Past Paper and Sample Solutions from Wednesday 7th November 2012
The Mathematics Admissions Test (MAT) is a paper based test that has been used by the University of
Oxford since 1996. This extract from the 2012 test and associated sample solutions constitute the test
undertaken by applicants to the Mathematics, Maths & Philosophy and Maths & Statistics undergraduate
degree courses at Oxford.
From 2013, the test will be used as part of the admissions process for applicants to the following courses
run by the Department of Mathematics at Imperial College.
UCAS code
G100
G103
G125
GG31
G104
G1F3
G102
G1G3
G1GH
Course title
Mathematics
Mathematics
Mathematics (Pure Mathematics)
Mathematics, Optimisation and Statistics
Mathematics with a Year in Europe
Mathematics with Applied Mathematics/Mathematical Physics
Mathematics with Mathematical Computation
Mathematics with Statistics
Mathematics with Statistics for Finance
IN 2013 THE ADMISSIONS TESTING SERVICE WILL BE ORGANIZING THE DISTRIBUTION AND RECEIPT OF
THE MATHEMATICS TEST. SEE THIS ADMISSIONS TESTING SERVICE PAGE FOR FULL DETAILS.
Extract from 2012 Mathematics Admissions Test
1. For ALL APPLICANTS.
For each part of the question on pages 3—7 you will be given four possible answers,
just one of which is correct. Indicate for each part A—J which answer (a), (b), (c),
or (d) you think is correct with a tick (X) in the corresponding column in the table
below. Please show any rough working in the space provided between the parts.
(a)
(b)
A
B
C
D
E
F
G
H
I
J
2
(c)
(d)
Extract from 2012 Mathematics Admissions Test
A. Which of the following lines is a tangent to the circle with equation
x2 + y 2 = 4 ?
(a) x + y = 2;
√
(b) y = x − 2 2;
(c) x =
√
2;
(d) y =
√
2 − x.
B. Let N = 2k × 4m × 8n where k, m, n are positive whole numbers. Then N will
definitely be a square number whenever
(a) k is even;
(b) k + n is odd;
(c) k is odd but m + n is even;
(d) k + n is even.
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3
Extract from 2012 Mathematics Admissions Test
C. Which is the smallest of the following numbers?
√ 3
π 2
,
3 ,
(a)
(b) log3 92 ,
(c)
3 sin
3
(d)
log2 log2 85 .
D. Shown below is a diagram of the square with vertices (0, 0) , (0, 1) , (1, 1) , (1, 0) and
the line y = x + c. The shaded region is the region of the square which lies below the
line; this shaded region has area A(c).
y
1.5
y=x+c
1.0
0.5
0.5
-0.5
1.0
1.5
x
-0.5
Which of the following graphs shows A(c) as c varies?
-1.5
-1.0
AHcL
AHcL
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.5
-0.5
1.0
1.5
c
-1.5
(a)
-1.5
-1.0
-0.5
0.5
1.0
1.5
0.5
1.0
1.5
c
(b)
-1.0
-0.5
AHcL
AHcL
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.5
1.0
1.5
c
-1.5
(c)
(d)
4
-1.0
-0.5
c
Extract from 2012 Mathematics Admissions Test
E. Which one of the following equations could possibly have the graph given below?
(a) y = (3 − x)2 (3 + x)2 (1 − x);
(b) y = −x2 (x − 9) (x2 − 3) ;
(c) y = (x − 6) (x − 2)2 (x + 2)2 ;
2
(d) y = (x2 − 1) (3 − x) .
y
x
F. Let
T =
π/2
cos x dx
−π/2
×
2π
π
sin x dx ×
π/8
0
dx
cos 3x
.
Which of the following is true?
(a) T = 0;
(b) T < 0;
(c) T > 0;
(d) T is not defined.
Turn over
5
Extract from 2012 Mathematics Admissions Test
G. There are positive real numbers x and y which solve the equations
2x + ky = 4,
x+y = k
for
(a) all values of k;
(b) no values of k;
(c)
k = 2 only;
(d)
only k > −2.
H. In the region 0 < x 2π, the equation
x
sin(sin t) dt = 0
0
has
(a) no solution;
(b) one solution;
(c)
6
two solutions;
(d)
three solutions.
Extract from 2012 Mathematics Admissions Test
I. The vertices of an equilateral triangle are labelled X, Y and Z. The points X, Y and
Z lie on a circle of circumference 10 units. Let P and A be the numerical values of the
triangle’s perimeter and area, respectively. Which of the following is true?
(a)
5
A
=
;
P
4π
(b) P < A;
(c)
10
P
=
;
A
3π
(d) P 2 is rational.
J. If two chords QP and RP on a circle of radius 1 meet in an angle θ at P , for example
as drawn in the diagram below,
Q
P
Θ
R
then the largest possible area of the shaded region RP Q is
θ
π
(1 − cos θ) ;
(a) θ 1 + cos
;
(b) θ + sin θ;
(c)
2
2
(d)
θ.
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7
Extract from 2012 Mathematics Admissions Test
2. For ALL APPLICANTS.
Let
f(x) = x + 1
and
g(x) = 2x.
We will, for example, write fg to denote the function "perform g then perform f " so
that
f g(x) = f (g(x)) = 2x + 1.
If i 0 is an integer we will, for example, write f i to denote the function which performs
f i times, so that
f i (x) = f f f · · · f (x) = x + i.
i times
(i) Show that
f 2 g(x) = gf (x).
(ii) Note that
gf 2 g(x) = 4x + 4.
Find all the other ways of combining f and g that result in the function 4x + 4.
(iii) Let i, j, k 0 be integers. Determine the function
f i gf j gf k (x).
(iv) Let m 0 be an integer. How many different ways of combining the functions f
and g are there that result in the function 4x + 4m?
8
Extract from 2012 Mathematics Admissions Test
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Extract from 2012 Mathematics Admissions Test
3.

