Copyright © UCLES 2018
TEST OF MATHEMATICS
FOR UNIVERSITY ADMISSION
D513/12
PAPER 2
Wednesday 31 October 2018 Time: 75 minutes
Additional materials: Answer sheet
INSTRUCTIONS TO CANDIDATES
Please read these instructions carefully, but do not open the question paper until
you are told that you may do so.
A separate answer sheet is provided for this paper. Please check you have one.
You also require a soft pencil and an eraser.
This paper is the second 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.
Any rough work should be done on this question paper. No extra paper is allowed.
Please complete the answer sheet with your candidate number, centre number,
date of birth, and full name.
Calculators and dictionaries must NOT be used.
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.
PV1
BLANK PAGE
2
1 Thefunctionfisgiven,forx>0, by
f(x)=
x
3
− 4x
2
√
x
Find the value of f
(4).
A 3
B 9
C 9.5
D 12
E 39.5
F 88
3
2 Find the value of the constant term in the expansion of
x
6
−
1
x
2
12
A −495
B −220
C −66
D 66
E 220
F 495
4
3 Consider the following statement:
A car journey consists of two parts. In the first part, the average speed is
u km/h. In the second part, the average speed is v km/h. Hence the average
speed for the whole journey is
1
2
(u + v)km/h.
Which of the following examples of car journeys provide(s) a counterexample to
the statement?
I In the first part of the journey, the car travels at a constant speed of
50 km/h for 100 km. In the second part of the journey, the car travels
at a constant speed of 40 km/h for 100 km.
II In the first part of the journey, the car travels at a constant speed
of 50 km/h for one hour. In the second part of the journey, the car
travels at a constant speed of 40 km/h for one hour.
III In the first part of the journey, the car travels at a constant speed of
50 km/h for 80 km. In the second part of the journey, the car travels
at a constant speed of 40 km/h for 100 km.
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III
5
4 The non-zero real number c is such that the equation cos x = c has two solutions for
0 <x<
3
2
π.
How many solutions of the equation cos
2
2x = c
2
are there in the range 0 <x<
3
2
π ?
A 2
B 3
C 4
D 6
E 7
F 8
6
5 The two diagonals of the quadrilateral Q are perpendicular.
Consider the following statements:
I One of the diagonals of Q is a line of symmetry of Q.
II The midpoints of the sides of Q are the vertices of a square.
Which of these statements is/are necessarily true for the quadrilateral Q ?
A neither of them
B I only
C II only
D IandII
7
6 Which one of the following functions provides a counterexample to the statement:
if f
(x) > 0 for all real x, then f(x) > 0 for all real x.
A f(x)=x
2
+1
B f(x)=x
2
− 1
C f(x)=x
3
+ x +1
D f(x)=1− x
E f(x)=2
x
8
7 Sequence 1 is an arithmetic progression with first term 11 and common difference 3.
Sequence 2 is an arithmetic progression with first term 2 and common difference 5.
Some numbers that appear in Sequence 1 also appear in Sequence 2. Let N be the
20thsuchnumber.
What is the remainder when N is divided by 7?
A 0
B 1
C 2
D 3
E 4
F 5
G 6
9
8 The diagram shows an example of a mountain profile.
upstroke:
downstroke:
This consists of upstrokes which go upwards from left to right, and downstrokes
which go downwards from left to right. The example shown has six upstrokes and six
downstrokes. The horizontal line at the bottom is known as sea level.
A mountain profile of order n consists of n upstrokes and n downstrokes, with the
condition that the profile begins and ends at sea level and never goes below sea
level (although it might reach sea level at any point). So the example shown is a
mountain profile of order 6.
Mountain profiles can be coded by using U to indicate an upstroke and D to indicate
a downstroke. The example shown has the code UDUUUDUDDUDD. A sequence of
U’s and D’s obtained from a mountain profile in this way is known as a valid code.
Which of the following statements is/are true?
I If a valid code is written in reverse order, the result is always a valid
code.
II If each U in a valid code is replaced by D and each D by U, the result
is always a valid code.
III If U is added at the beginning of a valid code and D is added at the
end of the code, the result is always a valid code.
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III
10
9 Consider the following attempt to solve the equation 4x
√
2x − 1=10x − 5:
4x
√
2x − 1=10x − 5
(I)
4x
√
2x − 1=5(2x − 1)
(II)
16x
2
(2x − 1) = 25(2x − 1)
2
(III)
16x
2
= 25(2x − 1)
(IV)
16x
2
− 50x +25=0
(V)
(8x − 5)(2x − 5) = 0
(VI)
The solutions of the original equation are x =
5
8
and x =
5
2
.
Which one of the following is true?
A The solution is correct.
B Only one of x =
5
8
and x =
5
2
is correct
and the error arises as a result of step (II).
C Only one of x =
5
8
and x =
5
2
is correct
and the error arises as a result of step (III).
D Only one of x =
5
8
and x =
5
2
is correct
and the error arises as a result of step (IV).
E There is another value of x that satisfies the original equation
and the error arises as a result of step (II).
F There is another value of x that satisfies the original equation
and the error arises as a result of step (III).
G There is another value of x that satisfies the original equation
and the error arises as a result of step (IV).
11
10 The function f(x) is defined for all real numbers.
Consider the following three conditions, where a is a real constant:
If(a − x)=f(a + x) for all real x.
II f(2a − x)=f(x) for all real x.
III f(a − x)=f(x) for all real x.
Which of these conditions is/are necessary and sufficient for the graph of y =f(x)
to have reflection symmetry in the line x = a ?
