CAIE P1 2014 June — Question 2 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeProduct with reciprocal term binomial
DifficultyStandard +0.3 This requires expanding a binomial with reciprocal terms and identifying which term combinations yield x², involving systematic application of the binomial theorem and algebraic manipulation. While it requires careful bookkeeping of powers and multiple terms, it's a standard textbook exercise with no novel insight needed, making it slightly above average difficulty.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).
AnswerMarks Guidance
\((1+x^2)\left(\frac{1}{2}-\frac{1}{4}\right)^n\)
Term in \(x^2 = 15 \times \frac{1}{16} \times (-4)^2 = 15\)B1 B1 B1 unsimplified. B1 15.
Constant term = \(20 \times \frac{1}{8} \times (-4)^3 = -160\)B1 B1 B1 unsimplified. B1 −160
Coefficient of \(x^2 = -145\)B1✓ [5] Uses 2 terms. ✓ on previous answers 
$(1+x^2)\left(\frac{1}{2}-\frac{1}{4}\right)^n$ | | |
Term in $x^2 = 15 \times \frac{1}{16} \times (-4)^2 = 15$ | B1 B1 | B1 unsimplified. B1 15.
Constant term = $20 \times \frac{1}{8} \times (-4)^3 = -160$ | B1 B1 | B1 unsimplified. B1 −160
Coefficient of $x^2 = -145$ | B1✓ [5] | Uses 2 terms. ✓ on previous answers
2 Find the coefficient of $x ^ { 2 }$ in the expansion of $\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }$.

\hfill \mbox{\textit{CAIE P1 2014 Q2 [5]}}
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Q1 5 Q2 5 Q3 5 Q4 5 Q5 7 Q6 7 Q7 7 Q8 8 Q9 11 Q10 15