Mathematics: Analysis and Approaches
Higher Level
Paper 1
Candidate Full Name
_________________________________________
Mathematics Teacher
January 2022
PHE
CCH
DGM
AHO
DGA
JBU
PRC
DAP
2 hours
Instructions to candidates
• Write your name in the box above and circle your teacher’s name.
• 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 in the answer boxes provided.
Section B: answer all questions on the file paper provided.
• 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 Analysis and Approaches formula booklet is required for
this paper.
• The maximum mark for this examination paper is [110 marks].
Full marks are not necessarily awarded for a correct answer with no working. Answers must
be supported by suitable working and/or explanations. Solutions found from a GDC should
be supported by suitable working. For example, if graphs are used to find a solution, you
should sketch these as part of your answer. 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
1.
[Maximum mark: 6]
(a)
Find the perimeter of the sector OAB in terms of π .
[2m]
(b)
Find the shaded region ABC in terms of π .
[4m]
2.
[Maximum mark: 5]
Solve the following simultaneous equations
2log x y = 1
xy = 64
3.
[Maximum mark: 9]
The function f is defined by f (x) =
x2 + 3
, where x ∈!, x ≠ 1 .
x −1
(a)
Find the equation(s) of all asymptotes.
[4m]
(b)
Find the coordinate(s) of any point(s) where the graph of y = f (x) crosses the axes.
[2m]
(c)
Sketch the graph of f on the axes below.
[3m]
4.
[Maximum mark: 7]
Four letters are arranged from the word SQUARE.
(a)
Find the number of arrangements of four different letters.
[2m]
(b)
Find the number of arrangements containing three vowels.
[2m]
(c)
Find the probability that an arrangement starts with the letter S given that it contains
three vowels.
[3m]
5.
[Maximum mark: 6]
Find the equation of the tangent to the curve 3x 2 − xy − 2 y 2 + 12 = 0 at point (2,3). Leave your
final answer in the form ax + by + c = 0 where a, b and c are integers.
6.
[Maximum mark: 5]
Use the identity for tan ( A ± B ) to find the value of arctan
in terms of π .
3
3
leaving your answer
+ arc tan
2
5
7.
[Maximum mark: 10]
(a)
(b)
Find the expansion of (1+ 2x ) 2 in ascending powers of x up to and including the
1
term in x 2 .
[4m]
Find the values of the positive constants a and b for which the expansion,
1
1+ ax
in ascending powers of x, of the two expressions (1+ 2x ) 2 and
,
1+ bx
up to and including the term in x 2 , are the same.
[6m]
8.
[Maximum mark: 7]
Find the particular solution of the differential equation
Give your answer in the form y = f (x) .
dy
= y tan x + 1 for which y(0) = 2 .
dx
SECTION B
9.
[Maximum mark: 17]
A particle moves in a straight line from point O where its velocity t seconds after leaving O is
given by v(t) = 2t 2 − 3t − 2 .
(a)
(b)
(i)
Find an expression in terms of t for the particles displacement from O.
[3m]
(ii)
Find an expression in terms of t for the particles acceleration.
[2m]
(iii) State the value of the particles initial displacement, velocity and acceleration.
[3m]
(i)
Show that the particle first comes to rest at t = 2 seconds.
[3m]
(ii)
Find the displacement of the particle when t = 3 seconds.
[2m]
(iii) Hence, or otherwise, find the total distance travelled in the first 3 seconds.
10.
[4m]
[Maximum mark: 12]
If α , β and γ are three roots of the cubic x 3 + 5x 2 − 4x − 10 = 0 .
(a)
(i)
Write down the value of αβγ .
[2m]
(ii)
Write down the value of α + β + γ .
[2m]
For any cubic equation
αβ + αγ + βγ =
(b)
11.
, with roots α , β and γ , it is given that
c
.
a
Hence, find the cubic equation having the roots of
1 1
1
, and .
α β
γ
[8m]
[Maximum mark: 26]
(a)
(i)
(ii)
(b)
(i)
(ii)
Use the complex number z = cosθ + isin θ and the binomial expansion to
deduce that sin3θ = 3sin θ − 4sin 3 θ .
Hence find the smallest positive solution of the equation 4x 3 − 3x = −
1
2
[6m]
.
Give your answer in the form a 6 + b 2, a,b ∈Q.
[8m]
Find the cube roots ω o , ω 1 and ω 2 of −64i .
Give your answer in Cartesian form.
[6m]
Points A, B and C correspond to the numbers ω o , ω 1 and ω 2 in an Argand
diagram (complex plane). Find the area of the triangle ABC.
[6m]