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# 2018 May Time Zone 2 Maths HL – Paper 1

### Paper 1 – Question 1

Topic: Vectors

Difficulty: Easy

Marks: 4

Question 1 (4 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a basic vectors question where two vectors are given. The question asks students to find the cosine of the angle between the two vectors.

This is a fairly common IB Mathematics Higher Level Exam Question and appears in 10% – 15% of IB Math HL Past Papers (Paper 1)

2018 Mathematics HL Paper 1 May TZ2 Question 1 – 18M.1.hl.TZ2.1 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 2

Topic: Functions

Difficulty: Medium

Marks: 7

Question 2 (7 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a functions question where two functions are given, one of which is a modulus function.

Part (a) asks students to sketch both functions.

Part (b) asks to solve when the functions are equal to each other.

This is a common IB Mathematics Higher Level Exam Question and appears in 30% – 35% of IB Math HL Past Papers (mostly Paper 1)

2018 Mathematics HL Paper 1 May TZ2 Question 2 – 18M.1.hl.TZ2.2 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 3

Topic: Probability Distribution

Difficulty: Medium

Marks: 6

Question 3 (6 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Probability Distribution question with a table provided.

Part (a) asks students to find the value of an unknown variable in the probability distribution table.

Part (b) asks to find the expected value of the discrete random variable as well as the probability of the random variable being greater than the mean.

This is a very common IB Mathematics Higher Level Exam Question and appears in 40% – 45% of IB Math HL Past Papers (mostly Paper 1)

2018 Mathematics HL Paper 1 May TZ2 Question 3 – 18M.1.hl.TZ2.3 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 4

Topic: Functions & Calculus

Difficulty: Medium

Marks: 6

Question 4 (6 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Functions & Calculus question.

A curve is given and the question asks to find the x-coordinates on the points of the curve where the gradient is zero. You need to know your deriving laws in order to correctly derive the function.

This is quite a common IB Mathematics Higher Level Exam Question and appears in 20% – 30% of IB Math HL Past Papers (mostly Paper 1).

2018 Mathematics HL Paper 1 May TZ2 Question 4 – 18M.1.hl.TZ2.4 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 5

Topic: Sequences & Series

Difficulty: Medium

Marks: 7

Question 5 (7 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is an interesting Sequences & Series question. The question has two sequences in it with u(n) being a Geometric Sequence and a(n) being an Arithmetic Sequence. The sequence u(n) is inside the sequence a(n). The a(n) sequence also involves logs.

Part (a) asks you to show that a(n) is an arithmetic sequence and to express the common difference d in terms of r.

Part (b) gives you u(1) and the sum to infinity and asks you to find the value of the common difference d.

This is an interesting twist on a common IB Mathematics Higher Level Exam Question. Sequence & Series questions typically appear in 80% – 90% of IB Math HL Past Papers (Paper 1 or Paper 2) but a question like this hasn’t been examined before. Watch the video solution for detailed steps on the solution.

2018 Mathematics HL Paper 1 May TZ2 Question 5 – 18M.1.hl.TZ2.5 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 6

Topic: Calculus

Difficulty: Medium / Hard

Marks: 7

Question 6 (7 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Calculus question where two functions are given, f(x) and g(x).

Part (a) asks you to find the derivative of both functions.

Part (b) is quite tricky where it asks to find the definite integral of a function that is linked to f(x) and g(x). You can either integrate normally by integration by parts or hopefully you can see the link between the answers found in part (a) and the question (part b). Watch the video solution for more details and step by step working.

This is a fairly common IB Mathematics Higher Level Exam Question and appear in 20% – 30% of IB Math HL Past Papers (mostly Paper 1).

2018 Mathematics HL Paper 1 May TZ2 Question 6 – 18M.1.hl.TZ2.6 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 7

Topic: Complex Numbers

Difficulty: Medium

Marks: 6

Question 7 (6 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Complex Numbers question where two general complex numbers are given ‘z’ and ‘c’.

Part (a) asks you to find the real part when z+w / z-w. Algebra is needed.

Part (b) asks to find the value of the real part of z+w / z-w when the modulus of z is equal to the modulus of w.

