SPS SPS FM (SPS FM) 2024 October

1. (a) (i) Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
   (ii) Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
   (b) In this question you must show detailed reasoning.

Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).
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2. (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]

3. (a) Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
(b) In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).
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4. The functions f and g are defined for all real values of \(x\) by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 6 x\) and \(\mathrm { g } ( x ) = 3 x + 2\).

1. Find the range of f.
2. Give a reason why f has no inverse.
3. Given that \(\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )\), where \(a\) is a constant, determine the value of \(a\).
4. Determine the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\). Give your answer in set notation.
   [0pt] [BLANK PAGE] \section*{5. In this question you must show detailed reasoning} Find the equation of the normal to the curve \(y = \frac { x ^ { 2 } - 32 } { \sqrt { x } }\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
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6. Given that the equation $$2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3 ,$$ only has one solution, find the value of \(x\).
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7. In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).
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8. Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geq 1\).
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9. a) Factorise \(8 x y - 4 x + 6 y - 3\) into the form \(( a x + b ) ( c y + d )\) where \(a , b , c\) and \(d\) are integers
b) Hence, or otherwise, solve $$8 \sin \left( x ^ { 2 } \right) \cos \left( e ^ { \frac { x } { 3 } } \right) - 4 \sin \left( x ^ { 2 } \right) + 6 \cos \left( e ^ { \frac { x } { 3 } } \right) - 3 = 0$$ where \(0 ^ { \circ } < x < 19 ^ { \circ }\), giving your answers to 1 decimal place.
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