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SPS SPS SM Pure 2025 February Q7
7. (i) Prove by counter example that the statement
"If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number."
is false.
(ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then $$m ^ { 2 } - 1$$ is divisible by 3
(3)
(Total for Question 7 is 5 marks)
SPS SPS SM Pure 2025 February Q8
8. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Show that \(\sin 3 x\) can be written in the form $$P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be found.
  2. Hence or otherwise, solve, for \(0 < \theta \leq 360 ^ { \circ }\), the equation $$2 \sin 3 \theta = 5 \sin 2 \theta$$ giving your answers, in degrees, to one decimal place as appropriate.
SPS SPS SM Pure 2025 February Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-18_542_551_212_790} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geq - \frac { 7 } { 4 }$$
  1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
  2. Find the coordinates of \(P\).
  3. Hence find the range of the function g defined by $$\operatorname { g } ( x ) = - 4 \mathrm { f } ( x ) \quad x \geq - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
  4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 f \left( x - \frac { 3 } { 2 } \right) - 8$$
SPS SPS SM Pure 2025 February Q10
10. The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\)
The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}
  1. State the value of \(a\)
    1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
    2. Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$
  3. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. CONTINUE YOUR ANSWER HERE CONTINUE YOUR ANSWER HERE
SPS SPS SM Pure 2025 February Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-26_462_586_148_593} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The heart rate of a horse is being monitored.
The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}
  1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
  2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
  3. Find the value of \(T\), giving your answer to 3 decimal places.
    (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
  4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
    1. find, to 4 decimal places, the value of \(t _ { 2 }\)
    2. find, to 4 decimal places, the value of \(M\)
SPS SPS SM Pure 2025 February Q12
12.
  1. Show that the first two terms of the binomial expansion of \(\sqrt { 4 - 2 x ^ { 2 } }\) are $$2 - \frac { x ^ { 2 } } { 2 }$$
  2. State the range of values of \(x\) for which the expansion found in part (a) is valid.
  3. Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } d x$$ giving your answer to five decimal places.
    Fully justify your answer.
  4. A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } d x$$ Explain why this may not be a suitable method.
SPS SPS SM Pure 2025 February Q13
13. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that $$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + C$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
(Total for Question 13 is 4 marks)
SPS SPS SM Pure 2025 February Q14
14.
  1. \(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\) Find \(\frac { d y } { d x }\) and simplify your answer.
  2. Hence, show that $$\int e ^ { - x } \sin x d x = a e ^ { - x } ( \sin x + \cos x ) + c$$ where \(a\) is a rational number.
  3. A sketch of the graph of \(y = \mathrm { e } ^ { - x } \sin x\) for \(x \geq 0\) is shown below. The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by \(A _ { 1 }\), \(A _ { 2 } , \ldots , A _ { n } , \ldots\)
    \includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-34_807_1246_959_406}
    1. Find the exact value of the area \(A _ { 1 }\)
    2. Show that $$\frac { A _ { 2 } } { A _ { 1 } } = e ^ { - \pi }$$
    3. Given that $$\frac { A _ { n + 1 } } { A _ { n } } = e ^ { - \pi }$$ show that the exact value of the total area enclosed between the curve and the \(x\)-axis is $$\frac { 1 + e ^ { \pi } } { 2 \left( e ^ { \pi } - 1 \right) }$$
SPS SPS SM Pure 2025 February Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-38_540_741_169_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leq x \leq q$$ where \(q\) is a constant to be found.
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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    SPS SPS FM 2025 October Q12
    12. Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ [BLANK PAGE]
    SPS SPS FM 2025 October Q13
    13. In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\) $$1 + \log _ { 3 } \left( 1 + \tan ^ { 2 } 2 x \right) = 2 \log _ { 3 } ( - 4 \sin 2 x )$$ [BLANK PAGE]
    SPS SPS FM 2025 October Q1
    6 marks
    1. The graph of \(y = f ( x )\), defined for \(- 3 \leq x \leq 7\), is shown below, along with the coordinates of the turning points and endpoints:
      \includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-02_1157_1584_1539_319}
      1. How many solutions are there to \(f ( x ) = 1\) ?
      2. If \(f ( x ) = k\) has three distinct solutions, find the possible values of \(k\).
      3. How many solutions are there to \(f \left( x ^ { 2 } \right) = 1\) ?
      4. If \(f \left( x ^ { 2 } \right) = k\) has five distinct solutions, find the value of \(k\).
      5. How many solutions are there to \([ f ( x ) ] ^ { 2 } = 2\) ?
      6. If \([ f ( x ) ] ^ { 2 } = k\) has six distinct solutions, find the range of possible values of \(k\).
      7. How many solutions are there to \(\log _ { 2 } f ( x ) = - 2025\) ?
      8. How many solutions are there to \(\log _ { 2 } \left( [ f ( x ) ] ^ { 2 } \right) = 0\) ?
      9. Show that, if \(n\) is a non-negative integer, \(4 ^ { 3 n } + 5 ^ { 2 n + 2 }\) cannot be a prime.
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      10. All of these questions concern the curve \(y = g ( x )\).
      Part of the graph of \(y = g ^ { \prime \prime } ( x )\) is shown below:
      \includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-06_1083_1744_258_258} You are given that \(y = g ( x )\) has exactly two local minima and one local maximum in this range.