This is an interesting IB Mathematics Higher Level Exam Question. Section A Complex Numbers questions typically appear in 60% – 70% of IB Math HL Past Papers (Paper 1 or Paper 2) however this has a unique feel to it. Quite challenging!

2018 Mathematics HL Paper 1 May TZ2 Question 7 – 18M.1.hl.TZ2.7 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 8

Topic: Calculus

Difficulty: Medium / Hard

Marks: 7

Question 8 (7 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Calculus question (Integration by substitution).

Part (a) is an indefinite integral question where you need to use substitution.

Part (b) is definite integration by substitution question that is directly related to part (a). In order to get part (b), you will need to get part (a) correct.

This is common IB Mathematics Higher Level Exam Question and typically appear in 40% – 50% of IB Math HL Past Papers (mostly Paper 1).

2018 Mathematics HL Paper 1 May TZ2 Question 8 – 18M.1.hl.TZ2.8 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 9

Topic: Vectors

Difficulty: Hard

Marks: 24

Question 9 (24 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a big Vectors question. Four points are given A,B,C,D and their respective position vectors a,b,c,d.

Part (a) asks to explain why ABCD is a parallelogram and to use vector algebra to show that the vector AD is equal to the vector BC.

Part (b) asks to find some unknown variables in the given position vectors using the fact that we know ABCD is a parallelogram.

Part (c) asks to find the area of the parallelogram ABCD.

Part (d) asks to find the vector equation of the straight line that passes through M (the intersection of the diagonals of ABCD) normal to the plane containing ABCD.

Part (e) asks to find the Cartesian equation of the plane containing ABCD.

Part (f) asks to find the coordinates of where the plane cuts the x, y, z axes and also to find the distance from the y-intercept to the z-intercept.

This is a more heavily weighted IB Mathematics Higher Level Exam Question than usual. Section B Vector questions typically appear in 40% – 50% of IB Math HL Past Papers (Paper 1 or Paper 2) however students aiming for a 5,6 or a 7 in this exam will need to be strong in vectors. See video solutions for step by step working.

2018 Mathematics HL Paper 1 May TZ2 Question 9 – 18M.1.hl.TZ2.9 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 10

Topic: Functions

Difficulty: Hard

Marks: 14

Question 10 (14 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is a Functions question. A rational function f(x) is defined.

Part (a) asks to find the inverse of f(x) and to state the domain.

Part (b) introduces a new rational function g(x). The question asks to express g(x) in a slightly different form and to also sketch the function stating all intercepts and asymptotes.

Part (c) introduces a third function, h(x). The question asks to state the domain and range of the composite function, h(g(x)). This is quite challenging without the use of a calculator. You need to carefully look at the domain and range of both g(x) and h(x) respectively first in order to get the answer to part (c), see video solutions for more details.

This is a fairly common IB Mathematics Higher Level Exam Question and appears in 30% – 40% of IB Math HL Past Papers (Paper 1 or Paper 2).

2018 Mathematics HL Paper 1 May TZ2 Question 10 – 18M.1.hl.TZ2.10 – M18/5/MATHL/HP1/ENG/TZ2/XX

### Paper 1 – Question 11

Topic: Logs & Calculus

Difficulty: Hard

Marks: 12

Question 11 (12 marks) in the 2018 May TZ2 Maths HL Paper 1 Exam is Logs & Calculus question.

Part (a) asks to show that two logarithmic expressions are equal to each other. Log laws are needed.

Part (b) defines an equation with variables ‘x’ and ‘y’ involving logs. The question asks to express ‘y’ in terms of ‘x’.

Part (c) is a definite integral question that is directly related to part (b). The question gives you the area bounded by the curve fond in part (b) between limits (one limit is unknown). The question asks to find the unknown limit given that we know that bounded area answer.

This is a challenging IB Mathematics Higher Level Exam Question. Part (b) is worth 5 marks and you need the answer to part (b) to do part (c) which is also worth 5 marks. Section B log questions typically only appear in 10% – 20% of IB Math HL Past Papers (Paper 1 or Paper 2) so students aiming for a 5 or above will need to be strong with their log laws.

