Substitution into binomial expansion

4

1. Expand \(( 1 + a ) ^ { 5 }\) in ascending powers of \(a\) up to and including the term in \(a ^ { 3 }\).
2. Hence expand \(\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying your answer.

3

1. Find the first three terms in the expansion of \(( 2 + u ) ^ { 5 }\) in ascending powers of \(u\).
2. Use the substitution \(u = x + x ^ { 2 }\) in your answer to part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + x + x ^ { 2 } \right) ^ { 5 }\).

1

1. Find the first 3 terms in the expansion of \(( 2 - y ) ^ { 5 }\) in ascending powers of \(y\).
2. Use the result in part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 - \left( 2 x - x ^ { 2 } \right) \right) ^ { 5 }\).

3

1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \(( 1 + x ) ^ { 5 }\). The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \left( p x + x ^ { 2 } \right) \right) ^ { 5 }\) is 95 .
2. Use the answer to part (i) to find the value of the positive constant \(p\).

1

1. Expand \(( 1 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).
2. In the expansion of \(\left( 1 + \left( p x - 2 x ^ { 2 } \right) \right) ^ { 6 }\) the coefficient of \(x ^ { 2 }\) is 48 . Find the value of the positive constant \(p\).

3. (i) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(ii) Hence expand \(\left( 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.

3

1. Find and simplify the first four terms in the expansion of \(( 2 - x ) ^ { 7 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(w ^ { 6 }\) in the expansion of \(\left( 2 - \frac { 1 } { 4 } w ^ { 2 } \right) ^ { 7 }\).

4

1. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
2. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).

6

1. Find the first four terms in the expansion of \(( 3 + 2 x ) ^ { 5 }\) in ascending powers of \(x\).
2. Hence determine the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 3 + 2 y + 4 y ^ { 2 } \right) ^ { 5 }\).

4. (a) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(b) Hence expand ( \(\left. 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).

1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
2. Evaluate this probability when \(p = 0.6\).

1.
a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 5 }$$ giving each term in its simplest form.
b) Hence write down the first 3 terms, in ascending powers of \(y\), of the binomial expansion of $$\left( 2 + 3 y ^ { \frac { 3 } { 2 } } \right) ^ { 5 }$$

6. In this question you must show detailed reasoning. The functions f and g are defined for all real values of \(x\) by $$f ( x ) = x ^ { 3 } \text { and } g ( x ) = x ^ { 2 } + 2$$

1. Write down expressions for
   1. \(\mathrm { fg } ( x )\),
   2. \(\mathrm { gf } ( x )\).
2. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).
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6

1. 1. Show that \(\cos \theta = \frac { 1 } { 2 }\) is one solution of the equation $$6 \sin ^ { 2 } \theta + 5 \cos \theta = 7$$ 6
2. (ii) Find all the values of \(\theta\) that solve the equation $$6 \sin ^ { 2 } \theta + 5 \cos \theta = 7$$ for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
   Give your answers to the nearest degree.
   6
3. Hence, find all the solutions of the equation $$6 \sin ^ { 2 } 2 \theta + 5 \cos 2 \theta = 7$$ for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
   Give your answers to the nearest degree.