SPS SPS FM (SPS FM) 2024 October

1. Given the function \(f ( x ) = x - x ^ { 2 }\), defined for all real values of \(x\),
   1. Show that \(f ^ { \prime } ( x ) = 1 - 2 x\) by differentiating \(f ( x )\) from first principles.
   2. Find the maximum value of \(f ( x )\).
   3. Explain why \(f ^ { - 1 } ( x )\) does not exist.
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   4. The quadratic equation \(k x ^ { 2 } + 2 k x + 2 k = 3 x - 1\), where \(k\) is a constant, has no real roots.
   5. Show that \(k\) satisfies the inequality
   $$4 k ^ { 2 } + 16 k - 9 > 0$$
2. Hence find the set of possible values of \(k\). Give your answer in set notation.
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3. (a) Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
(b) In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.
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4. The curve \(y = \sqrt { 2 x - 1 }\) is stretched by scale factor \(\frac { 1 } { 4 }\) parallel to the \(x\)-axis and by scale factor \(\frac { 1 } { 2 }\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt { a x - b }\) where \(a\) and \(b\) are rational numbers.
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1. (a) Show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
   (b) Hence find the exact roots of the equation \(\mathrm { f } ( x ) = 0\).
2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( x ) = 0\).
   (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
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6. The first three terms of a geometric sequence are $$u _ { 1 } = 3 k + 4 \quad u _ { 2 } = 12 - 3 k \quad u _ { 3 } = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of \(k\), giving a reason for your answer.
2. Find the value of \(\sum _ { r = 2 } ^ { \infty } u _ { r }\)
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7. The diagram shows part of the graph of \(y = x ^ { 2 }\). The normal to the curve at the point \(A ( 1,1 )\) meets the curve again at \(B\). Angle \(A O B\) is denoted by \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-16_506_741_283_217}

1. Determine the coordinates of \(B\).
2. Hence determine the exact value of \(\tan \alpha\).
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8. Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).
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9. A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(A B\) is \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-20_776_730_280_214} The equation of the circle is \(x ^ { 2 } - 6 x + y ^ { 2 } + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of M and hence show that the area of triangle ABC is \(\frac { 3 } { 2 } \sqrt { 9 - 2 a }\).
2. 1. Find the value of \(a\) when the area of triangle \(A B C\) is zero.
   2. Give a geometrical interpretation of the case in part (b)(i).
3. Give a geometrical interpretation of the case where \(a = 5\).
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