IB Maths AA SL AA 1.3 Proof 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, calculatorMedium7 marks

Let $A(x)=(x+1)^2-1$ and $B(x)=x(x+2)$ where $x\in\mathbb{R}$

(a)

Find $A(3)$ and $B(3)$

[2]

(b)

Starting with $A(x)$ show by a chain of algebraic equalities that $A(x)$ has the same value as $B(x)$ for every $x\in\mathbb{R}$

[3]

(c)

Complete each statement using either $=$ or $\equiv$

A(3)\ \square\ B(3)

A(x)\ \square\ B(x),\quad \text{for all }x\in\mathbb{R}

[2]

Question 2

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 3

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]

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