2018 Mathematics HL Paper 1 May TZ2 Question 11 – 18M.1.hl.TZ2.11 – M18/5/MATHL/HP1/ENG/TZ2/XX

# 2018 May Time Zone 2 Maths HL – Paper 2

### Paper 2 – Question 1

Topic 1: Complex Numbers

Difficulty: Easy

Marks: 6

Question 1 (6 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is a fairly straight forward Complex Numbers question. A complex number is given.

Part (a) asks to express z (our complex number) in the form a+ib.

Part (b) asks to find the exact value of the modulus of z.

Part (c) asks to find the argument of z.

This is very common IB Mathematics Higher Level Exam Question and typically appear in Section A of IB Math HL Past Papers (Paper 1 or Paper 2).

2018 Mathematics HL Paper 2 May TZ2 Question 1 – 18M.2.hl.TZ2.1 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 2

Topic: Functions

Difficulty: Easy

Marks: 5

Question 2 (5 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam gives you a polynomial with 3 unknown constants p,q,r. It is given that the polynomial is exactly divisible by (x-1), (x-2) and (x-3). The question asks to find the 3 unknown constants.

The remainder & factor theorem is needed to form your 3 equations and then you can use your GDC to solve for the 3 unknowns (see video solution for more details).

This is a fairly easy IB Mathematics Higher Level Exam Question. Similar questions to this typically appear in Section A of IB Math HL Past Papers (mostly Paper 2). Calculator skills are required.

2018 Mathematics HL Paper 2 May TZ2 Question 2 – 18M.2.hl.TZ2.2 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 3

Topic: Normal Distribution

Difficulty: Medium

Marks: 6

Question 3 (6 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is a Normal Distribution question. The mean and the variance of the random variable X is given.

Part (a) asks to sketch the probability density function and to shade in a certain region (see video solution for more details).

Part (b) asks to find the probability that our random variable is between two values.

Part (c) is quite challenging. The question tells you that the probability that our random variable is between two values in equal to 0.5 and the question asks to find the value of k (see video solution for more details).

This is tricky IB Mathematics Higher Level Exam Question. Similar questions to this typically appear in Section A of IB Math HL Past Papers (mostly Paper 2). Calculator skills are required.

2018 Mathematics HL Paper 2 May TZ2 Question 3 – 18M.2.hl.TZ2.3 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 4

Topic: Trigonometry

Difficulty: Medium

Marks: 8

Question 4 (6 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is a Trigonometry question. A diagram of a triangle attached to a circle sector is provided with known lengths.

Part (a) asks to find the length of a line connecting a corner of the triangle to the centre of a different line (see video solution for more details). Part (a) also asks to find an angle in the diagram.

Part (b) asks to find the shaded region of the circle sector (see video solution for more details).

This is challenging but fairly common IB Mathematics Higher Level Exam Question. Similar questions to this typically appear in Section A of IB Math HL Past Papers (mostly Paper 2).

2018 Mathematics HL Paper 2 May TZ2 Question 4 – 18M.2.hl.TZ2.4 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 5

Topic: Binomial Theorem

Difficulty: Hard

Marks: 6

Question 5 (6 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is an interesting Binomial Theorem question.

Part (a) asks to express a given binomial coefficient as a polynomial (in terms of n). This is achieved by using the ncr formula.

Part (b) asks to hence find the least value of n for which the binomial coefficient (now expressed as a polynomial) is greater than 10^6.

This is fairly challenging IB Mathematics Higher Level Exam Question. When looking at IB Math HL Past Papers, this type of question has not been seen recently and will challenge most students. Calculator skills are required for part (b).

2018 Mathematics HL Paper 2 May TZ2 Question 5 – 18M.2.hl.TZ2.5 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 6

Topic: Proof by Induction

Difficulty: Hard

Marks: 7

Question 6 (7 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is a classic Proof by Mathematical Induction question. The proof involve powers and inequalities (see video solution for step by step working).

This is a common IB Mathematics Higher Level Exam Question. Induction questions typically appear towards the end of Section A when looking at IB Math HL Past Papers. Students need to practice these types of questions as they appear in every exam.

2018 Mathematics HL Paper 2 May TZ2 Question 6 – 18M.2.hl.TZ2.6 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 7

Topic: Calculus

Difficulty: Medium / Hard

Marks: 5

Question 7 (5 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is Kinematics question. The velocity function of a particle is given (see video solution for more details).

