Monday 1 November 2021 (afternoon)

Candidate session number

2 hours

Write your session number in the boxes above.

Do not open this examination paper until instructed to do so.
You are not permitted access to any calculator for this paper.
Section A: answer all questions. Answers must be written within the answer boxes provided.
Section B: answer all questions in the answer booklet provided. Fill in your session number
on the front of the answer booklet, and attach it to this examination paper and your
cover sheet using the tag provided.
Unless otherwise stated in the question, all numerical answers should be given exactly or

A clean copy of the is required for

this paper.
The maximum mark for this examination paper is .

8821 – 7101
14 pages © International Baccalaureate Organization 2021

16EP01
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Please write on this page.

Answers written on this page

will not be marked.

16EP02
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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.

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

[Maximum mark: 4]

dy 3
Given that cos x and y 2 when x , find y in terms of x .
dx 4 4

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16EP03
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[Maximum mark: 9]

2x + 4
The function f is defined by f ( x ) = , where x , x 3.
3− x
(a) Write down the equation of

(i) the vertical asymptote of the graph of f ;

(ii) the horizontal asymptote of the graph of f . [2]

(b) Find the coordinates where the graph of f crosses

(i) the x-axis;

(ii) the y-axis. [2]

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16EP04
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(c) Sketch the graph of f on the axes below. [1]

15

10

x
15 10 5 0 5 10 15

10

15

ax + 4
The function g is defined by g ( x ) = , where x , x 3 and a .
3− x
1
(d) Given that g x g x , determine the value of a . [4]

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16EP05
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[Maximum mark: 5]

1
Solve the equation log 3 x = + log 3 ( 4 x3 ) , where x 0.
2 log 2 3

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16EP06
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[Maximum mark: 5]

Box 1 contains 5 red balls and 2 white balls.

Box 2 contains 4 red balls and 3 white balls.

(a) A box is chosen at random and a ball is drawn. Find the probability that the ball is red. [3]

Let A be the event that “box 1 is chosen” and let R be the event that “a red ball is drawn”.

(b) Determine whether events A and R are independent. [2]

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16EP07
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[Maximum mark: 7]

The function f is defined for all x . The line with equation y 6x 1 is the tangent to
the graph of f at x 4 .

(a) Write down the value of f 4 . [1]

(b) Find f 4 . [1]

The function g is defined for all x where g x x2 3x and h x f g x .

(c) Find h 4 . [2]

(d) Hence find the equation of the tangent to the graph of h at x 4. [3]

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16EP08
 –9– 8821 – 7101

[Maximum mark: 7]

6 2 x2 5x 3
(a) Show that 2 x 3 , x , x 1. [2]
x 1 x 1
6 π
(b) Hence or otherwise, solve the equation 2 sin 2θ − 3 − = 0 for 0 ,θ≠ . [5]
sin 2θ − 1 4

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16EP09
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[Maximum mark: 7]

The equation 3px2 2px 1 p has two real, distinct roots.

(a) Find the possible values for p . [5]

(b) Consider the case when p 4 . The roots of the equation can be expressed in
a 13
the form x , where a . Find the value of a . [2]
6

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16EP10
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[Maximum mark: 7]

dy ln 2 x 2y 1
Solve the differential equation , x 0 , given that y 4 at x .
dx x2 x 2
Give your answer in the form y f x .

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16EP11
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[Maximum mark: 7]

1
Consider the expression 1 x where a , a 0.
1 ax
The binomial expansion of this expression, in ascending powers of x , as far as the term in x2
is 4bx bx2 , where b .

(a) Find the value of a and the value of b . [6]

(b) State the restriction which must be placed on x for this expansion to be valid. [1]

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16EP12
 – 13 – 8821 – 7101

Do write solutions on this page.

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

[Maximum mark: 16]

A particle P moves along the x-axis. The velocity of P is v m s 1 at time t seconds,

where v t 4 4t 3t2 for 0 t 3 . When t 0 , P is at the origin O .

(a) (i) Find the value of t when P reaches its maximum velocity.

88
(ii) Show that the distance of P from O at this time is metres. [7]
27
(b) Sketch a graph of v against t , clearly showing any points of intersection
with the axes. [4]

(c) Find the total distance travelled by P . [5]

[Maximum mark: 14]

dn 2 x
n (
(a) Prove by mathematical induction that x e ) =  x 2 + 2nx + n ( n − 1)  e x for n . [7]
dx
(b) Hence or otherwise, determine the Maclaurin series of f x x2ex in ascending powers
4
of x , up to and including the term in x . [3]

 ( x 2e x − x 2 )3 
(c) Hence or otherwise, determine the value of lim  . [4]
x →0  x9 
 

16EP13
 – 14 – 8821 – 7101

Do write solutions on this page.

[Maximum mark: 22]

Consider the equation z 1 3 i, z . The roots of this equation are 1, 2 and 3,

where Im 2 0 and Im 3 0.
π
i
(a) (i) Verify that ω1 = 1 + e 6
is a root of this equation.

(ii) Find 2 and 3, expressing these in the form a ei , where a and 0. [6]

The roots 1 , 2 and 3 are represented by the points A, B and C respectively on an

Argand diagram.

(b) Plot the points A, B and C on an Argand diagram. [4]

(c) Find AC. [3]

3
Consider the equation z 1 iz3 , z .

1
(d) By using de Moivre’s theorem, show that α = π
is a root of this equation. [3]
i
6
1− e
(e) Determine the value of Re . [6]

16EP14
Please write on this page.

Answers written on this page

will not be marked.

16EP15
Please write on this page.

Answers written on this page

will not be marked.

16EP16