Generalised Binomial Theorem

Find the first four terms in the binomial expansion of \(\sqrt{1+2x}\). State the set of values of \(x\) for which the expansion is valid. [5]

2

1. Expand \(( 2 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

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

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

13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).

1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
   \end{figure}
5. Explain why the trapezium rule gives an underestimate.

Given the binomial expansion \((1 + qx)^p = 1 - x + 2x^2 + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

2. \(\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2\), where \(k\) is a positive constant The binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where \(A\) and \(B\) are constants.

1. Write down the value of \(A\).
2. Find the value of \(k\).
3. Find the value of \(B\).

In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]

2. (a) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(b) State the set of values of \(x\) for which your expansion is valid.
(c) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting \(x = 1\) into the answer to part (a),
(b) find an approximation for \(\sqrt { 7 }\), giving your answer to 4 decimal places.

2. $$f ( x ) = ( 125 - 5 x ) ^ { \frac { 2 } { 3 } } \quad | x | < 25$$

1. Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) as simplified fractions.
2. Use your expansion to find an approximate value for \(120 ^ { \frac { 2 } { 3 } }\), stating the value of \(x\) which you have used and showing your working. Give your answer to 5 decimal places.

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

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

1. 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]
2. 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]

3

1. Find the first three terms in the expansion of \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), where \(| x | < \frac { 1 } { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).

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]

2

1. Expand \(\frac { 1 } { \sqrt { } ( 1 - 4 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }\).

1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).

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.

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.

Write down the first three terms in the binomial expansion of \((1-4x)^{-\frac{1}{2}}\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac{1}{13}\) in your expansion, find an approximate value for \(\sqrt{13}\) in the form \(\frac{a}{b}\), where \(a\), \(b\) are integers. [5]

1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$( 1 + 8 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
(b) Explain how you could use \(x = \frac { 1 } { 32 }\) in the expansion to find an approximation for \(\sqrt { 5 }\) There is no need to carry out the calculation.

4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for \(\sqrt [ 3 ] { 7100 }\), giving your answer to 4 decimal places. State the value of \(x\), which you use in your expansion, and show all your working.

5. $$f ( x ) = \left( 8 + 27 x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 2 } { 3 }$$ 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.

5. $$f ( x ) = \left( 8 + 27 x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 2 } { 3 }$$ 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.

5. $$f ( x ) = \left( 8 + 27 x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 2 } { 3 }$$ 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.

8

1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).

5 When \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.

1. Find the value of \(a\).
2. When \(a\) has this value, find the term in \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), simplifying the coefficient.

5 When \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.

1. Find the value of \(a\).
2. When \(a\) has this value, find the term in \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), simplifying the coefficient.

6 When \(( a + b x ) \sqrt { 1 + 4 x }\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are 3 and - 6 respectively. Find the values of \(a\) and \(b\).

6 When \(( a + b x ) \sqrt { 1 + 4 x }\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are 3 and - 6 respectively. Find the values of \(a\) and \(b\).

4

1. Expand \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of \(x\) is the term in \(x ^ { 3 }\). Find the values of the constants \(a\) and \(b\).

1 Find the coefficient of the term in \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }\).

1. Expand \((1 - 3x)^{-\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^3\). [4]
2. Hence find the coefficient of \(x^3\) in the expansion of \(\left(1 - 3(x + x^3)\right)^{-\frac{1}{2}}\). [3]

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

2 Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 4 + x } - 2 } { x + x ^ { 2 } } \right]$$ (3 marks)

2. (a) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(b) Hence, or otherwise, show that for small \(x\), $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

Expand \((9 - 3x)^{\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients. [4]

3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 3 + x ) \sqrt { 1 + 4 x }\).

1

1. Expand \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
2. Hence expand \(\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).

5

1. Simplify \(( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )\), showing your working, and deduce that $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
2. Using this result, or otherwise, obtain the expansion of $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.

1. Find the exact value of \(a\).
2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)

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

Find the first three terms in the binomial expansion of \(\frac { 2 - x } { \sqrt { 1 + 3 x } }\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac { 1 } { 22 }\) in your expansion, find an approximate value for \(\sqrt { 22 }\) in the form \(\frac { a } { b }\), where \(a , b\) are integers whose values are to be found.

1.(a)For \(| y | < 1\) ,write down the binomial series expansion of \(( 1 - y ) ^ { - 2 }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) (b)Show that when it is convergent,the series $$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$ can be written in the form \(( 1 + a x ) ^ { n }\) ,where \(a\) and \(n\) are constants to be found.
(c)Find the set of values of \(x\) for which the series in part(b)is convergent.

3

1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).

1. Expand \((2+x)^{-2}\) in ascending powers of \(x\) up to and including the term in \(x^3\), and state the set of values of \(x\) for which the expansion is valid. [5]
2. Hence find the coefficient of \(x^3\) in the expansion of \(\frac{1+x^2}{(2+x)^2}\). [2]

1. (a) Find the binomial expansion of

$$\frac { 1 } { ( 4 + 3 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction. In the binomial expansion of $$\frac { 1 } { ( 4 - 9 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 9 }$$ the coefficient of \(x ^ { 2 }\) is \(A\).
(b) Using your answer to part (a), or otherwise, find the value of \(A\). Give your answer as a simplified fraction.

1.(a)Show that \(\sqrt { \frac { 1 + x } { 1 - x } }\) can be written in the form \(\frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } }\) for \(| x | < 1\) (b)Hence,or otherwise,find the expansion,in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\) ,of \(\sqrt { \frac { 1 + x } { 1 - x } }\)