IB Maths AA SL Topic 1: Number and algebra 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 1, no calculatorHard6 marks

For $n\in\mathbb Z$ define

S_n=(n+1)^5+(n-1)^5.

A student conjectures that $S_n$ is divisible by $32$ for every odd integer $n$

(a)

Calculate $S_1$ directly.

[1]

(b)

Starting from $S_n$ prove the identity

S_n\equiv 2n^5+20n^3+10n

for all $n\in\mathbb Z$ Use $=$ between successive expressions and $\equiv$ for the final identity.

[2]

(c)

Hence prove the student’s conjecture for every odd integer $n$

[3]

Question 2

Paper 1, no calculatorHard6 marks

Let $m\in\mathbb Z^+$ Figure 1 shows a rectangle with side lengths $m+1$ and $m+2$ divided into four smaller rectangles.

It is claimed that integers $A$ and $B$ can be chosen so that, for every $m\in\mathbb Z^+$

\frac{1}{m+1}+\frac{1}{m+2}+\frac{Am+B}{m^2+3m+2} =\frac{3}{m+1}.

(a)

Using the areas shown in Figure 1, write an identity relating the area of the whole rectangle to the areas of the four smaller rectangles. Use $\equiv$ to denote an identity.

[1]

(b)

If the claim is true, the equality for $m=1$ requires a particular relation between $A$ and $B$ Determine this relation.

[1]

(c)

Determine $A$ and $B$ so that the claim is an identity for every $m\in\mathbb Z^+$ Then, starting with the left-hand side, prove the resulting identity. Use your answer to part (b).

[4]

Question 3

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]

The best questions are still waiting

Unlock every question in this topic, then type, photograph or draw your answer and have it marked against the markscheme. Find out where you are dropping marks.

Unlock all questions Compare Student and Student Plus