# IB Maths Past Paper Overviews

## 2017 November Maths Studies – Paper 1

### Paper 1 – Question 1

Topic: Descriptive Statistics

Sub Topic 1: Grouped Data Tables

Sub Topic 2: Box and Whisker Plots

Difficulty: Easy

Marks: 6

Paper 1 starts with a fairly easy Descriptive Question.  A table is presented with one variable – age of a sample population – and their respective frequencies.

Part (a) asks to determine the mean and media of this data set.  This will require knowledge of setting up a spreadsheet in your GDC and performing a one-variable statistics calculation.  Alternately, these values could be found by hand calculations, however this is time consuming in test conditions.

Part (b) asks to draw of box and whisker plot for the data, with the lower and upper quartiles provided.  This requires the knowledge of the five key pieces of information of a box and whisker plot; minimum, Q1, Median, Q3 and Max.

### Paper 1 – Question 2

Topic: Geometry and Trigonometry

Sub Topic: Coordinate Geometry

Difficulty: Medium

Marks: 6

Question 2 is a step up in difficulty from question 1, and involves coordinate geometry.  Two coordinates are provided, with the following questions:

Part (a) asks to find the midpoint.  This will require knowledge of how to apply the midpoint formula.

Part (b) asks to find the gradient between the two coordinates.  This requires an understanding of the gradient formula, or using rise / run.

Part (c) introduces a new line that is perpendicular to the line in part (b).  The questions asks to find both the gradient of this new line (requiring understanding of negative reciprocals) and then to determine the equation of the line in the form of y=mx+c.

### Paper 1 – Question 3

Topic: Number and Algebra

Sub Topic: Scientific Notation

Difficulty: Medium

Marks: 6

Question 3 is an somewhat unusual IB Maths Studies exam question, and is not a straightforward application of a concept or topic.  The question looks at distances between two objects and requires calculating:

Part (a) – The time taken to travel between the two objects.  This can be calculated through conceptual understanding, or applying the rule speed=(distance / time).

Part (b) also requires a step of calculation that arrives at a very large number, which then needs to be converted into scientific notation.

### Paper 1 – Question 4

Topic: Logic, sets and probability

Sub Topic: Logic

Difficulty: Medium

Marks: 6

Question 4 is a three part question entirely focused on the topic of logic.

Part (a) requires writing a compound proposition – involving negation, disjunction and implication – in word form.

Part (b) involves the completion of a truth table, again involving negation, disjunction and implication.  You will require a sound understanding of these three terms respective logic tables.

Part (c) involves determining if provided compound proposition is the converse, inverse or contrapositive or the original compound proposition at the start of the question.

### Paper 1 – Question 5

Topic: Statistical Application

Sub Topic: Scatter plots, correlation and line of regression

Difficulty: Easy

Marks: 6

Question 5 presents a scatter diagram with a data sample involving the age of a group of people and the number of hours they participate in an activity each week.  The line of regression equation is also provided, which is a bit unusual for Studies exams, whereby questions usually require determining the equation.

For four subsequent parts in this question all relate to the scatter diagram and line of regression equation provided.

Part (a) is a fairly simple questions, requiring a direct substitution into the line of regression equation

Part (b) involves plotting the line of regression on top of the scatter diagram.  This requires your knowledge of plotting linear lines.

Part (c) involves determining if the nature of the correlation of the data sample; whether it is positive, negative or neither.

Part (d) asks to determine if the line of regression equation can be used for an additional data point.

### Paper 1 – Question 6

Topic: Geometry & Trigonometry

Sub Topics: Triangle Trigonometry

Sub Topics: Mathematical Models

Difficulty: Medium

Marks: 6

Question 6 is an interesting question that involves both trigonometry and mathematical models.  The question involves a rectangular shape in which the diagonal measurements is provided, however the width and height is not (they are assigned unknown variables).

Parts (a) and (b) involve determining two equations for the unknown variables (width and height).  A knowledge of Pythagoras theorem is required to determine one of these equations.

Part (c) uses the above two equations to solve for the unknown variables.  This will require an understanding of how to solve a straightforward simultaneous equation.

### Paper 1 – Question 7

Topic: Logic, sets and probability

Sub Topics: Venn Diagrams

Difficulty: Medium

Marks: 6

Question 7 is a probability question whereby a scenario is presented with an incomplete Venn diagram.

Part (a) asks to complete the Venn diagram.  This is fairly straight forward, however requires an understand to start at the intersection of the sets first when calculating set elements

Part (b) involves conditional probability, however the conditional probability formula is not required if you are able to calculate it off of the Venn diagram.

