Edexcel C4 — Question 2 9 marks

Exam BoardEdexcel
ModuleC4 (Core Mathematics 4)
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeFactoring out constants before expansion
DifficultyStandard +0.3 This is a standard binomial expansion question requiring factoring out the constant (4) before applying the formula, then using the expansion for approximation. It's slightly above average difficulty due to the multi-step nature and the need to recognize how to set up part (c), but all techniques are routine for C4 students.
Spec1.04c Extend binomial expansion: rational n, |x|<1
2. (a) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(b) State the set of values of \(x\) for which your expansion is valid.
(c) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.
AnswerMarks Guidance
(a) \(4^{\frac{1}{2}}(1 - \frac{1}{4}x)^{\frac{1}{2}} = 2(1 - \frac{1}{4}x)^{\frac{1}{2}}\)B1
\(= 2[1 + (\frac{1}{4})(-\frac{1}{4}x) + \frac{(\frac{1}{4})(-\frac{3}{4})}{2}(-\frac{1}{4}x)^2 + ...] = 2 - \frac{1}{4}x - \frac{1}{64}x^2 + ...\)M1 A2 B1
(b) \(x < 4\)
(c) \(x = 0.01 \Rightarrow (4-x)^{\frac{1}{2}} = \sqrt{3.99} = \sqrt{\frac{399}{100}} = \frac{1}{10}\sqrt{399}\)M1
\(x = 0.01 \Rightarrow (4-x)^{\frac{1}{2}} = 2 - \frac{1}{400} - \frac{1}{640000} = 1.997498438\)M1
\(\therefore \sqrt{399} = 10 \times 1.997498438 = 19.9749844\) (9sf)M1 A1 (9) 
**(a)** $4^{\frac{1}{2}}(1 - \frac{1}{4}x)^{\frac{1}{2}} = 2(1 - \frac{1}{4}x)^{\frac{1}{2}}$ | B1 |

$= 2[1 + (\frac{1}{4})(-\frac{1}{4}x) + \frac{(\frac{1}{4})(-\frac{3}{4})}{2}(-\frac{1}{4}x)^2 + ...] = 2 - \frac{1}{4}x - \frac{1}{64}x^2 + ...$ | M1 A2 | B1

**(b)** $|x| < 4$ | B1 |

**(c)** $x = 0.01 \Rightarrow (4-x)^{\frac{1}{2}} = \sqrt{3.99} = \sqrt{\frac{399}{100}} = \frac{1}{10}\sqrt{399}$ | M1 |

$x = 0.01 \Rightarrow (4-x)^{\frac{1}{2}} = 2 - \frac{1}{400} - \frac{1}{640000} = 1.997498438$ | M1 |

$\therefore \sqrt{399} = 10 \times 1.997498438 = 19.9749844$ (9sf) | M1 A1 | (9)

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2. (a) Expand $( 4 - x ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$, simplifying each coefficient.\\
(b) State the set of values of $x$ for which your expansion is valid.\\
(c) Use your expansion with $x = 0.01$ to find the value of $\sqrt { 399 }$, giving your answer to 9 significant figures.\\

\hfill \mbox{\textit{Edexcel C4  Q2 [9]}}
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