IB Maths AA HL AA 2.5 Polynomials and inequalities 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

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Question 1

Paper 2, calculatorEasy6 marks

For $x\in\mathbb{R}$ let $f(x)=x^2-2x$ and $g(x)=3$

(a)

Use a graphic display calculator to determine the $x$ -coordinates of the points of intersection of the graphs of $f$ and $g$ Hence solve $g(x)\geq f(x)$ giving your answer as an interval.

[3]

(b)

Solve $g(x)\geq f(x)$ algebraically.

[3]

Question 2

Paper 1, no calculatorHard6 marks

Let $f:\mathbb R\to\mathbb R$ be odd and have period $T>0$ so that $f(x+T)=f(x)$ for all $x\in\mathbb R$

(a)

Explain why $f$ cannot have an inverse function when its domain is all of $\mathbb R$

[2]

(b)

Let $I\subseteq\mathbb R$ be symmetric about $0$ and suppose the restriction of $f$ to $I$ is one-to-one. Let $g$ be the inverse of this restriction, with domain $f(I)$ Show that $g$ is odd.

[4]

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