MATHEMATICS



MATHEMATICS & STATISTICS
For APPLICANTS IN
MATHEMATICS & PHILOSOPHY



MATHEMATICS & COMPUTER SCIENCE







ONLY.
Computer Science and Computer Science & Philosophy applicants should turn to
page 14.
Let f (x) = x3 + ax2 + bx + c, where the coefficients a, b and c are real numbers. The
figure below shows a section of the graph of y = f (x). The curve has two distinct turning
points; these are located at A and B, as shown. (Note that the axes have been omitted
deliberately.)
A
B
(i) Find a condition on the coefficients a, b, c such that the curve has two distinct turning
points if, and only if, this condition is satisfied.
It may be assumed from now on that the condition on the coefficients in (i) is satisfied.
(ii) Let x1 and x2 denote the x coordinates of A and B, respectively. Show that
x2 − x1 =
2√ 2
a − 3b.
3
(iii) Suppose now that the graph of y = f(x) is translated so that the turning point at
A now lies at the origin. Let g(x) be the cubic function such that y = g(x) has the
translated graph. Show that
√
g(x) = x2 x − a2 − 3b .
(iv) Let R be the area of the region enclosed by the x-axis and the graph y = g(x).
Show that if a and b are rational then R is also rational.
(v) Is it possible for R to be a non-zero rational number when a and b are both irrational?
Justify your answer.
10
Extract from 2012 Mathematics Admissions Test
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Extract from 2012 Mathematics Admissions Test
4.


 MATHEMATICS

MATHEMATICS & STATISTICS
For APPLICANTS IN
ONLY.


