CAIE P1 2011 November — Question 1 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2011
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeSingle coefficient given directly
DifficultyModerate -0.5 This is a straightforward application of the binomial theorem requiring identification of the correct term and solving a simple equation. While it involves a reciprocal term that students must handle carefully, it's a standard textbook exercise with no conceptual difficulty beyond basic binomial expansion mechanics.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n

The coefficient of \(x^2\) in the expansion of \(\left(k + \frac{1}{x}\right)^5\) is 30. Find the value of the constant \(k\). [3]

\(k^3 \times \left(\frac{1}{3}\right)^2 \times 10\) (or correct factorials)
AnswerMarks Guidance
\(10 \times k^3 \times \frac{1}{9} = 30 \Rightarrow k = 3\)B2, B1 B1 for 2/3 terms correct; cao
$k^3 \times \left(\frac{1}{3}\right)^2 \times 10$ (or correct factorials)

$10 \times k^3 \times \frac{1}{9} = 30 \Rightarrow k = 3$ | B2, B1 | B1 for 2/3 terms correct; cao

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The coefficient of $x^2$ in the expansion of $\left(k + \frac{1}{x}\right)^5$ is 30. Find the value of the constant $k$. [3]

\hfill \mbox{\textit{CAIE P1 2011 Q1 [3]}}