SPS SPS SM (SPS SM) 2022 October

1 Simplify \(\left( \frac { x ^ { 12 } } { 16 } \right) ^ { - \frac { 3 } { 4 } }\)

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.
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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 \(k\).
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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]

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 \(d\).
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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]

8. \begin{figure}[h] \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 \(p\).
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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]

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.
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