    2. Identify which of the labelled points could correspond to the local maximum.
    3. Identify two of the labelled points which could correspond to the local minima. There is more than one possible pair of answers but you are only required to give one.
    4. Identify all of the labelled points which correspond to points of inflection.
    5. As \(x \rightarrow - \infty , g ^ { \prime \prime } ( x ) \rightarrow 0\). What does this tell you about the shape of the curve \(y = g ( x )\) as \(x \rightarrow - \infty\) ?
      [0pt] [BLANK PAGE] \section*{4. In this question you must show detailed reasoning} The non-zero coefficients of \(x , x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 1 + x ) ^ { n }\) form the first, second and third terms of an arithmetic sequence (in that order).
    6. Determine the possible value(s) of \(n\).
    7. For the same value(s) of \(n\), there is another value of \(a\) for which \(( 1 + a x ) ^ { n }\) has this property. Determine this value of \(a\).
    SPS SPS FM 2025 October Q5
    5. Circles \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) have collinear centres.
    \(C _ { 3 }\) is tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
    The equations for \(C _ { 1 }\) and \(C _ { 2 }\) are as follows:
    \(C _ { 1 } : x ^ { 2 } + y ^ { 2 } + 4 x - 60 = 0\)
    \(C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 14 x + 40 = 0\)
    Find all possible equations for \(C _ { 3 }\).
    [0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} You are given that \(P _ { 0 } , P _ { 1 }\) and \(Q\) are the vertices of a right-angled triangle with hypotenuse \(P _ { 0 } Q\). The length of \(P _ { 0 } Q\) is 1 .
    \(P _ { 2 }\) is the foot of the perpendicular from \(P _ { 1 }\) to \(P _ { 0 } Q\).
    \(P _ { 3 }\) is the foot of the perpendicular from \(P _ { 2 }\) to \(P _ { 1 } Q\).
    The infinite set of points \(P _ { 4 } , P _ { 5 } , P _ { 6 } , \ldots\) is defined similarly.
    The angle at \(P _ { 0 }\) is \(\theta\).
    The diagram below shows the points up to \(P _ { 5 }\) :
    \includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-12_848_1065_662_571}
    1. Given that $$\sum _ { r = 0 } ^ { \infty } P _ { 2 r } P _ { 2 r + 1 } = 2$$ find the value of \(\theta\).
    2. For this value of \(\theta\), evaluate the following: $$\sum _ { r = 0 } ^ { \infty } P _ { 2 r + 1 } P _ { 2 r + 2 }$$
    3. If instead \(\theta = 45 ^ { \circ }\), determine the smallest value of \(n\) for which: $$\sum _ { r = 0 } ^ { n } P _ { 2 r + 1 } P _ { 2 r + 2 } > 0.999$$ [BLANK PAGE]
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    SPS SPS SM 2025 October Q1
    1. Express each of the following in the form \(p x ^ { q }\), where \(p\) and \(q\) are constants.
      1. \(\frac { 2 } { \sqrt [ 4 ] { x } }\)
      2. \(\quad ( 5 x \sqrt { x } ) ^ { 3 }\)
      3. \(\sqrt { 2 x ^ { 3 } } \times \sqrt { 8 x ^ { 5 } }\)
      4. \(\quad x ^ { 5 } \left( 27 x ^ { 6 } \right) ^ { \frac { 2 } { 3 } }\)
      5. In this question you must show detailed reasoning.
      Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).
    SPS SPS SM 2025 October Q3
    3. The line \(l\) passes through the points \(A ( - 3,0 )\) and \(B \left( \frac { 5 } { 2 } , 22 \right)\)
    1. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2fa9e78c-8210-456c-9b70-5378609ac47d-04_728_959_447_625} \captionsetup{labelformat=empty} \caption{Figure 2}
      \end{figure} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\).
      Given that
      • \(C\) has equation \(y = 2 x ^ { 2 } + 5 x - 3\)
      • the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)
      • use inequalities to define \(R\).
        (2)
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    SPS SPS SM 2025 October Q4
    1. (a) A sequence has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = u _ { n } ^ { 2 } - 5\) for \(n \geqslant 1\).
      1. Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
      2. Describe the behaviour of the sequence.
        (b) The second, third and fourth terms of a geometric progression are 12,8 and \(\frac { 16 } { 3 }\). Determine the sum to infinity of this geometric progression.
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      3. In this question you must show detailed reasoning.
        \includegraphics[max width=\textwidth, alt={}, center]{2fa9e78c-8210-456c-9b70-5378609ac47d-08_464_645_210_283}
      The diagram shows the cuboid \(A B C D E F G H\) where \(A D = 3 \mathrm {~cm} , A F = ( 2 x + 1 ) \mathrm { cm }\) and \(D C = ( x - 2 ) \mathrm { cm }\). The volume of the cuboid is at most \(9 \mathrm {~cm} ^ { 3 }\).
      Find the range of possible values of \(x\). Give your answer in interval notation.
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    SPS SPS SM 2025 October Q6
    6. Sketch the graph of $$y = ( x - k ) ^ { 2 } ( x + 2 k )$$ where \(k\) is a positive constant.
    Label the coordinates of the points where the graph meets the axes.
    \includegraphics[max width=\textwidth, alt={}, center]{2fa9e78c-8210-456c-9b70-5378609ac47d-10_1253_1207_596_395}
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