SPS SPS FM 2024 February — Question 5

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionFebruary
TopicPartial Fractions

5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the position vector of \(P\).
  2. Find, correct to 1 decimal place, the acute angle between \(/ _ { 1 }\) and \(/ _ { 2 }\).
    \(Q\) is a point on \(/ 1\) which is 12 metres away from \(P \cdot R\) is the point on \(/ 2\) such that \(Q R\) is perpendicular to \(/ 1\).
  3. Determine the length \(Q R\).
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    5. (a) Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.

  4. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
  5. State the set of values for which the expansion in part (b) is valid.
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