SPS SPS FM (SPS FM) 2022 October

1.
a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 5 }$$ giving each term in its simplest form.
b) Hence write down the first 3 terms, in ascending powers of \(y\), of the binomial expansion of $$\left( 2 + 3 y ^ { \frac { 3 } { 2 } } \right) ^ { 5 }$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$ [BLANK PAGE]

3.

1. Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
2. Sketch the graph of \(y = \frac { 3 } { x }\) in the space provided and write down the equations of any asymptotes.
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4. Prove, from first principles, that if \(f ( x ) = 2 x ^ { 2 } - 5 x + 2\) then \(f ^ { \prime } ( x ) = 4 x - 5\).
(3)
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5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$ [BLANK PAGE]

6. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm}\), \(C D = 8 \mathrm {~cm} , A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).

1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
2. Find the size of \(\angle B C D\) in degrees.
3. Find the area of quadrilateral \(A B C D\).
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7. (i) Solve the inequality \(| 2 x + 1 | \leqslant | x - 3 |\).
(ii) Given that \(x\) satisfies the inequality \(| 2 x + 1 | \leqslant | x - 3 |\), find the greatest possible value of \(| x + 2 |\).
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8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),

1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
2. find the value of \(a\),
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9. Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
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10. In this question you must show detailed reasoning. The centre of a circle C is the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\).
Show that L is a tangent to C .
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