SPS SPS FM (SPS FM) 2023 October

1. This question requires detailed reasoning.

Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
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2. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(\sin \frac { 1 } { 2 } x = 0.8\)
2. \(\sin x = 3 \cos x\)
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3.

1. Sketch the curve \(y = - \frac { 1 } { x }\).
2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).
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4. In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), band \(c\) are integers.
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1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.
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6. In this question you must show detailed reasoning. The functions f and g are defined for all real values of \(x\) by $$f ( x ) = x ^ { 3 } \text { and } g ( x ) = x ^ { 2 } + 2$$

1. Write down expressions for
   1. \(\mathrm { fg } ( x )\),
   2. \(\mathrm { gf } ( x )\).
2. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).
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7. The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q \sqrt { r }\) where \(p , q\) and \(r\) are integers.
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8. Prove that \(2 ^ { 3 n } - 3 ^ { n }\) is divisible by 5 for all integers \(n \geq 1\).
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9. (a)
\includegraphics[max width=\textwidth, alt={}, center]{9c377549-1fbd-4790-b57e-37ec9707c9d8-20_540_529_157_278} The shape \(A B C\) shown in the diagram is a student's design for the sail of a small boat. The curve \(A C\) has equation \(y = 2 \log _ { 2 } x\) and the curve \(B C\) has equation \(y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3\). State the \(x\)-coordinate of point \(A\).
(b) Determine the \(x\)-coordinate of point \(B\).
(c) By solving an equation involving logarithms, show that the \(x\)-coordinate of point \(C\) is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units \(^ { 2 }\).
(d) Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.
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