CAIE P1 2011 June — Question 1 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2011
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeFind constant from coefficient condition
DifficultyModerate -0.5 This is a straightforward application of the binomial theorem requiring students to find two coefficients using the standard formula, set up an equation, and solve for a. While it involves two expansions and equation solving, the steps are routine and mechanical with no conceptual challenges beyond basic binomial coefficient calculation.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 1 - 2 x ) ^ { 6 }\), where \(a\) is positive, is 90 . Find the value of \(a\).
\((a+x)^5 + (1-2x)^6\)
AnswerMarks Guidance
Coeff of \(x^3\) in \(1^st = 10 \times a^2\)B1 co
Coeff of \(x^3\) in \(2^{nd} = 20 \times (-2)^3\)B1 + B1 co
\(\rightarrow 10a^2 - 160 = 90\)M1 Forming an equation for \(a\) + solution
\(\rightarrow a = 5\)A1 co (condone ±)
[5] 
$(a+x)^5 + (1-2x)^6$

Coeff of $x^3$ in $1^st = 10 \times a^2$ | B1 | co
Coeff of $x^3$ in $2^{nd} = 20 \times (-2)^3$ | B1 + B1 | co
$\rightarrow 10a^2 - 160 = 90$ | M1 | Forming an equation for $a$ + solution
$\rightarrow a = 5$ | A1 | co (condone ±)
| [5] |
1 The coefficient of $x ^ { 3 }$ in the expansion of $( a + x ) ^ { 5 } + ( 1 - 2 x ) ^ { 6 }$, where $a$ is positive, is 90 . Find the value of $a$.

\hfill \mbox{\textit{CAIE P1 2011 Q1 [5]}}
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