Standard +0.3 This is a straightforward binomial coefficient problem requiring students to equate coefficients and solve simultaneous equations. The first two coefficients give nk=36 and n(n-1)k²/2=126k, leading directly to the given result and then to n=9, k=4. It's slightly easier than average as it's a standard textbook exercise with clear algebraic steps and no conceptual surprises.
10. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is
$$1 + 36 x + 126 k x ^ { 2 } + \ldots$$
where \(k\) is a non-zero constant and \(n\) is a positive integer.
a) Show that \(n k ( n - 1 ) = 252\)
b) Find the value of \(k\) and the value of \(n\). [0pt]
10. The binomial expansion, in ascending powers of $x$, of $( 1 + k x ) ^ { n }$ is
$$1 + 36 x + 126 k x ^ { 2 } + \ldots$$
where $k$ is a non-zero constant and $n$ is a positive integer.\\
a) Show that $n k ( n - 1 ) = 252$\\
b) Find the value of $k$ and the value of $n$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q10 [7]}}