SPS SPS FM (SPS FM) 2022 October

1. a) Rationalise the denominator for \(\frac { \sqrt { 8 } + 2 } { 5 - \sqrt { 2 } }\)
   b) Solve

$$( \sqrt { 2 } ) ^ { x + 1 } = \frac { 1 } { 4 ^ { 4 - 3 x } }$$ [BLANK PAGE]

2. Given that $$f ( x ) = \ln x , x > 0$$ Sketch on separate axes the graphs of
i) \(y = f ( x )\)
ii) \(\quad y = f ( x - 4 )\) Show on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.

3. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
a) Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\) The sum of the first n terms of the series is greater than 300 .
b) Calculate the smallest possible value of n
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4. Let \(f ( x )\) be given by: $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$ a) Use the factor theorem to show that ( \(x + 3\) ) is a factor of \(f ( x )\)
b) Factorise \(f ( x )\) into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation \(f ( x ) = 0\), showing detailed reasoning with your working
c) Hence write down the one solution to the equation $$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$ in the form \(\ln ( a + \sqrt { b } )\)
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5. Solve, for \(0 < \theta < 360 ^ { \circ }\),
a) \(5 \cos ( \theta + 30 ) = 3\)
b) \(\cos ^ { 2 } ( x ) + 4 \sin ^ { 2 } ( x ) + 4 \sin ( x ) = 0\) Give each non-exact solution to one decimal place.
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6. The curve \(C\) has the equation \(y = 6 x ^ { 2 } + 2 \sqrt { x }\). Find the equation of the normal of the curve at the point where \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a x + b y = k\) where \(a , b\) and \(k\) are positive integers. For this question, show detailed reasoning with your working
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7. A sequence of positive integers is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = u _ { n } + n ( 3 n + 1 ) , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that $$u _ { n } = n ^ { 2 } ( n - 1 ) + 1 , \quad n \in \mathbb { Z } ^ { + }$$ [BLANK PAGE]

8. Given that \(k\) is a positive constant,
a) sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis
b) find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$ [BLANK PAGE]

9. a) Write the following as a single logarithm $$3 \log ( x ) - \frac { 1 } { 2 } \log ( y ) + 2$$ b) Solve \(2 ^ { x } e ^ { 3 x + 1 } = 10\) Giving your answer to (b) in the form \(\frac { \ln a + b } { \ln c + d }\), where \(a , b , c\) and \(d\) are integers.
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10. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.
a) Show that \(n k ( n - 1 ) = 252\)
b) Find the value of \(k\) and the value of \(n\).
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11.
a) Without using a calculator, show that \(5 > 3 \sqrt { 2 }\)
b) Two circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 5 y = \frac { 39 } { 4 } \text { and } x ^ { 2 } + y ^ { 2 } + 2 x - y = \frac { 3 } { 4 }$$ respectively. Show that \(C _ { 2 }\) lies completely inside \(C _ { 1 }\)
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