1.04c Extend binomial expansion: rational n, |x|<1

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.

10

1. The complex numbers \(v\) and \(w\) satisfy the equations $$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$ Solve the equations for \(v\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z - 2 - 3 \mathrm { i } | = 1\).
   2. Calculate the least value of \(\arg z\) for points on this locus.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
2. State the set of values of \(x\) for which the expansion in part (a) is valid.

9 Let \(\mathrm { f } ( x ) = \frac { 8 + 5 x + 12 x ^ { 2 } } { ( 1 - x ) ( 2 + 3 x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

2

1. Expand \(\sqrt [ 3 ] { 1 + 6 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

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 \(b\).

10 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 7 x + 8 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552} In the diagram, \(O A B C D\) is a solid figure in which \(O A = O B = 4\) units and \(O D = 3\) units. The edge \(O D\) is vertical, \(D C\) is parallel to \(O B\) and \(D C = 1\) unit. The base, \(O A B\), is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(B C\) is such that \(C N = 2 N B\).
   1. Express vectors \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
   2. Calculate the angle in degrees between the directions of \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\).
   3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\sqrt { \frac { 22 } { 5 } }\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

10 Let \(\mathrm { f } ( x ) = \frac { 24 x + 13 } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid.

9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.

7
Let \(f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\mathrm { f } ( x )\).

8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
3. State the set of values of \(x\) for which the expansion in part (b) is valid.

6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the coefficient of \(x ^ { 6 }\)

1. (a) Use the binomial expansion, in ascending powers of \(x\), of \(\frac { 1 } { \sqrt { } ( 1 - 2 x ) }\) to show that

$$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } \approx 2 + 5 x + 6 x ^ { 2 } , \quad | x | < 0.5$$ (b) Substitute \(x = \frac { 1 } { 20 }\) into $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } = 2 + 5 x + 6 x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 10 }\) Give your answer as a fraction in its simplest form.

1. $$f ( x ) = ( 3 - 2 x ) ^ { - 4 } , \quad | x | < \frac { 3 } { 2 }$$ Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.

3. (a) Express \(\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }\) in partial fractions.
(b) Hence, or otherwise, find the series expansion of $$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.

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.

2. Given that $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \equiv A + \frac { B } { x + 1 } + \frac { C } { x - 3 }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence, or otherwise, find the series expansion of $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) 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.

2. Given that $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } }$$

1. find the values of the constants \(A\) and \(C\) and show that \(B = 0\) (4)
2. Hence, or otherwise, find the series expansion of $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying each term.
   (5)

4. $$f ( x ) = \frac { 27 } { ( 3 - 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }$$

1. Find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
   (5) Use your answer to part (a) to find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of
2. \(g ( x ) = \frac { 27 } { ( 3 + 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }\)
3. \(\mathrm { h } ( x ) = \frac { 27 } { ( 3 - x ) ^ { 2 } } \quad | x | < 3\)

4. (a) Find the binomial expansion of $$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(p\). $$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$ where \(p\) is a positive constant. In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\).
(b) Find the value of \(p\).
(c) Hence find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

1. (a) Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction.
(b) Hence or otherwise, find the coefficient of \(x ^ { 2 }\) in the series expansion of

1. \(\frac { 1 } { ( 2 + 6 x ) ^ { 3 } } \quad | x | < \frac { 1 } { 3 }\)
2. \(\frac { 4 - x } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }\)

1. (a) Use the binomial series to expand

$$\frac { 1 } { ( 2 - 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. $$f ( x ) = \frac { 4 + k x } { ( 2 - 3 x ) ^ { 3 } } \quad \text { where } k \text { is a constant and } | x | < \frac { 2 } { 3 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$\frac { 1 } { 2 } + A x + \frac { 81 } { 16 } x ^ { 2 } + \cdots$$ where \(A\) is a constant,
(b) find the value of \(k\),
(c) find the value of \(A\).

6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\) (b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$