANK PAGE]