SPS SPS SM (SPS SM) 2025 October

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 \(a + b \sqrt { 5 }\).

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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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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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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7. In this question you must show detailed reasoning. Solve the following equations.

1. \(\quad y ^ { 6 } + 7 y ^ { 3 } - 8 = 0\)
2. \(\quad 9 ^ { x + 1 } + 3 ^ { x } = 8\)
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8. In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.

1. \(2 \log _ { 3 } ( x + 1 ) = 1 + \log _ { 3 } ( x + 7 )\)
2. \(\log _ { y } \left( \frac { 1 } { 8 } \right) = - \frac { 3 } { 2 }\)
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9.

1. Show that the equation \(x ^ { 2 } + k x - k ^ { 2 } = 0\) has real roots for all real values of \(k\).
2. Show that the roots of the equation \(x ^ { 2 } + k x - k ^ { 2 } = 0\) are \(\left( \frac { - 1 \pm \sqrt { 5 } } { 2 } \right) k\).
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10. \(f ( x ) = x ^ { 4 } + b x + c\)
\(( x - 2 )\) is a factor of \(f ( x )\).
\(f ( - 3 ) = 35\).

1. Find b and c.
2. Hence express \(\mathrm { f } ( \mathrm { x } )\) as the product of linear and cubic factors.
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11. A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M \mathrm {~kg}\) of salt remaining after \(t\) minutes by \(M = a k ^ { t }\) where \(a\) and \(k\) are constants.

1. Show that the model for \(M\) can be rewritten in the form \(\log _ { 10 } M = t \log _ { 10 } k + \log _ { 10 } a\). The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.

   \(t\) minutes	8	13	21	35	50
   \(M \mathrm {~kg}\)	0.4	0.3	0.2	0.1	0.05

   The student uses this data and plots \(y = \log _ { 10 } M\) against \(x = t\) using graph drawing software. The software gives \(y = - 0.0214 x - 0.2403\) for the equation of the line of best fit.
2. 1. Find the values of \(a\) and \(k\) that follow from the equation of the line.
   2. Interpret the value of \(k\) in context.
3. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg . Comment on the accuracy when the model is used to estimate the initial mass of the salt.
4. Use the model to predict the value of \(t\) at which \(M = 0.01 \mathrm {~kg}\).
5. Rewrite the model for \(M\) in the form \(M = a \mathrm { e } ^ { - h t }\) where \(h\) is a constant to be determined.
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12. An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).

1. 1. Show that \(r = \frac { a + 2 d } { a }\).
   2. Find \(d\) in terms of \(a\).
2. Find the common ratio of the geometric progression.
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13. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 4 y + 1 = 0$$

1. Find
   1. the coordinates of the centre of \(C\)
   2. the exact radius of \(C\) The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
2. Find the range of values for \(k\), giving your answer in set notation.
3. The line with equation \(y = m x + 4\) is a tangent to C . Find possible exact values of m .
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