Condition I is
necessary and
sufficient
Condition II is
necessary and
sufficient
Condition III is
necessary and
sufficient
A yes yes yes
B yes yes no
C yes no yes
D yes no no
E no yes yes
F no yes no
G no no yes
H no no no
12
11 Consider the equation 2
x
= mx + c,wherem and c are real constants.
Which of the following statements is/are true?
I The equation has a negative real solution only if c>1.
II The equation has two distinct real solutions if c>1.
III The equation has two distinct positive real solutions if and only if c ≤ 1.
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III
13
12 Consider the following statement:
For any positive integer N there is a positive integer K such that
N(Km +1)− 1 is not prime for any positive integer m.
Which one of the following is the negation of this statement?
A For any positive integer N there is a positive integer K such that
there is a positive integer m for which N(Km +1)−1 is prime.
B For any positive integer N there is a positive integer K such that
there is a positive integer m for which N(Km +1)−1 is not prime.
C For any positive integer N there is a positive integer K such that
for any positive integer m, N(Km +1)− 1 is not prime.
D For any positive integer N, any positive integer K
and any positive integer m, N(Km +1)− 1 is not prime.
E There is a positive integer N such that for any positive integer K
there is a positive integer m for which N(Km +1)−1 is not prime.
F There is a positive integer N such that for any positive integer K
there is a positive integer m for which N(Km +1)−
1 is prime.
G There is a positive integer N such that for any positive integer K
and any positive integer m, N(Km +1)− 1 is prime.
H There is a positive integer N and a positive integer K for which
there is no positive integer m for which N(Km +1)−1 is prime.
14
13 The following is an attempted proof of the conjecture:
if tan θ>0, then sin θ +cosθ>1.
Suppose tan θ>0, so in particular cos θ =0.
Since tan θ =
sin θ
cos θ
, then sin θ cos θ =tanθ cos
2
θ>0. (I)
It follows that 1 + 2 sin θ cos θ>1. (II)
Therefore sin
2
θ +2sinθ cos θ +cos
2
θ>1, (III)
which factorises to give (sin θ +cosθ)
2
> 1. (IV)
Therefore sin θ +cosθ>1. (V)
Which one of the following is the case?
A The proof is correct.
B The proof is incorrect, and the first error occurs in line (I).
C The proof is incorrect, and the first error occurs in line (II).
D The proof is incorrect, and the first error occurs in line (III).
E The proof is incorrect, and the first error occurs in line (IV).
F The proof is incorrect, and the first error occurs in line (V).
15
14 In the triangle PQR, PR =2,QR = p and ∠RP Q =30
◦
.
What is the set of all thevaluesofp for which this information uniquely determines
the length of PQ?
A p =1
B p =
√
3
C 1 ≤ p<2
D
√
3 ≤ p<2
E p =1 or p ≥ 2
F p =
√
3orp ≥ 2
G p<2
H p ≥ 2
16
15 It is given that f(x)=x
3
+3qx
2
+2,whereq is a real constant.
The equation f(x) = 0 has 3 distinct real roots.
Which of the following statements is/are necessarily true?
I The equation f(x) + 1 = 0 has 3 distinct real roots.
II The equation f(x + 1) = 0 has 3 distinct real roots.
III The equation f(−x) − 1 = 0 has 3 distinct real roots.
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III
17
16 In this question, x
1
, x
2
, x
3
,... isanarithmetic progression, all of whose terms
are integers.
Let n be a positive integer. If the median of the first n terms of the sequence is an
integer, which of the following three statements must be true?
I The median of the first n + 2 terms is an integer.
II The median of the first 2n terms is an integer.
III The median of x
2
, x
4
, x
6
, ..., x
2n
is an integer.
A none of them
B I only
C II only
D III only
E I and II only
F I and III only
G II and III only
H I, II and III
18
17 A positive integer is called a squaresum if and only if it can be written as the sum
of the squares of two integers. For example, 61 and 9 are both squaresums since
61 = 5
2
+6
2
and9=3
2
+0
2
.
A prime number is called awkward if and only if it has a remainder of 3 when
divided by 4. For example, 23 is awkward since 23 = 5 × 4+3.
A (true) theorem due to Fermat states that:
A positive integer is a squaresum if and only if each of its awkward prime
factors occurs to an even power in its prime factorisation.
It follows that 5 × 23
2
is a squaresum, since 23 occurs to the power 2, but 5 × 23
3
is
not, since 23 occurs to the power 3.
Which one of the following statements is not true?
A Every square number is a squaresum.
B If N and M are squaresums, then so is NM.
C If NM is a squaresum, then N and M are squaresums.
D If N is not a squaresum, then kN is a squaresum for some number k which is a
product of awkward primes.
19
18 f(x) is a polynomial function defined for all real x.
Which of the following is a necessary condition for the inequality
f(a)+f(b)
2
≥ f
a + b
2
to be true for all real numbers a and b with a<b?
A f(x) ≥ 0 for all real x
B f
(x) ≥ 0 for all real x
C f
(x) ≥ 0 for all real x
D f(x) ≤ 0 for all real x
E f
(x) ≤ 0 for all real x
F f
(x) ≤ 0 for all real x
20
19 Three real numbers x, y and z satisfy x>y>z>1.
Which one of the following statements must be true?
A
2
z+1
2
x
>
2
x
+2
z
2
y
B 2 >
3
x
+3
z
3
y
C
2 × 5
x
5
z
>
5
x
+5
z
5
y
D 2 <
7
x
+7
z
7
y
21
20 It is given that the equation
√
x + p +
√
x = p has at least one real solution for x,
where p is a real constant.
What is the complete set of possible values for p ?
A p =0orp =1
B p =0orp ≥ 1
C p ≥−x
D p ≥
√
x
E p ≥ 0
F p ≥ 1
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