Product with linear term

Questions where a simple linear term (a+bx) or (a-bx) is multiplied by a binomial expansion (1+cx)^n, requiring straightforward multiplication of the linear factor with the first few terms of the expansion.

12 questions · Moderate -0.2

1.04c Extend binomial expansion: rational n, |x|<1
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CAIE P3 2013 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 + 3 x } { \sqrt { } ( 1 + 2 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2019 June Q1
4 marks Standard +0.3
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
CAIE P3 2008 November Q2
4 marks Moderate -0.3
2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2016 November Q2
4 marks Moderate -0.3
2 Expand \(( 2 - x ) ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2024 June Q1
4 marks Moderate -0.3
1 Expand \(( 3 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
Pre-U Pre-U 9794/1 2012 June Q7
9 marks Moderate -0.3
7
  1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
  2. State the range of values of \(x\) for which this expansion is valid.
  3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).
WJEC Unit 3 2019 June Q2
Standard +0.8
Expand \(\frac { 4 - x } { \sqrt { 1 + 2 x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). State the range of values of \(x\) for which the expansion is valid.
Edexcel C4 Q4
8 marks Moderate -0.3
$$f(x) = (1 + 3x)^{-1}, \quad |x| < \frac{1}{3}.$$
  1. Expand \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\). [3]
  2. Hence show that, for small \(x\), $$\frac{1 + x}{1 + 3x} \approx 1 - 2x + 6x^2 - 18x^3.$$ [2]
  3. Taking a suitable value for \(x\), which should be stated, use the series expansion in part \((b)\) to find an approximate value for \(\frac{101}{103}\), giving your answer to 5 decimal places. [3]
Edexcel C4 Q29
8 marks Moderate -0.3
  1. Expand \((1 + 3x)^{-2}\), \(|x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  2. Hence, or otherwise, find the first three terms in the expansion of \(\frac{x + 4}{(1 + 3x)^2}\) as a series in ascending powers of \(x\). [4]
OCR C4 Q4
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}, |x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  2. Hence, or otherwise, show that for small \(x\), $$\left(\frac{2-x}{1-3x}\right)^2 \approx 4 + 20x + 85x^2 + 330x^3.$$ [3]
SPS SPS FM 2022 February Q8
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
  2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]
SPS SPS FM 2025 February Q2
5 marks Moderate -0.3
  1. Find the first three terms in the expansion of \((1-2x)^{-1}\) in ascending powers of \(x\), where \(|x| < \frac{1}{2}\). [3]
  2. Hence find the coefficient of \(x^2\) in the expansion of \(\frac{x+3}{\sqrt{1-2x}}\). [2]