1.04a Binomial expansion: (a+b)^n for positive integer n

375 questions

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Edexcel C2 Q2
8 marks Standard +0.3
For the binomial expansion, in descending powers of \(x\), of \(\left( x^3 - \frac{1}{2x} \right)^{12}\),
  1. find the first 4 terms, simplifying each term. [5]
  2. Find, in its simplest form, the term independent of \(x\) in this expansion. [3]
OCR C2 Q6
8 marks Moderate -0.8
  1. Find the binomial expansion of \(\left(x^2 + \frac{1}{x}\right)^3\), simplifying the terms. [4]
  2. Hence find \(\int \left(x^2 + \frac{1}{x}\right)^3 dx\). [4]
OCR C2 2007 January Q6
7 marks Moderate -0.8
  1. Find and simplify the first four terms in the expansion of \((1 + 4x)^7\) in ascending powers of \(x\). [4]
  2. In the expansion of $$(3 + ax)(1 + 4x)^7,$$ the coefficient of \(x^2\) is 1001. Find the value of \(a\). [3]
OCR C2 Specimen Q1
5 marks Easy -1.2
Expand \((1-2x)^4\) in ascending powers of \(x\), simplifying the coefficients. [5]
Edexcel C2 Q1
4 marks Moderate -0.3
Find the coefficient of \(x^2\) in the expansion of $$(1 + x)(1 - x)^6.$$ [4]
Edexcel C2 Q1
4 marks Easy -1.2
Expand \((3 - 2x)^4\) in ascending powers of \(x\) and simplify each coefficient. [4]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
  2. Using your expansion, express each of the following in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
    1. \((1 + \sqrt{2})^4\)
    2. \((1 - \sqrt{2})^8\) [7]
Edexcel C2 Q6
9 marks Moderate -0.8
  1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
  2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
  3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
OCR C2 Q7
9 marks Moderate -0.8
  1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
  2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
  3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
OCR MEI C2 Q3
13 marks Moderate -0.3
  1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
  2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
    1. Expand \((2 + h)^4\). [3]
    2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
    3. Show how your result in part (iii) (B) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]
Edexcel C4 Q1
5 marks Easy -1.2
  1. Expand \((1 + 4x)^5\) in ascending powers of \(x\) up to and including the term in \(x^5\), simplifying each coefficient. [4]
  2. State the set of values of \(x\) for which your expansion is valid. [1]
AQA AS Paper 1 2018 June Q4
5 marks Moderate -0.8
  1. Find the first three terms in the expansion of \((1 - 3x)^4\) in ascending powers of \(x\). [3 marks]
  2. Using your expansion, approximate \((0.994)^4\) to six decimal places. [2 marks]
AQA AS Paper 1 2019 June Q7
6 marks Challenging +1.2
Given that \(y \in \mathbb{R}\), prove that $$(2 + 3y)^4 + (2 - 3y)^4 \geq 32$$ Fully justify your answer. [6 marks]
AQA AS Paper 1 2020 June Q4
3 marks Moderate -0.5
In the binomial expansion of \((\sqrt{3} + \sqrt{2})^4\) there are two irrational terms. Find the difference between these two terms. [3 marks]
AQA AS Paper 1 2021 June Q1
1 marks Easy -1.8
Find the coefficient of the \(x\) term in the binomial expansion of \((3 + x)^4\) Circle your answer. [1 mark] 12 27 54 108
AQA AS Paper 1 2022 June Q3
3 marks Easy -1.2
Find the coefficient of the \(x^3\) term in the expansion of \(\left(3x + \frac{1}{2}\right)^4\) [3 marks]
AQA AS Paper 1 2023 June Q3
3 marks Moderate -0.3
The coefficient of \(x^2\) in the binomial expansion of \((1 + ax)^6\) is \(\frac{20}{3}\) Find the two possible values of \(a\) [3 marks]
AQA AS Paper 1 Specimen Q8
6 marks Moderate -0.8
  1. Find the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) [3 marks]
  2. Carly has lost her calculator. She uses the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) to evaluate \(0.998^{10}\) Find Carly's value for \(0.998^{10}\) and show that it is correct to five decimal places. [3 marks]
AQA AS Paper 2 2018 June Q10
5 marks Standard +0.3
In the binomial expansion of \((1 + x)^n\), where \(n \geq 4\), the coefficient of \(x^4\) is \(\frac{1}{2}\) times the sum of the coefficients of \(x^2\) and \(x^3\) Find the value of \(n\). [5 marks]
AQA AS Paper 2 2020 June Q5
4 marks Standard +0.3
Joseph is expanding \((2 - 3x)^7\) in ascending powers of \(x\). He states that the coefficient of the fourth term is 15120 Joseph's teacher comments that his answer is almost correct. Using a suitable calculation, explain the teacher's comment. [4 marks]
AQA AS Paper 2 2024 June Q6
7 marks Standard +0.3
In the expansion of \((3 + ax)^n\), where \(a\) and \(n\) are integers, the coefficient of \(x^2\) is 4860
  1. Show that $$3^n a^2 n (n - 1) = 87480$$ [3 marks]
  2. The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative.
    1. Verify that \(n = 6\) [1 mark]
    2. Find the value of \(a\) [3 marks]
AQA Paper 1 2024 June Q8
5 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Find the first three terms, in ascending powers of \(x\), in the expansion of $$(2 + kx)^5$$ where \(k\) is a positive constant. [3 marks] \item Hence, given that the coefficient of \(x\) is four times the coefficient of \(x^2\), find the value of \(k\) [2 marks]
AQA Paper 2 2018 June Q2
1 marks Easy -1.8
Find the coefficient of \(x^2\) in the expansion of \((1 + 2x)^7\) Circle your answer. [1 mark] 42 4 21 84
AQA Paper 2 2020 June Q3
3 marks Moderate -0.3
Find the coefficient of \(x^2\) in the binomial expansion of \(\left(2x - \frac{3}{x}\right)^8\) [3 marks]
AQA Paper 3 2020 June Q7
7 marks Moderate -0.8
  1. Using \({}^n C_r = \frac{n!}{r!(n-r)!}\) show that \({}^n C_2 = \frac{n(n-1)}{2}\) [2 marks]
    1. Show that the equation $$2 \times {}^n C_4 = 51 \times {}^n C_2$$ simplifies to $$n^2 - 5n - 300 = 0$$ [3 marks]
    2. Hence, solve the equation $$2 \times {}^n C_4 = 51 \times {}^n C_2$$ [2 marks]