Part (c) asks to determine if the two events are independent or not.  This requires an understanding of how to test for independence.

### Paper 1 – Question 8

Topic: Number and Algebra

Sub Topic: Currency Conversion

Difficulty: Easy

Marks: 6

Question 8 is all about currency conversion, with three different types of questions:

Part (a) is a straightforward conversion from one currency to another with the exchange rate provided.

Part (b) follows on from part (a) however introduces a commission charge from the exchange provider.

Part (c) is now the reverse of (a).  A new scenario is presented whereby the values of the amount exchanged and received is provided, however the exchange rate is unknown.

### Paper 1 – Question 9

Topic: Number and Algebra

Sub Topic: Compound Interest

Difficulty: Easy

Marks: 6

Question 9 continues the topic 1 questions and moves to compound interest.  This is a fairly typical IB Maths Studies compound interest type question and requires an understanding of how to apply the compound interest formula.

Part (a) is a lead in question and doesn’t involve compound interest yet.  This question involves calculating the original price of an item, given the discount and sale price.  This requires an understanding of percentages.

Part (b) is where compound interest comes in, and it is a direct application of the compound interest formula.  After determining the Future Value (FV), the question also asks to compare your answer to the original price from Part (a).

### Paper 1 – Question 10

Topic: Geometry and Trigonometry

Sub Topic: Triangle Trigonometry

Difficulty: Medium

Marks: 6

A diagram is presented in Question 10 with numerous measurements (angles and lengths).  From this diagram, a series of trigonometry questions are asked.

Part (a) requires the sine rule to determine an unknown side length

Part (b) requires an understanding of angles of elevation to determine an unknown angle

Part (c) follows on from Part (b), using the angle of elevation found to determine another unknown side length using right angle triangle trigonometry.

This question was ranked a Medium in difficulty due to the level of interpretation and comprehension of the diagram.  The trigonometry mathematics is direct applications of the formulas.

### Paper 1 – Question 11

Topic: Mathematical Models

Difficulty: Hard

Marks: 6

A sound understanding of Quadratics is required for Question 11.  The standard form of a quadratic is provided (with unknowns a, b and c) and two coordinates that the quadratic passes through.

Part (a) asks to determine the axis of symmetry of the quadratic, with no additional information provided.  This requires an understanding of the symmetrical nature of quadratics and conceptually determine the equation from the two coordinates provided.

Part (b) is a relatively easy question, asking to determine the value of c (the y intercept).  This can be read directly from one of the coordinates provided.

Part (c) presents a new piece of information, then asks to determine the values of a and b, and thus the equation of the quadratic.  Finding a and b requires the use of simultaneous equations, or a sound understanding how the values of a, b and c change a shape of a quadratic.

### Paper 1 – Question 12

Topic: Mathematical Models

Sub Topic: Exponential Models

Difficulty: Medium

Marks: 6

Question 12 continues the theme of mathematical models, this time an exponential model involving temperature that reaches a maximum then plateaus.  The question is quite similar to past IB Maths studies exponential model questions, with the following information requiring to be found:

Part (a) asks describe what the horizontal asymptote of the exponential model means in the context of this question (where the temperature plateaus).

Part (b) asks to determine the value of the unknown variable in the exponential model equation, given the temperature at a given time.  This is a very standard and recurring type of question from past papers, requiring substitution of values and algebraic solving.

Part (c) is similar to Part (b) however slightly more difficult.  The question involves finding the time taken to reach a certain temperature.  Since t (the temp) is power, solving for t is more difficult and can be found using a number of different methods.  In the video solutions, the method of plotting two functions using a GDC is used (to visually understand the scenario).

### Paper 1 – Question 13

Topic: Statistical Application

Sub Topic: Normal Distribution

Difficulty: Medium

Marks: 6

Question 13 is entirely focused on the Normal Distribution, and somewhat unusually in comparison to Maths Studies past papers, a Normal Distribution curve is neither presented or required to be drawn.  A sound understanding of the two Normal Distribution functions on the GDC is required, and if you have this understanding, this question is actually quite easy.

Part (a) asks to determine a probability from a given value – requiring a normCDF calculation

Part (b) is the opposite of Part (a), whereby a percentage (probability) is provided and a value is asked – requiring a invNorm calculation

Part (c) then outlines the population size and asks to determine the estimated number of people for a given value, again requiring a normCDF calculation.

### Paper 1 – Question 14

Topic: Introduction to Differential Calculus

Difficulty: Hard

Marks: 6

Question 14 is the first question in the paper involving differential calculus.  There are only two parts to this question:

Part (a) requires finding the derivative of a function involving standard terms, and a fraction with the variable on the denominator.

