IB Maths AA HL 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.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 2, calculatorEasy6 marks

(a)

Expand $(1+2x)^3$

[2]

(b)

For integer $r$ where $0\leq r\leq 6$ use your GDC to generate a table of $\binom{6}{r}$ Hence determine the value of $r$ for which $\binom{6}{r}=20$ Using the formula for $\binom{6}{r}$ verify your answer.

[4]

Question 2

Paper 2, calculatorMedium12 marks

Let $S$ be a set of 9 distinct elements. Let $N$ be the number of subsets of $S$ that contain exactly 4 elements, and define

k=\left(1+\frac{1}{N}\right)^{1/2}.

(a)

Determine $N$

[2]

(b)

Find the first four non-zero terms, in descending powers of $N$ in the binomial expansion of $k$

[4]

(c)

Hence use the first three non-zero terms of this expansion to estimate $k$ Give your answer to 9 significant figures.

[2]

(d)

Use your GDC to determine $k$ and hence find the absolute difference between this value and your estimate in part (c), correct to 2 significant figures.

[3]

(e)

State whether your estimate in part (c) gives $k$ correctly to 3 decimal places.

[1]

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