CAIE P1 2006 November — Question 1 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2006
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeBinomial with negative or fractional powers of x
DifficultyModerate -0.8 This is a straightforward application of the binomial theorem requiring identification of the correct term where powers of x sum to x². The calculation involves one binomial coefficient and simple arithmetic, making it easier than average with minimal problem-solving required.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 6 }\).

AnswerMarks Guidance
\(\left(x + \frac{2}{x}\right)^3\) Term in \(x^2\) has \(\binom{3}{C_2}\) - needs factorials or 15. \((3)^1 \times (2x)^2\) \(\rightarrow 60\) (needs selecting) (first 2 marks can be obtained from expansion only)B1 B1 B1 [3] Binomial coeff. Needs \(2^3 \cdot (2x)^2 = 2x^2\) gets B1. Needs () co.
$\left(x + \frac{2}{x}\right)^3$ Term in $x^2$ has $\binom{3}{C_2}$ - needs factorials or 15. $(3)^1 \times (2x)^2$ $\rightarrow 60$ (needs selecting) (first 2 marks can be obtained from expansion only) | B1 B1 B1 [3] | Binomial coeff. Needs $2^3 \cdot (2x)^2 = 2x^2$ gets B1. Needs () co.

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1 Find the coefficient of $x ^ { 2 }$ in the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 6 }$.

\hfill \mbox{\textit{CAIE P1 2006 Q1 [3]}}