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**You can download all 17 of the Paper 3 questions for free here:** [**PDF**].

**The full typed mark scheme is available to download at the bottom of the page.**

**Seventeen IB Higher Level Paper 3 Practice Questions**

With the new syllabus just started for IB Mathematics we currently don’t have many practice papers to properly prepare for the Paper 3 Higher Level exam. As a result I’ve put together 17 full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages of questions and **600 marks** worth of content. This has been specifically written for the Analysis and Approaches syllabus – though some parts would also be suitable for Applications.

Below I have split the questions into individual pdfs, with more detail about each one. For each investigation question I have combined several areas of the syllabus in order to create some level of discovery – and in many cases I have introduced some new mathematics (as will be the case on the real Paper 3).

**Topics explored:**

**Paper 1: Rotating curves: **[Individual question download** here**. Mark-scheme download** here.**]

Students explore the use of parametric and Cartesian equations to rotate a curve around the origin. You can see a tutorial video on this above. The mathematics used here is trigonometry (identities and triangles), functions and transformations.

**Paper 12: Circumscribed and inscribed polygons **[Individual question download **here**].

Students explore different methods for achieving an upper and lower bound for pi using circumscribed and inscribed polygons. You can see a video solution to this investigation above. The mathematics used here is trigonometry and calculus (differentiation and L’Hopital’s rule).

**Paper 2: Who killed Mr. Potato? **[Individual question download **here****.**]

Students explore Newton’s Law of Cooling to predict when a potato was removed from an oven. The mathematics used here is logs laws, linear regression and solving differential equations.

**Paper 3: Graphically understanding complex roots **[Individual question download **here.**]

Students explore graphical methods for finding complex roots of quadratics and cubics. The mathematics used here is complex numbers (finding roots), the sum and product of roots, factor and remainder theorems, equations of tangents.

**Paper 4: Avoiding a magical barrier **[Individual question download **here**.]

Students explore a scenario that requires them to solve increasingly difficult optimization problems to find the best way of avoiding a barrier. The mathematics used here is creating equations, optimization and probability.

**Paper 5 : Circle packing density** [Individual question download **here**.]

** **Students explore different methods of filling a space with circles to find different circle packing densities. The mathematics used here is trigonometry and using equations of tangents to find intersection points.

**Paper 6: A sliding ladder investigation **[Individual question download **here**.]

Students find the general equation of the midpoint of a slipping ladder and calculate the length of the astroid formed. The mathematics used here is trigonometry and differentiation (including implicit differentiation). Students are introduced to the ideas of parametric equations.

Paper** 7: Exploring the Si(x) function **[Individual question download **here**.]

** **Students explore methods for approximating non-integrable functions and conclude by approximating pi squared. The mathematics used here is Maclaurin series, integration, summation notation, sketching graphs.

**Paper 8: Volume optimization of a cuboid **[Individual question download **here.**]

** **Students start with a simple volume optimization problem but extend this to a general case of an m by n rectangular paper folded to make an open box. The mathematics used here is optimization, graph sketching, extended binomial series, limits to infinity.

**Paper 9: Exploring Riemann sums **[Individual question download **here**.]

Students explore the use of Riemann sums to find upper and lower bounds of functions – finding both an approximation for pi and also for ln(1.1). The mathematics used here is integration, logs, differentiation and functions

**Paper 10 : Optimisation of area **[Individual question download **here**.]

Students start with a simple optimisation problem for a farmer’s field then generalise to regular shapes. The mathematics used here is trigonometry and calculus (differentiation and L’Hopital’s rule)

**Paper 11: Quadruple Proof **[Individual question download **here.**]

Students explore 4 different ways of proving the same geometrical relationship. The mathematics used here is trigonometry (identities) and complex numbers.

**Paper 13: Using the binomial expansion for bounds of accuracy **[Individual question download **here**.]

Students explore methods of achieving lower and upper bounds for and non-calculator methods for calculating logs. The mathematics used here is the extended binomial expansion for fractional and negative powers and integration.

**Paper 14: Radioactive Decay **[Individual question download **here**.]

Students explore discrete decay models, using probability density functions to investigate the decay of Carbon-14 and then explore the use of Euler’s method to approximate more complex decay chains. The mathematics used here is integration, probability density functions and Euler’s method of approximation

** Paper 15: Probability generating functions [**Individual question download **here**.**]**

Students explore the use of probability generating functions to find probabilities, expected values and variance for the binomial distribution and Poisson distribution for predicting the eruption of a volcano.

**Paper 16: Finding the Steiner inellipse using complex numbers [**Individual question download **here]**

Students use a beautiful relationship between complex numbers and an ellipse tangent to the midpoints of a triangle. This relationship allows you to find the equation of an ellipse from coordinate points of a triangle.

**Paper 17: Elliptical curves** [Individual question download **here**]

Students explore a method for adding points on an elliptical curve. This has links with elliptical curve cryptography.

**Mark-scheme download**

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IB HL Paper 3 Practice Questions and markscheme.

100 pages of preparatory questions with answers for the IB HL Analysis P3 exam. Please note this is not an automatic download and will be sent the same day.

$7.50

IB HL Paper 3 Practice Questions and markscheme AND Exploration Guide

All the Paper 3 questions and mark scheme AND the 63 page Exploration Guide. The Exploration Guide includes: Investigation essentials, Marking criteria guidance, 70 hand picked interesting topics, Useful websites for use in the exploration, A student checklist for top marks, Avoiding common student mistakes, A selection of detailed exploration ideas, Advice on using Geogebra, Desmos and Tracker. And more!

$12.00