answer---(b) In the expansion of $(x + \frac{2}{x})^n$ in decreasing powers of $x$, the 6th term is a constant.
(i) Find the value of the positive integer $n$. [2]
(ii) Find the value of the 6th term. [2]
2 The function f is defined by f(x) = 1 - 4x - x^2 for all real values of x.
(a) Write f(x) in the form a - (x + b)^2, where a and b are constants. [2]
(b) Find the range of f. [1]
The function g is defined by g(x) = 1 - 4x - x^2 for x >= k, where k is a constant.
(c) State the least possible value of k such that g has an inverse. [1]
(d) Using your value of k, find g^-1 (x), stating its domain and range. [5]
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We need to find the value of n where the sixth term in the binomial expansion of x plus two over x to the power n is constant. Using the general term formula for binomial expansion, we substitute a equals x and b equals two over x. For the sixth term where r equals five, we get the expression with x to the power n minus ten. For this to be constant, the exponent must be zero, so n minus ten equals zero, giving us n equals ten.