Copyright © UCLES 2020
TMUA/CTMUA D513/01
PAPER 1
November 2020 75 minutes
Additional materials: Answer sheet
INSTRUCTIONS TO CANDIDATES
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You also require a soft pencil and an eraser.
Please complete the answer sheet with your candidate number, centre number,
date of birth, and full name.
This paper is the first of two papers.
There are 20 questions on this paper. For each question, choose the one answer you
consider correct and record your choice on the separate answer sheet. If you make a
mistake, erase thoroughly and try again.
There are no penalties for incorrect responses, only marks for correct answers, so you
should attempt all 20 questions. Each question is worth one mark.
You can use the question paper for rough working or notes, but no extra paper is
allowed.
You must complete the answer sheet within the time limit.
Calculators and dictionaries are NOT permitted.
There is no formulae booklet for this test.
Please wait to be told you may begin before turning this page.
This question paper consists of 21 printed pages and 3 blank pages.
PV2
BLANK PAGE
2
1 Which of the following is an expression for the first derivative with respect to x of
x
3
− 5x
2
2x
√
x
A −
√
x
2
B
√
x
4
C
3x − 5
4
√
x
D
3
√
x − 5
4
√
x
E
3
√
x − 10
3
√
x
F
3x
2
− 10x
3
√
x
3
2 (2x +1)and(x − 2) are factors of 2x
3
+ px
2
+ q
What is the value of 2p + q ?
A −10
B −
38
5
C −
22
3
D
22
3
E
38
5
F 10
4
3 Find the complete set of values of x for which
(x +4)(x + 3)(1 − x) > 0 and (x +2)(x − 2) < 0
A 1 <x<2
B −2 <x<1
C −2 <x<2
D x<−2orx>1
E x<−4orx>2
F x<−4or − 3 <x<1
G −4 <x<−2orx>1
5
4 The 1
st
,2
nd
and 3
rd
terms of a geometric progression are also the 1
st
,4
th
and 6
th
terms, respectively, of an arithmetic progression.
The sum to infinity of the geometric progression is 12.
Find the 1
st
term of the geometric progression.
A 1
B 2
C 3
D 4
E 5
F 6
6
5 The curve S has equation
y = px
2
+6x − q
where p and q are constants.
S has a line of symmetry at x = −
1
4
and touches the x-axis at exactly one point.
What is the value of p +8q ?
A 6
B 18
C 21
D 25
E 38
7
6 Find the maximum value of the function
f(x)=
1
5
2x
− 4(5
x
)+7
A
1
7
B
1
4
C
1
3
D 3
E 4
F 7
8
7 Given that
2
3x
=8
(y+3)
and
4
(x+1)
=
16
(y+1)
8
(y+3)
what is the value of x + y ?
A −23
B −22
C −15
D −14
E −11
F −10
9
8 The function f is defined for all real x as
f(x)=(p − x)(x +2)
Find the complete set of values of p for which the maximum value of f(x)is
less than 4.
A −2 − 4
√
2 <p<−2+4
√
2
B −2 − 2
√
2 <p<−2+2
√
2
C −2
√
5 <p<2
√
5
D −6 <p<2
E −4 <p<0
F −2 <p<2
10
9 The quadratic expression x
2
− 14x + 9 factorises as (x − α)(x − β), where α and β
are positive real numbers.
Which quadratic expression can be factorised as (x −
√
α )(x −
√
β )?
A x
2
−
√
10x +3
B x
2
−
√
14x +3
C x
2
−
√
20x +3
D x
2
− 178x +81
E x
2
− 176x +81
F x
2
+ 196x +81
11
10 The following sequence of transformations is applied to the curve y =4x
2
1. Translation by
3
−5
2. Reflection in the x-axis
3. Stretch parallel to the x-axis with scale factor 2
What is the equation of the resulting curve?
A y = −x
2
+12x − 31
B y = −x
2
+12x − 41
C y = x
2
+12x +31
D y = x
2
+12x +41
E y = −16x
2
+48x − 31
F y = −16x
2
+48x − 41
G y =16x
2
− 48x +31
H y =16x
2
− 48x +41
12
11 The quadratic function shown passes through (2, 0) and (q, 0), where q>2.
2
q
R
S
x
y
What is the value of q such that the area of region R equals the area of region S ?
A
√
6
B 3
C
18
5
D 4
E 6
F
33
5
13
12 How many real solutions are there to the equation
3cosx =
√
x
where x is in radians?
A 0
B 1
C 2
D 3
E 4
F 5
G infinitely many
14
13 Find the coefficient of x
2
y
4
in the expansion of (1 + x + y
2
)
7
A 6
B 10
C 21
D 35
E 105
F 210
15
14 The area enclosed between the line y = mx and the curve y = x
3
is 6.
What is the value of m ?
A 2
B 4
C
√
3
D
√
6
E 2
√
3
F 2
√
6
16
15 Find the positive difference between the two real values of x for which
(log
2
x)
4
+12
log
2
1
x
2
− 2
6
=0
A 4
B 16
C
15
4
D
17
4
E
255
16
F
257
16
17
16 The circle C
1
has equation (x +2)
2
+(y − 1)
2
=3
The circle C
2
has equation (x − 4)
2
+(y − 1)
2
=3
The straight line l is a tangent to both C
1
and C
2
and has positive gradient.
The acute angle between l and the x-axis is θ
Find the value of tan θ
A
1
2
B 2
C
√
2
2
D
√
2
E
√
6
2
F
√
6
3
G
√
3
3
H
√
3
18
17 Find the complete set of values of m in terms of c such that the graphs of
y = mx + c and y =
√
x have two points of intersection.
A 0 <m<
1
4c
B 0 <m<4c
2
C m>
1
4c
D m<
1
4c
E m>4c
2
F m<4c
2
19
18 Find the number of solutions and the sum of the solutions of the equation
1 − 2cos
2
x = |cos x |
where 0 ≤ x ≤ 180
◦
A Number of solutions = 2 Sum of solutions = 180
◦
B Number of solutions = 2 Sum of solutions = 240
◦
C Number of solutions = 3 Sum of solutions = 180
◦
D Number of solutions = 3 Sum of solutions = 360
◦
E Number of solutions = 4 Sum of solutions = 240
◦
F Number of solutions = 4 Sum of solutions = 360
◦
20
19 Find the lowest positive integer for which x
2
− 52x − 52 is positive.
A 26
B 27
C 51
D 52
E 53
F 54
21
20 For how many values of a is the equation
(x − a)(x
2
− x + a)=0
satisfied by exactly two distinct values of x ?
A 0
B 1
C 2
D 3
E 4
F more than 4
22
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