SPS SPS SM (SPS SM) 2022 October

1. (a) Sketch the curve \(y = 3 ^ { - x }\)
   (b) Solve the inequality \(3 ^ { - x } < 27\)
2. (a) Complete the square for \(1 - 4 x - x ^ { 2 }\)
   (b) Sketch the curve \(y = 1 - 4 x - x ^ { 2 }\), including the coordinates of any maximum or minimum points and the y intercept only.
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3. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010 .

The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
(a) the number of houses built in 1986, the first year of the building programme,
(b) the total number of houses built in the 25 years of the programme.
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4. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$ [BLANK PAGE]

5. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .

1. Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\).
2. Find, to 2 decimal places, the difference between the 5th and 6th term.
3. Calculate the sum of the first 7 terms. The sum of the first \(n\) terms of the series is greater than 300 .
4. Calculate the smallest possible value of \(n\).
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6. \begin{displayquote} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Solve the equation \end{displayquote} $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$ [BLANK PAGE]

7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 61 }\)
3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)
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8. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
2. Find the range of possible values for \(k\).
   [0pt] [BLANK PAGE] \section*{9. In this question you must show detailed reasoning.} The centre of a circle C is the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\).
   Show that L is a tangent to C .
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