11 ib aa hl mathematics semester 2 2020 revision …

11 IB AA HL MATHEMATICS SEMESTER 2 2020 REVISION PROGRAM

Examination dates

o Technology Free (Paper 1) Friday Nov 20 8:30 a.m. (SS Level 2) o Technology Active (Paper 2) Friday Nov 20 10:30 a.m. (SS Level 2)

Topics to be examined

o Everything from the year. A checklist given as a handout and OneNote. Format

• Technology Free Examination – Paper 1 (75 marks). o 90 minutes duration (plus 15 minutes reading time). o No calculators or summary books are allowed. o A formula sheet will be provided. o There are two sections: o Section A (35 marks) solutions will be written in the spaces provided o Section B (40 marks) solutions will be written in the answer booklet provided. o Detailed working to questions is required for full marks.

• Technology Active Examination – Paper 2 (70 marks). o 90 minutes duration (plus 15 minutes reading time). o A formula sheet will be provided. o There are two sections: o Section A (40 marks) solutions will be written in the spaces provided o Section B (30 marks) solutions will be written in the answer booklet provided. o A graphical calculator is allowed.

Revision material

• Text: Complete all questions from the work schedule. End of chapter reviews and all additional questions from each chapter.

• 2020 Tests and Quizzes – go back through these and write out corrected versions of any mistakes made. Fresh copies of the each is available on the class OneNote.

� 1. Sequences and Series Test

� 2. Functions Test

� 5. Integration Assignment

� 3. Exponentials and Logs Test

� 4. Differentiation Test

� 6. x8 Quizzes

• Semester 1 Exams (complete under timed conditions)

� Semester 1 Paper 1

� Semester 1 Paper 2

• 2019 Semester 2 exams (complete under timed conditions)

� 2019 Semester 2 Paper 1

� 2019 Semester 2 Paper 2 General notes

• Students should attempt and follow up on questions early. • It is most important that you have a detailed summary book of your own written notes that

supports your revision. • Make sure that you have a charged calculator for the technology active examination and that

you know how to use it. • Ensure you have made full corrections to all work handed back during the year.

SEMESTER 2 EXAM CHECKLIST Haese Textbook

o Sets and Venn diagrams Core Chapter 2

o Sequences and series Core Chapter 5

o Circles and measurement Core Chapter 6

o Radians Core Chapter 8

o Non-right-angled trigonometry Core Chapter 9

o Laws of probability and Bayes’ theorem Core Chapter 11

o Quadratic functions Core Chapter 14

o Functions Core Chapter 15

o Graph transformations Core Chapter 16

o The unit circle and trigonometric equations Core Chapter 17

o Reciprocal and inverse trigonometric functions AA Chapter 1

o Exponentials AA Chapter 2

o Logarithms AA Chapter 3

o Operations with complex numbers AA Chapter 4

o Real polynomials AA Chapter 5

o Odd and even functions, the modulus function AA Chapter 6

o Rational functions AA Chapter 6

o Systems of linear equations AA Chapter 11

o Polar form of complex numbers AA Chapter 14

o De Moivre’s theorem and roots of complex numbers AA Chapter 14

o Limits at infinity AA Chapter 15

o Trigonometric limits AA Chapter 15

o Rates of change AA Chapter 16

o Differentiability and continuity AA Chapter 16

o Rules of differentiation AA Chapter 17

o Second and higher derivatives AA Chapter 17

o Implicit differentiation AA Chapter 17

o Sketch a curve, stationary and inflection points AA Chapter 18

o L’Hôpital’s rule AA Chapter 18

o Related rates AA Chapter 19

o The Fundamental Theorem of Calculus AA Chapter 20

o Integration by substitution AA Chapter 21

o Integration by parts AA Chapter 21

o Definite integrals AA Chapter 22

o The area between two curves AA Chapter 22

o Solids of revolution AA Chapter 22

o Improper integrals AA Chapter 22

Students are not permitted to bring mobile phones and / or any other unauthorized electronic devices into the examination room.

