SPS SPS FM (SPS FM) 2022 November

1. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
   (b) Find the value of \(x\) such that

$$\frac { 1 + x } { x } = \sqrt { 3 }$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
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3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).
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4. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
2. Find an equation for \(l\).
3. Find the coordinates of the point where \(l\) intersects \(C\) again.
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5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$

6. A sequence of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(\left\{ \begin{array} { c } u _ { 1 } = 1 \\ u _ { n + 1 } = 3 u _ { n } + 2 \end{array} ( n \geq 1 ) \right.\) Prove by induction that \(u _ { n } = 2 \left( 3 ^ { n - 1 } \right) - 1\).
(4)
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7. (a) Sketch on the same diagram in the space provided the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

8. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.

1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on circle \(C\),
2. find the coordinates of the centre of \(C\),
3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y = k$$ where \(a , b\) and \(k\) are integers to be found.
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