Part (b) asks to determine the point on the function at which the gradient of the tangent at that point is equal to 6.  This requires an understanding that the relationship between the original function and its derivative, substituting in the gradient and solving for x.  Although this question was rated as Hard (primarily as it involved gradients of tangents and a decent level of algebraic solving), if you are comfortable with this concept, it is a fairly straightforward differential calculus question – there are no tricks to the question.

### Paper 1 – Question 15

Topic: Mathematical Models

Difficulty: Hard

Marks: 6

Paper 1 finishes with a fairly difficult mathematical models question involving a the sales and profits of a business selling a product.  A function describing the quantity of goods sold in relation to the currency in which the business operates in is presented at the start of the question

Part (a) starts with a fairly easy question, asking for the quantity of product sold for a given currency value.  This is a direct substitution into the function provided.

Part (b) now introduces a new function, the sales (revenue) of the business, again in relation to the currency value.  Similar to Part (a), this question requires substitution of the given currency value into this new function.

Part (c) is where this question starts to get difficult.  This part asks to present an expression for the profit in terms of currency value.

Part (d) then asks to find the currency value that maximizes the profit.  This can be found either through optimisation (differential calculus) or by finding the turning point of the quadratic function found in Part (c).  The video solution to this question demonstrates the latter option.

## 2017 November Maths Studies – Paper 2

### Paper 2 – Question 1

Topic 1: Descriptive Statistics

Sub Topic: Grouped Frequency Tables

Topic 2: Statistical Application

Sub Topic: Two Variable Statistic

Difficulty: Hard

Marks: 15

Paper 2 starts with a substantial Statistics question, rated as Hard due to the high number of concepts required to obtain the full 15 marks.  It starts with descriptive statistics (grouped data tables) and then transitions into statistical application questions (null hypothesis, testing for independence etc.).

Part (a) asks to determine if the variables are discrete or continuous.

Part (b) asks to calculate the modal and mid intervals of the grouped table.

Part (c) follows on from Part (b) and asks to calculate (using your GDC) the Mean and Standard Deviation of one of the variables.  This requires an understanding of how to set up a frequency table in a GDC spreadsheet and perform a one-variable statistic calculation.

Part (d) is where the question transitions into the topic of two variable statistics (Statistical Application) and asks to calculate the expected frequency of one of the data cells.

Part (e) requires calculation of the null hypothesis and degrees of freedom of the data sample.

Part (f) requires calculation using GDC of the p-value and chi-squared statistic of a test at the 5% significance level.  This requires an understanding of how to set up a matrix on the GDC and perform a chi-squared 2-way test.

Part (g) concludes this long question asking to state the result of the test (a follow on from Part (f)) with a reason provided.  This requires what constitutes a rejection of the Null Hypothesis.

### Paper 2 – Question 2

Topic: Number and Algebra

Sub Topic: Sequences & Series

Difficulty: Medium

Marks: 15

Question 2 is a moderately difficult Sequences and Series question that involves both Arithmetic and Geometric questions.  The question starts with an Arithmetic scenario and provides the first term (u1) and common difference (d), although these key pieces are not directly expressed, they require comprehension and extraction from the scenario.

Part (a) asks to calculate the third term and an expression for the nth term of the Arithmetic Sequence

Part (b) asks to calculate n (number of terms) for a given term value (un).  This requires an understanding of substitution of the term value into the Arithmetic Sequence Formula (un) and solving for n.

Part (c) is a fairly straight forward sum of terms (Sn) question.  No new information is presented or problem solving required.

Prior to Part (d), a new scenario is presented, which turns the question into a Geometric Sequence question.  The first term and common ratio is provided, however the common ratio is provided as a percentage (increasing by 20%, which means r=1.2).  Part (d) is a straightforward question asking to calculate the fifth term in the sequence.

Part (e) then concludes question 2 with a Geometric Sequence Sum of Terms (Sn) question.

### Paper 2 – Question 3

Topic: Geometry and Trigonometry

Sub Topic: Trigonometry

Difficulty: Easy

Marks: 13

Question 3 in Paper 2 is a fairly straightforward Trigonometry question that presents a Quadrilateral diagram and requires the following calculations.

Part (a) requires the calculation of a side length that can be found  using Pythagoras Theorem.

Part (b) asks to calculate an angle in the Quadrilateral that can be found be dividing the shape into two parts and applying the cosine rule.

Part (c) asks to calculate the total area of the quadrilateral, which again requires dividing the shape into two parts (same split as Part (b)) and performing an Area of a Triangle calculation on each part, then summing the results.

