IB Maths AA SL AA 1.4 Binomial theorem and counting questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 2, calculatorHard6 marks

In the expansion of $(1+x)^n$ where $n$ is an integer with $1\leq n\leq 12$ the coefficients of $x^r$ and $x^{r+1}$ are 120 and 210, respectively. Here $r$ is an integer with $0\leq r<n$

(a)

Determine $n$ and $r$

[4]

(b)

Hence, for this value of $n$ use your GDC to generate a table of $\binom{n}{k}$ for integer $k$ from 0 to $n$ Use the table to determine all powers of $x$ whose coefficients are 120. Verify your answers using the formula for $\binom{n}{k}$

[2]

Question 2

Paper 1, no calculatorHard6 marks

For positive integers $n$ and $a$ consider the expansion of $(1+ax)^n$

The coefficient of $x^2$ is five times the coefficient of $x$ and the coefficient of $x^3$ is $\frac{8}{3}$ times the coefficient of $x^2$

(a)

Determine $n$ and $a$

[4]

(b)

Hence determine the sum of the coefficients of all even powers of $x$ including the constant term, in the expansion.

[2]

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