Form (1+bx)^n expansion - Generalised Binomial Theorem

CAIE P3 2002 June Q2

4 marks Moderate -0.8

2 Expand $( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2005 June Q1

4 marks Moderate -0.8

1 Expand $( 1 + 4 x ) ^ { - \frac { 1 } { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2011 June Q1

4 marks Moderate -0.8

1 Expand $\sqrt [ 3 ] { } ( 1 - 6 x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2014 June Q2

4 marks Moderate -0.8

2 Expand $( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2016 June Q2

4 marks Moderate -0.3

2 Expand $\frac { 1 } { \sqrt { ( 1 - 2 x ) } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2017 June Q2

4 marks Moderate -0.3

2 Expand $\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

CAIE P3 2010 November Q1

3 marks Easy -1.2

1 Expand $( 1 + 2 x ) ^ { - 3 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.

CAIE P3 2015 November Q2

3 marks Moderate -0.8

2 Given that $\sqrt [ 3 ] { } ( 1 + 9 x ) \approx 1 + 3 x + a x ^ { 2 } + b x ^ { 3 }$ for small values of $x$, find the values of the coefficients $a$ and $b$.

CAIE P3 2021 June Q1

4 marks Moderate -0.8

1 Expand $( 1 + 3 x ) ^ { \frac { 2 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.

Edexcel P4 2024 January Q1

4 marks Moderate -0.8

Find, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, the binomial expansion of

$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$ fully simplifying each term.

Edexcel C4 2010 January Q1

9 marks Moderate -0.3

(a) Find the binomial expansion of

$$\sqrt { } ( 1 - 8 x ) , \quad | x | < \frac { 1 } { 8 }$$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
(b) Show that, when $x = \frac { 1 } { 100 }$, the exact value of $\sqrt { } ( 1 - 8 x )$ is $\frac { \sqrt { } 23 } { 5 }$.
(c) Substitute $x = \frac { 1 } { 100 }$ into the binomial expansion in part (a) and hence obtain an approximation to $\sqrt { } 23$. Give your answer to 5 decimal places.

OCR C4 2010 June Q1

5 marks Moderate -0.8

1 Expand $( 1 + 3 x ) ^ { - \frac { 5 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.

Edexcel C4 Q1

5 marks Easy -1.2

Expand $(1 + 4x)^5$ in ascending powers of $x$ up to and including the term in $x^5$, simplifying each coefficient. [4]
State the set of values of $x$ for which your expansion is valid. [1]

Pre-U Pre-U 9794/1 2011 June Q8

8 marks Standard +0.3

Find and simplify the first three terms in the expansion of $(1 - 4a)^{\frac{1}{2}}$ in ascending powers of $a$, where $|a| < \frac{1}{4}$. [4]
Hence show that the roots of the quadratic equation $x^2 - x + a = 0$ are approximately $1 - a - a^2$ and $a + a^2$, where $a$ is small. [4]