Year 11 IB Mathematics Higher Level

EXAMINATION Paper 1

Semester 2 2019

Question and Answer Booklet

STUDENT NAME: _________________________________________________

TEACHER(S): Mr T Allott

TIME ALLOWED: Reading time 5 minutes

Writing time 90 minutes

INSTRUCTIONS

Do not open this examination paper until instructed to do so. Section A - all answers in the spaces provided Section B - answer in the separate answer booklets. You are not permitted access to any calculator for this paper. Write your answers clearly with relevant working shown. Put your name at the top of each piece of paper. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. A clean copy of the mathematics HL formula booklet is required for this paper.

STRUCTURE OF BOOKLET / MARKINGSCHEME

Exam Section Number of questions to be answered

Total marks

A B

6 2

35 35

IB Higher Level Paper 1 – Semester 2 2019

1

Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.

Section A Answer all questions. Answers must be written within the answer lines provided. Working may be continued below the lines, if necessary.

1. [Maximum mark: 6]

The roots of the equation 2 7 2 0x x+ − = are α and β .

(a) Find the values of 2 2α +β and 2 2α β . [5]

(b) Hence, find a quadratic equation whose roots are 2α and 2β . [1]


2


A geometric sequence 1 2 3, , ,...u u u has 1 27u = and a sum to infinity of812

.

(a) Find the common ratio of the geometric sequence. [2]

An arithmetic sequence 1 2 3, , ,...v v v is such that 2 2 4 4 and v u v u= = .

(b) Find the greatest value of N such that1

0N

nn

v=

>∑ . [5]


3

3. [Maxmum mark: 6]

The discrete random variable X has the following distribution, where p is a constant.

(a) Find the value of p.

(b) (i) Find µ , the expected value of X.

(ii) Find ( )P .X > µ [6]

Not covered in 2020


4

4. [Maxmum mark: 5]

Solve the equation 14 2 8x x− = + . [5]


5

5. [Maxmum mark: 6]

(a) Show that . [2]

(b) Hence find the value of k such that2

20

5 11 d ln4 3

x x kx x

+=

+ +∫ . [4]

34115

32

13

2 +++

=+

++ xx

xxx


6

6. [Maxmum mark: 5]

Consider the data set {k − 2, k, k +1, k + 4}, where k∈ .

(a) Find the mean of this data set in terms of k. [3]

Each number in the above data set is now decreased by 3.

(b) Find the mean of this new data set in terms of k. [2]

Not covered in 2020


7

Do not write solutions on this page. Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.


A window is made in the shape of a rectangle with semicircle of radius r metres on top as shown

in the diagram. The perimeter of the window is constant P metres.

(a) (i) Find the area of the window in terms of P and r.

(ii) Find the width of the window in terms of P when the area is a maximum,

justifying that this is a maximum. [9]

(b) Show that in this case the height of the rectangle is equal to the radius of the semicircle. [2]

Turn over


8

Do not write solutions on this page.


(a) (i) Sketch the graphs of y = sin x and y = sin 2x, on the same set of axes,

for 0 ≤ x ≤ .

(ii) Find the x-coordinates of the points of intersection of the graphs in the

domain 0 ≤ x ≤ .

(iii) Find the area enclosed by the graphs. [9]

(b) Find the value of using the substitution 24sinx = θ . [8]

(c) The increasing function f satisfies ( )0 0f = and ( )f a b= , where a > 0 and b > 0.

(i) By reference to a sketch, show that .

(ii) Hence find the value of . [7]

.

2π

2π

xx

x d4

1

0∫ −

∫∫ −−=ba

xxfabxxf0

1

0d)(d)(

xx d4

arcsin2

0∫

Students are not permitted to bring mobile phones and / or any other unauthorized electronic devices into the examination room.

Year 11 IB Mathematics Higher Level

EXAMINATION Paper 2

Semester 2 2019

Question and Answer Booklet

STUDENT NAME: _________________________________________________

TEACHER(S): Mr T Allott

TIME ALLOWED: Reading time 5 minutes

Writing time 90 minutes

INSTRUCTIONS

Do not open this examination paper until instructed to do so. Section A - all answers in the spaces provided Section B - answer in the separate answer booklets. A graphic display calculator is required for this paper. Write your answers clearly with relevant working shown. Put your name at the top of each piece of paper. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. A clean copy of the mathematics HL formula booklet is required for this paper.

