A-Level Maths | Binomial Expansion
Question 1
a) Show that $(1-4x)^{\frac{1}{2}} = 1-2x-2x^2-4x^3+O(x^4)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{1}{4} < x < \dfrac{1}{4}$
Question 2
a) Show that $\dfrac{1}{(1-3x)^2} = 1+6x+27x^2+108x^3+405x^4+O(x^5)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{1}{3} < x < \dfrac{1}{3}$
Question 3
a) Show that $\dfrac{1}{\sqrt{1-2x}} = 1+x+\dfrac{3}{2}x^2+\dfrac{5}{2}x^3+O(x^4)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{1}{2} < x < \dfrac{1}{2}$
Question 4
a) Show that $\sqrt{4-9x} = 2-\dfrac{9}{4}x-\dfrac{81}{64}x^2-\dfrac{729}{512}x^3+O(x^4)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{4}{9} < x < \dfrac{4}{9}$
Question 5
a) Show that $\dfrac{1}{(2-5x)^2} = \dfrac{1}{4}+\dfrac{5}{4}x+\dfrac{75}{16}x^2+\dfrac{125}{8}x^3+O(x^4)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{2}{5} < x < \dfrac{2}{5}$
Question 6
a) Show that $\dfrac{1}{(3+2x)^3} = \dfrac{1}{27}-\dfrac{2}{27}x+\dfrac{8}{81}x^2-\dfrac{80}{729}x^3+O(x^4)$.
b) State the range of values of $x$ for which the expansion is valid.
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b) $-\dfrac{3}{2} < x < \dfrac{3}{2}$
Question 7
$f(x) = \sqrt{1+\dfrac{1}{8}x},\ |x| < 8$
a) Show that $f(x) = 1+\dfrac{1}{16}x-\dfrac{1}{512}x^2+O(x^3)$.
b) By substituting $x=1$ in the expansion, show that $\sqrt{2} \approx \dfrac{256}{181}$ or $\sqrt{2} \approx \dfrac{181}{128}$.
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a) Given
b) Given
Question 8
$f(x) = \dfrac{15}{\sqrt{1-x}},\ |x| < 1$
a) Show that $f(x) = 15+\dfrac{15}{2}x+\dfrac{45}{8}x^2+\dfrac{75}{16}x^3+O(x^4)$.
b) By substituting $x=0.1$ in the expansion, show that $\sqrt{10} \approx 3.162$.
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a) Given
b) Given
Question 9
$f(x) = \dfrac{20}{\sqrt{4+2x}},\ |x| < 2$
a) Show that $f(x) = 10-\dfrac{5}{2}x+\dfrac{15}{16}x^2-\dfrac{25}{64}x^3+O(x^4)$.
b) By substituting $x=\dfrac{1}{12}$ in the above expansion, show that $\sqrt{6} \approx 2.45$.
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a) Given
b) Given
Question 10
$f(x) = \sqrt{225+15x},\ |x| < 15$
a) Show that $f(x) = 15+\dfrac{1}{2}x-\dfrac{1}{120}x^2+O(x^3)$.
b) By substituting $x=1$ in the expansion, show that $\sqrt{15} \approx \dfrac{1859}{480}$.
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a) Given
b) Given
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