SPS SPS FM (SPS FM) 2026 November

4. (a) The curves \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 1\) and \(2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { 2 y } = 41\) intersect at the point \(P\). Show that the \(y\)-coordinate of \(P\) satisfies the equation \(3 \mathrm { e } ^ { 2 y } + 4 \mathrm { e } ^ { y } - 39 = 0\).
(b) In this question you must show detailed reasoning. Hence find the exact coordinates of \(P\).
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8. Prove by induction that \(7 \times 9 ^ { n } - 15\) is divisible by 4 , for all integers \(n \geq 0\).
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10. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
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