Coefficient relationship between terms

The relationship between coefficients of two different terms is given (e.g., one coefficient is double another, or coefficients are equal), and the question asks to find the unknown parameter.

4 questions

Edexcel C2 2007 June Q3
3. (a) Find the first four terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 6 }\), where \(k\) is a non-zero constant. Given that, in this expansion, the coefficients of \(x\) and \(x ^ { 2 }\) are equal, find
(b) the value of \(k\),
(c) the coefficient of \(x ^ { 3 }\).
Edexcel C2 2008 June Q3
3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).
Edexcel C2 2009 June Q2
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) is 6 times the coefficient of \(x\),
(b) find the value of \(k\).
Edexcel C2 2011 June Q2
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + b x ) ^ { 5 }$$ where \(b\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
(b) find the value of \(b\).