Two unknowns from two coefficient conditions

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. (a) Use the binomial theorem to expand $$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\) Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.

1. (a) Use the binomial theorem to expand

$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\). Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.

1. The binomial expansion of

$$( 3 + k x ) ^ { - 2 } \quad | k x | < 3$$ where \(k\) is a non-zero constant, may be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } + \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that \(C = 3 B\)
2. show that $$k ^ { 2 } + 6 k = 0$$
3. Hence (i) find the value of \(k\)
   (ii) find the value of \(D\)

6

1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), 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 \(b\).

5.

1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).
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5

1. Find the first three terms in the expansion of \(( 1 + p x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. The expansion of \(( 1 + q x ) ( 1 + p x ) ^ { \frac { 1 } { 3 } }\) is \(1 + x - \frac { 2 } { 9 } x ^ { 2 } + \ldots\). Find the possible values of the constants \(p\) and \(q\).