SPS SPS FM (SPS FM) 2025 October

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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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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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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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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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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9. \begin{figure}[h] \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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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. 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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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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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]

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]