SPS SPS SM (SPS SM) 2023 October

1 In this question you must show detailed reasoning.
Find the smallest positive integers m and n such that \(\left( \frac { 64 } { 49 } \right) ^ { - \frac { 3 } { 2 } } = \frac { m } { n }\)

2 In this question you must show detailed reasoning. Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that

• \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression in \(x\)
• \(C\) cuts the \(x\)-axis at 0 and 6
• \(l\) cuts the \(y\)-axis at 60 and intersects \(C\) at the point \(( 10,80 )\)
  use inequalities to define the region \(R\) shown shaded in Figure 3.
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1. In this question you must show detailed reasoning.

A curve has equation $$y = 2 x ^ { 2 } + p x + 1$$ A line has equation $$y = 5 x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points.
Give your answer in exact form using set notation.
(6)
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5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$

1. Show that \(u _ { 5 } = 20\).
2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
3. State what type of sequence it is.
4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).
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6. In part (ii) of this question you must show detailed reasoning.

1. Use logarithms to solve the equation \(8 ^ { 2 x + 1 } = 24\), giving your answer to 3 decimal places.
   (2)
2. Find the values of \(y\) such that $$\log _ { 2 } ( 11 y - 3 ) - \log _ { 2 } 3 - 2 \log _ { 2 } y = 1 , \quad y > \frac { 3 } { 11 }$$ [BLANK PAGE]

7. (a) Sketch the curve with equation $$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$ [BLANK PAGE] \section*{8. In this question you must show detailed reasoning.} The curve \(C _ { 1 }\) has equation \(y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }\)
The curve \(C _ { 2 }\) has equation \(y = x ^ { 2 } - 12 x + 14\)
(a) Verify that when \(x = 1\) the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect. The curves also intersect when \(x = k\).
Given that \(k < 0\)
(b) use algebra to find the exact value of \(k\).
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9. The first term of a geometric progression is 10 and the common ratio is 0.8 .

1. Find the fourth term.
2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002$$ and use logarithms to find the smallest possible value of \(N\).
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10. In this question you must show detailed reasoning.
A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 12 = 0\). Two tangents to this circle pass through the point \(( 0,1 )\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same.
Find the angle between these two tangents.
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