STRUCTURE OF BOOKLET / MARKINGSCHEME

Exam Section Number of questions to be answered

Total marks

A B

8 3

40 35


1

Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.

Section A Answer all questions. Answers must be written within the answer lines provided. Working may be continued below the lines, if necessary.

1. [Maximum marks: 5]

Given that 4ln 2 3ln 4 ln k− = − , find the value of k . [5]


2


A recruitment company tests the aptitude of 100 applicants applying for jobs in engineering.

Each applicant does a puzzle and the time taken, t, is recorded.

The cumulative frequency curve for these data is shown below.

Using the cumulative frequency curve,

(a) write down the value of the median; [1]

(b) determine the interquartile range; [2]

(c) complete the frequency table below. [2]

Time to complete puzzle in seconds Number of applicants

20 < t ≤ 30

30 < t ≤ 35

35 < t ≤ 40

40 < t ≤ 45

45 < t ≤ 50

50 < t ≤ 60

60 < t ≤ 80

Not covered in 2020


3


4

3. [Maxmum mark: 5]

(a) Show that = tan θ. [2]

(b) Hence find the value of cot in the form a + , where a, b . [3]

θθ2cos1

2sin+

8π

2b ∈


5

4. [Maxmum mark: 6]

Solve the following system of equations

1

1

log 21log4

x

y

y

x

+

+

=

=

[6]


6

5. [Maxmum mark: 5]

The diagram below shows a sketch of the gradient function ( )'f x of the curve ( )f x .

On the graph below, sketch the curve ( )y f x= given that ( )0 0f = .

Clearly indicate on the graph any maximum, minimum or inflexion points. [5]


7

6. [Maxmum mark: 5]

Jenny goes to school by bus every day. When it is not raining, the probability that the bus is late is320

.

When it is raining, the probability that the bus is late is 720

.

The probability that it rains on a particular day is 920

.

On one particular day the bus is late. Find the probability that it is not raining on that day. [5]


8

7. [Maxmum mark: 5]

(a) Sketch the graph of ( ) 3 2 2f x x x x= + − + . [3]

(b) Find the values of k such that the equation 3 2 2x x x k+ − + = has three distinct real solutions.[2]


9

8. [Maxmum mark: 4]

In a class of 20 students, 12 study Biology, 15 study History and 2 students study neitherBiology nor History.

(a) Illustrate this information on a Venn diagram. [2]

(b) Find the probability that a randomly selected student from this class is studying

both Biology and History. [1]

(c) Given that a randomly selected student studies Biology, find the probability that this

student also studies History. [1]


10

Do not write solutions on this page. Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.


The following graph shows the two parts of the curve defined by the equation 2 45x y y= − ,

and the normal to the curve at the point P ( )2,1 .

(a) Show that there are exactly two points on the curve where the gradient is zero. [6]

(b) Find the equation of the normal to the curve at the point P. [5]

(c) The normal at P cuts the curve again at the point Q. find the x-coordinate of Q. [3]

(d) The shaded region is rotated by 2π about the y-axis. Find the volume of the solid formed. [7]

Turn over


11

Do not write solutions on this page.


A jet plane travels horizontally along a straight path for one minute, starting at time 0t = , where t is measured in seconds. The acceleration, a, measured in 2m s− , of the jet plane is given by the straight line graph below.

(a) Find an expression for the acceleration of the jet plane during this time, in terms of t. [1]

(b) Given that when 0t = the jet plane is travelling at 1125m s− , find its maximum velocity

in 1m s− during the minute that follows. [4]

(c) Given that the jet plane breaks the sound barrier at 1295m s− , find out for how long the

jet plane is travelling greater than this speed. [2]


12


The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.

The points P and Q lie on the larger circle and POQ , where 02

x x π∠ = < < .

diagram not to scale

(a) Show that the area of the shaded region is 8sin 2x x− . [3]

(b) Find the maximum area of the shaded region. [4]

11 ib aa hl mathematics semester 2 2020 revision …

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