Binomial expansions：練習問題

Find the coefficient of \(x^4\) in the expansion of \((2 - x)(1 + 2x)^6\).

Given that the expansion of \(\left(x^2 + \frac{k}{x}\right)^{12}\) contains a constant term of 21120, find the possible values of the constant \(k\).

In the binomial expansion of \((1 + kx)^n\), where \(n\) is a positive integer and \(k\) is a non-zero constant, the first three terms are \(1 + 30x + 375x^2\). Find the value of \(n\) and \(k\).

The coefficient of \(x^2\) in the expansion of \(\left(1 + \frac{x}{2}\right)^n\) is 7. Given that \(n\) is a positive integer, find the value of \(n\).

In the expansion of \((1 + ax)^n\), the first three terms are \(1 - 24x + 252x^2\). Determine the value of the fourth term.

Find the coefficient of \(x^2\) in the binomial expansion of \((1 + 3x)^6\).

Find the constant term in the binomial expansion of \(\left( x^2 - \frac{2}{x} \right)^9\).

Consider the expansion of \( (1 + kx)^n \) in ascending powers of \( x \), where \( n \) is a positive integer greater than 2 and \( k \) is a non-zero constant.

(a) Given that the first three terms in the expansion are \( 1 \), \( 12x \), and \( 60x^2 \), find the value of \( n \) and the value of \( k \).

(b) Using the values of \( n \) and \( k \) found in part (a), determine the coefficient of \( x^3 \) in the expansion of \( (2 - x)(1 + kx)^n \).