SPS SPS SM (SPS SM) 2020 October

1. Simplify fully the following expressions:
   i. \(\frac { 7 y ^ { 13 } } { 35 y ^ { 7 } }\)
   ii. \(6 x ^ { - 2 } \div x ^ { - 5 }\)
2. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { 1 } = 7\) and \(u _ { n + 1 } = u _ { n } + 4\) for \(n \geq 1\).
   i. State what type of sequence this is.
   ii. Find \(u _ { 17 }\).
3. i. Write \(3 x ^ { 2 } - 6 x + 1\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
   ii. Solve \(3 x ^ { 2 } - 6 x + 1 \leq 0\), giving your answer in set notation. In this question you must show detailed reasoning.
   i. Express \(\frac { \sqrt { 2 } } { 1 - \sqrt { 2 } }\) in the form \(c + d \sqrt { } e\), where \(c\) and \(d\) are integers and \(e\) is a prime number.
   ii. Solve the equation \(\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8\).
4. Let \(a = \log _ { 2 } x , b = \log _ { 2 } y\) and \(c = \log _ { 2 } z\).

Express \(\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)\) in terms of \(a , b\) and \(c\).

6. i. A student was asked to solve the equation \(2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0\). The student's attempt is written out below. $$\begin{gathered} 2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0
2 ^ { 2 x } + 2 ^ { 4 } - 9 \left( 2 ^ { x } \right) = 0
\text { Let } y = 2 ^ { x }
y ^ { 2 } - 9 y + 8 = 0
( y - 8 ) ( y - 1 ) = 0
y = 8 \text { or } y = 1
\text { So } x = 3 \text { or } x = 0 \end{gathered}$$ Identify the two mistakes that the student has made.
ii. Solve the equation \(2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0\), giving your answer in exact form.

7. i. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided below.
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ii. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
iii. Hence, solve the inequality \(\frac { 3 } { x ^ { 2 } } \leq x ^ { 2 } - 2\), giving your answer in interval notation.

9. In this question you must show detailed reasoning. Solve the following simultaneous equations: $$\begin{gathered} \left( \log _ { 3 } x \right) ^ { 2 } + \log _ { 3 } \left( y ^ { 2 } \right) = 5
\log _ { 3 } \left( \sqrt { 3 } x y ^ { - 1 } \right) = 2 \end{gathered}$$

1. In this question you must show detailed reasoning.

A sequence \(t _ { 1 } , t _ { 2 } , t _ { 3 } \ldots\) is defined by \(t _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that $$\sum _ { n = 1 } ^ { \infty } t _ { n } - \sum _ { n = 1 } ^ { N } t _ { n } < 10 ^ { - 4 }$$