Edexcel C2 — Question 4 7 marks

Exam BoardEdexcel
ModuleC2 (Core Mathematics 2)
Marks7
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TopicBinomial Theorem (positive integer n)
TypeSubstitution into binomial expansion
DifficultyModerate -0.3 Part (a) is a straightforward binomial expansion requiring calculation of four terms using the binomial theorem with simple coefficients. Part (b) requires the insight to substitute y = x - x² into part (a) and then expand/collect terms, which adds a layer of algebraic manipulation beyond routine expansion. This is a standard C2 question testing both direct application and substitution technique, slightly easier than average due to limited terms required.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
4. (a) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(b) Hence expand ( \(\left. 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
(a)
AnswerMarks
\(= 2^6 + 6(2^5)(y) + \binom{6}{2}(2^4)(y^2) + \binom{6}{3}(2^3)(y^3) + \ldots\)M1 A1
\(= 64 + 192y + 240y^2 + 160y^3 + \ldots\)B1 A1
(b)
AnswerMarks
Let \(y = x - x^2\)M1
\((2 + x - x^2)^6 = 64 + 192(x - x^2) + 240(x - x^2)^2 + 160(x - x^2)^3 + \ldots\)M1
\(= 64 + 192(x - x^2) + 240(x^2 - 2x^3 + \ldots) + 160(x^3 - \ldots) + \ldots\)M1
\(= 64 + 192x + 48x^2 - 320x^3 + \ldots\)A1 
**(a)**

$= 2^6 + 6(2^5)(y) + \binom{6}{2}(2^4)(y^2) + \binom{6}{3}(2^3)(y^3) + \ldots$ | M1 A1 |
$= 64 + 192y + 240y^2 + 160y^3 + \ldots$ | B1 A1 |

**(b)**

Let $y = x - x^2$ | M1 |
$(2 + x - x^2)^6 = 64 + 192(x - x^2) + 240(x - x^2)^2 + 160(x - x^2)^3 + \ldots$ | M1 |
$= 64 + 192(x - x^2) + 240(x^2 - 2x^3 + \ldots) + 160(x^3 - \ldots) + \ldots$ | M1 |
$= 64 + 192x + 48x^2 - 320x^3 + \ldots$ | A1 |
4. (a) Expand $( 2 + y ) ^ { 6 }$ in ascending powers of $y$ as far as the term in $y ^ { 3 }$, simplifying each coefficient.\\
(b) Hence expand ( $\left. 2 + x - x ^ { 2 } \right) ^ { 6 }$ in ascending powers of $x$ as far as the term in $x ^ { 3 }$, simplifying each coefficient.\\

\hfill \mbox{\textit{Edexcel C2  Q4 [7]}}
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