Question 2 - A-Level Maths

Edexcel C4 2011 June — Question 2 6 marks

Exam Board	Edexcel
Module	C4 (Core Mathematics 4)
Year	2011
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Generalised Binomial Theorem
Type	Non-zero terms only
Difficulty	Standard +0.3 This is a straightforward application of the binomial expansion with a fractional power. Students need to factor out the constant, apply the standard binomial formula (1+u)^n with n=-1/2, and simplify coefficients. While it requires careful algebraic manipulation and knowledge of the generalized binomial theorem, it's a routine C4 question with no conceptual surprises—slightly easier than average due to its mechanical nature.
Spec	1.04c Extend binomial expansion: rational n, \|x\|<1

2. $$f ( x ) = \frac { 1 } { \sqrt { } \left( 9 + 4 x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { 2 }$$ Find the first three non-zero terms of the binomial expansion of $\mathrm { f } ( x )$ in ascending powers of $x$. Give each coefficient as a simplified fraction.

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Answer	Marks	Guidance
$f(x) = (\ldots + \ldots)^{-\frac{1}{3}}$	M1
$= 9^{-\frac{1}{3}}(\ldots + \ldots)^{-\frac{1}{3}}$	B1	$3^{-1}, \frac{1}{3}$ or $\frac{1}{9}$
$(1+kx^2)^n = 1 + nkx^2 + \ldots$	M1	$n$ not a natural number, $k \neq 1$
$(1+kx^2)^{-\frac{1}{3}} = \ldots + \frac{(-\frac{1}{3})(-\frac{4}{3})}{2}(kx^2)^2$	A1 ft	ft their $k \neq 1$
$\left(1+ \frac{4}{9}x^2\right)^{-\frac{1}{3}} = 1 - \frac{2}{9}x^2 + \frac{2}{27}x^4$	A1
$f(x) = \frac{1}{3} - \frac{2}{27}x^2 + \frac{2}{81}x^4$	A1	(6) [6]

$f(x) = (\ldots + \ldots)^{-\frac{1}{3}}$ | M1 |

$= 9^{-\frac{1}{3}}(\ldots + \ldots)^{-\frac{1}{3}}$ | B1 | $3^{-1}, \frac{1}{3}$ or $\frac{1}{9}$

$(1+kx^2)^n = 1 + nkx^2 + \ldots$ | M1 | $n$ not a natural number, $k \neq 1$

$(1+kx^2)^{-\frac{1}{3}} = \ldots + \frac{(-\frac{1}{3})(-\frac{4}{3})}{2}(kx^2)^2$ | A1 ft | ft their $k \neq 1$

$\left(1+ \frac{4}{9}x^2\right)^{-\frac{1}{3}} = 1 - \frac{2}{9}x^2 + \frac{2}{27}x^4$ | A1 |

$f(x) = \frac{1}{3} - \frac{2}{27}x^2 + \frac{2}{81}x^4$ | A1 | (6) [6] |

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2.

$$f ( x ) = \frac { 1 } { \sqrt { } \left( 9 + 4 x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { 2 }$$

Find the first three non-zero terms of the binomial expansion of $\mathrm { f } ( x )$ in ascending powers of $x$. Give each coefficient as a simplified fraction.\\

\hfill \mbox{\textit{Edexcel C4 2011 Q2 [6]}}

This paper (7 questions)

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Q1 4 Q2 6 Q3 6 Q4 15 Q5 7 Q6 14 Q7 15

Answer	Marks	Guidance
\(f(x) = (\ldots + \ldots)^{-\frac{1}{3}}\)	M1
\(= 9^{-\frac{1}{3}}(\ldots + \ldots)^{-\frac{1}{3}}\)	B1	\(3^{-1}, \frac{1}{3}\) or \(\frac{1}{9}\)
\((1+kx^2)^n = 1 + nkx^2 + \ldots\)	M1	\(n\) not a natural number, \(k \neq 1\)
\((1+kx^2)^{-\frac{1}{3}} = \ldots + \frac{(-\frac{1}{3})(-\frac{4}{3})}{2}(kx^2)^2\)	A1 ft	ft their \(k \neq 1\)
\(\left(1+ \frac{4}{9}x^2\right)^{-\frac{1}{3}} = 1 - \frac{2}{9}x^2 + \frac{2}{27}x^4\)	A1
\(f(x) = \frac{1}{3} - \frac{2}{27}x^2 + \frac{2}{81}x^4\)	A1	(6) [6]