Part (d) is an interesting finish to this trigonometry question.  It provides an alternate and simplified method to approximating the area of the Quadrilateral (the calculation is required) and to then calculate the percentage error of the approximation.  This requires an understanding of the the percentage error formula, which is provided in the formula booklet.

### Paper 2 – Question 4

Topic: Logic, set and probability

Sub Topic: Tree Diagrams

Difficulty: Medium

Marks: 17

Question 4 is a long question (7 parts, 17 marks) focused on entirely on probability.  It is moderately difficult question, however was rated as Medium as there were no tricks or question types unusual to IB Maths Studies Past Papers.  If you have a confident understanding of probability calculations and completing Tree Diagrams, the question isn’t too bad.

Part (a) starts with a fairly easy probability question (favorable outcomes divided by total outcomes).  All information directly provided.

Part (b) follows on from Part (a) and asks a ‘successive probability’ type question. This requires an understanding of reducing the favorable and/or total outcomes by the outcome in Part (a) before performing the probability calculation.

Part (c) introduces a large chunk of information and then requires completing a tree diagram.  The tree diagram is drawn with some of the probabilities listed.  Simply understanding that the sum of probabilities of branches coming from the one point must add to 1 will get you 2 out of the 3 marks available.

Part (d) and (e) are probability questions based on the completed tree diagram.  The key understanding is to multiply the branch probabilities for the required end outcome.

Part (f) is a conditional probability question requiring an application of the conditional probability formula.  Fortunately both the numerator and denominator of the formula were found in Parts (d) and (e) respectively.

Part (g) follows on from Part (f) and asks to determine the expected number given the population size and probability found in Part (f).  This requires an understanding of multiplying the population size by the probability to find the expected number.

### Paper 2 – Question 5

Topic: Differential Calculus

Difficulty: Medium

Marks: 16

Question 5 is the first differential calculus question of the paper.  It is a fairly standard IB Maths Studies calculus question, with a focus on deriving the function, understanding when the function is increasing and decreasing (using the derivative) and using the GDC to find intersections.

A factorised function (2 brackets with 2 terms each) is provided at the start of the question with the following questions asked:

Part (a) asks to find the exact values of the zeros (the x intercepts) of the function.  The word exact is bolded, which provides a hint to use algebraic techniques instead of your GDC.  This can be achieved by applying the Null Factor Law and solving for x.  There are three solutions.

Part (b) asks to expand the function and to then find the derivative.  If is a fairly straightforward differentiation as there are no fractions.

Part (c) asks to use the derivative found in Part (b) to determine when the original function is ‘increasing’ (meaning when the derivative is positive).  This can be found by sketching the derivative function using a GDC and identifying the x values for which this derivative function is positive (above the x axis).

Part (d) asks to draw a sketch of the original function for a given domain and range (x values and y values).  This can be achieved by plotting the function on the GDC, setting the window settings equal to the domain and range provided, and then simply copying the function onto your page.

Part (e) presents a new function and asks to find where the new function intersects the original function.  This can be found by plotting both functions on the GDC and using the analyse > intersection function.

### Paper 2 – Question 6

Topic: Geometry and Trigonometry

Sub Topic: Geometry

Difficulty: Hard

Marks: 14

As usual, the final question in Paper 2 is a challenging one, for this paper focused on Geometry.  Two shapes are presented at the start of the question; one a cone and the other a hemisphere (half a sphere).

Part (a) asks to calculate the volume of the cone.  This is fairly straightforward using the Volume of a Cone formula in the formula book.  The only consideration is the diameter is provided in the diagram, which needs to be converted to the radius.

Part (b) asks to calculate the radius of the hemisphere with the volume provided.  This is the reverse of Part (a).  It requires using the Volume of a Sphere equation found in the formula book, dividing it by 2, making it equal to the provided volume and solving for the radius.

Prior to Part (c), the cost to fill the cone with a particular material is provided, then Part (c) asks to therefore determine how much 100 cm cubed of that material costs.  This is a ratio type question, by dividing the total cost by the cone volume, then multiplying by 100.

Part (d) asks to calculate the volume of half the height of the cone.

Part (e) is similar to Part (c), asking to calculate the cost to fill the cone with two different types of material, the bottom half (volume found in part (d)) with one material type and the top half with another.

The paper concludes with a simultaneous equations question with Part (f).  Two pieces of information are provided about the number of cones and hemispheres sold, and secondly the respective costs to sell this quantity.  This information is to be set up with two pairs of equations and solved, either using algebraic methods or using Numerical Solve with the GDC.

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