Part (a) asks to determine the first time (t1) the particle has zero velocity.

Part (b) asks to find the acceleration of the particle at any time t. Differentiation laws are required. Part (b) also asks to find the acceleration of the particle at  t1.

This is a common IB Mathematics Higher Level Exam Question and appears in 30% – 40% of IB Math HL Past Papers (mostly Paper 2).

2018 Mathematics HL Paper 2 May TZ2 Question 7 – 18M.2.hl.TZ2.7 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 8

Topic: Probability

Difficulty: Medium / Hard

Marks: 7

Question 8 (7 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is Binomial Distribution question where the parameters n and p are not given but the expected value is provided.

Part (a) is tricky and asks to find the least value for n [2 marks]. This is a unique question when comparing to other IB Math HL Past Papers.

Part (b) gives you the probability of the random variable being less than or equal to 1. With this information, you are asked to find the value of n and p.

This is an interesting IB Mathematics Higher Level Exam Question. When comparing to other IB Math HL Past Papers, this question has a different feel and will challenge a lot of students.

2018 Mathematics HL Paper 2 May TZ2 Question 8 – 18M.2.hl.TZ2.8 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 9

Topic: Poisson Distribution

Difficulty: Medium / Hard

Marks: 13

Question 9 (13 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is a Poisson Distribution question looking at the number of taxis arriving at a railway station. During a ‘busy’ period, the Poisson mean is given.

Part (a) asks a series of questions asking for specific probabilities using our given information. Part (a iii) is a conditional probability question.

Part (b) introduces a ‘quite’ period with a new Poisson mean. The question then asks to find the probability of a certain number of taxis arriving in a time period which has both a ‘busy’ and ‘quiet’ period.

This is an interesting and fairly common IB Mathematics Higher Level Exam Question. When comparing to other IB Math HL Past Papers, this question is similar in difficulty to most Section B Poisson Distribution questions.

2018 Mathematics HL Paper 2 May TZ2 Question 9 – 18M.2.hl.TZ2.9 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 10

Topic: Functions & Trigonometry

Difficulty: Hard

Marks: 18

Question 10 (18 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is big a Functions & Trigonometry question. A function f(x) is given with tan and cot in the function.

Part (a) asks to sketch f(x) within a specific domain. The question also asks to (ii) explain why f(x) is a function on the given domain, (iii) explain why f(x) has no inverse on the given domain, (iv) explain why f(x) is not a function for a different given domain.

Part (b) introduces g(t) which is related to f(x). The question asks to find an expression for g(t) (see video solution for more details).

Part (c) asks to sketch g(t) and to state intercepts & asymptotes.

Part (d) tells you that alpha and beta are the roots when g(t)=k (where k is a constant between 0 and 1). The question then asks to find an expression for alpha and beta in terms of k and to also show that alpha + beta is less than -2. (see video solution for more details)

This is a very challenging IB Mathematics Higher Level Exam Question. When comparing to other IB Math HL Past Papers, this question is similar / slightly more difficult to most Section B functions questions.

2018 Mathematics HL Paper 2 May TZ2 Question 10 – 18M.2.hl.TZ2.10 – M18/5/MATHL/HP2/ENG/TZ2/XX

### Paper 2 – Question 11

Topic: Calculus

Difficulty: Hard

Marks: 19

Question 11 (19 marks) in the 2018 May TZ2 Maths HL Paper 2 Exam is challenging Calculus question. A curve is given by an implicit equation.

Part (a) asks you to find dy/dx using implicit differentiation.

Part (b) introduces a second curve and tells you that the two curves intersect at points P and Q. The question asks to find the points P and Q and to show that the gradients of the tangents at P and Q when multiplied are equal to 1.

Part (c) is very challenging. The questions asks to find the three points on our original curve (nearest to the origin) when the tangents are parallel to a given line (see video solution for step by step working).

This is a very challenging IB Mathematics Higher Level Exam Question, especially part (c). When comparing to other IB Math HL Past Papers, this question is similar / slightly more difficult to most Section B questions.

2018 Mathematics HL Paper 2 May TZ2 Question 11 – 18M.2.hl.TZ2.11 – M18/5/MATHL/HP2/ENG/TZ2/XX