MATHEMATICS & PHILOSOPHY
Mathematics & Computer Science, Computer Science and Computer Science & Philosophy applicants should turn to page 14.
The diagram below shows the parabola y = x2 and a circle with centre (0, 2) just ‘resting’
on the parabola. By ‘resting’ we mean that the circle and parabola are tangential to
each other at the points A and B.
y
4
3
2
A
B
1
-2
1
-1
2
x
(i) Let (x, y) be a point on the parabola such that x = 0. Show that the gradient of the
line joining this point to the centre of the circle is given by
x2 − 2
.
x
(ii) With the help of the result from part (i), or otherwise, show that the coordinates of
B are given by
3 3
,
.
2 2
(iii) Show that the area of the sector of the circle enclosed by the radius to A, the minor
arc AB and the radius to B is equal to
7
1
−1
√ .
cos
4
7
(iv) Suppose now that a circle with centre (0, a) is resting on the parabola, where a > 0.
Find the range of values of a for which the circle and parabola touch at two distinct
points.
(v) Let r be the radius of a circle with centre (0, a) that is resting on the parabola.
Express a as a function of r, distinguishing between the cases in which the circle is, and
is not, in contact with the vertex of the parabola.
12
Extract from 2012 Mathematics Admissions Test
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Extract from 2012 Mathematics Admissions Test
5. For ALL APPLICANTS.
A particular robot has three commands:
F: Move forward a unit distance;
L: Turn left 90o ;
R: Turn right 90o .
A program is a sequence of commands. We consider particular programs Pn (for n 0)
in this question. The basic program P0 just instructs the robot to move forward:
P0 = F.
The program Pn+1 (for n 0) involves performing Pn , turning left, performing Pn again,
then turning right:
Pn+1 = Pn L Pn R.
So, for example, P1 = F L F R.
(i) Write down the program P2 .
(ii) How far does the robot travel during the program Pn ? In other words, how many F
commands does it perform?
(iii) Let ln be the total number of commands in Pn ; so, for example, l0 = 1 and l1 = 4.
Write down an equation relating ln+1 to ln . Hence write down a formula for ln in terms
of n. No proof is required. Hint: consider ln + 2.
(iv) The robot starts at the origin, facing along the positive x-axis. What direction is
the robot facing after performing the program Pn ?
(v) The left-hand diagram on the opposite page shows the path the robot takes when it
performs the program P1 . On the right-hand diagram opposite, draw the path it takes
when it performs the program P4 .
(vi) Let (xn , yn ) be the position of the robot after performing the program Pn , so
(x0 , y0 ) = (1, 0) and (x1 , y1 ) = (1, 1). Give an equation relating (xn+1 , yn+1 ) to (xn , yn ).
What is (x8 , y8 )? What is (x8k , y8k )?
14
Extract from 2012 Mathematics Admissions Test
-4
4
4
2
2
2
-2
4
-4
2
-2
-2
-2
-4
-4
P1
4
P4
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15
Sample Solutions for Extract from 2012 Mathematics Admissions Test
SOLUTIONS FOR ADMISSIONS TEST IN
MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS
WEDNESDAY 7 NOVEMBER 2012
Mark Scheme:
Each part of Question 1 is worth four marks which are awarded solely for the correct answer.
Each of Questions 2-7 is worth 15 marks
QUESTION 1:
A.
can proceed by elimination. Point (1, 1) lies on line (a) and is clearly inside the given circle.
√ One
2, 0 lies on both lines (c) and (d) which is again inside the given circle. The answer is (b).
B. We can rewrite N as
N = 2k × 4m × 8n = 2k+2m+3n .
Now 2r is a square when r is even and is not a square when r is odd. So we need
k + 2m + 3n = k + n + 2 (m + n)
to be even , which is equivalent to needing k + n to be even. The answer is (d).
C. Note that log3 (92 ) = 2 log3 9 = 4. By comparison with this number
The answer is (d).
√ 3
√
3
= 3 3>4
as on squaring 27 > 16;
√ 2
27
3 3
π 2
=
=
> 4;
3 sin
3
2
4
log2 log2 85
= log2 (5 log2 8) = log2 15 < log2 16 = 4.
D. If −1 < c < 0 then the horizontal and vertical sides of the shaded triangle have length 1 + c and
hence
1
A(c) = (1 + c)2 .
2
This is a parabolic curve with minimum at (−1, 0) and so the answer is (a).
E. The curve is that of a quintic with a negative leading coefficient which eliminates (c). Two of
its roots are repeated which eliminates (b). Finally the smallest and middle of these roots are the
repeated ones which eliminates (a) which has its smallest and largest roots repeated. The answer
is (d).
1
Sample Solutions for Extract from 2012 Mathematics Admissions Test
F. None of the integrals needs to be calculated. We need only note
π/2
π
π
cos x dx > 0 as cos x > 0 for − < x < ;
2
2
−π/2
2π
sin x dx < 0 as sin x < 0 for π < x < 2π;
π
and
0
π/8
3π
3π
dx
is defined and positive as cos 3x > cos
> 0 for 0 < 3x <
;
cos 3x
8
8
The product of two positive numbers and a negative one is negative. The answer is (b).
G. Using the second equation to eliminate x we see that
2 (k − y) + ky = 4 =⇒ (k − 2) y = (4 − 2k) = −2 (k − 2) .
If k = 2 then we can divide by k − 2 and we see y = −2. So positive solutions aren’t possible when
k = 2. When k = 2 then the equations are
2x + 2y = 4,
x+y =2
which are both clearly satisfied by x = y = 1. The answer is (c).
H. A sketch of y = sin (sin t) in the range 0 < t 2π looks like
y
1.0
0.5
0.0
1
2
3
4
5
6
t
-0.5
-1.0
To appreciate this we need to realise how the graph y = sin (sin t) relates to the graph y = sin t. The
value sin t is in the range −1 sin t 1 and for values in the range −1 θ 1 then sin θ is a value
which is smaller than but of the same sign as θ. Importantly sin is also odd.
So for 0 < x π then
x
sin(sin t) dt > 0.
0
Also, by the oddness of sin we have that
2π
sin(sin t) dt = −
π
So
π
sin(sin t) dt.
0
x
sin(sin t) dt
0
remains positive for π < x < 2π, the integral only becoming 0 when the signed area between π and
2π cancels out the area between 0 and π. The only root of the equation in the given range is x = 2π
and the answer is (b).
2
Sample Solutions for Extract from 2012 Mathematics Admissions Test
I. If the radius of the circle is r then we have 2πr = 10 and r = 5/π. This distance r is also the
distance from the centre of the triangle to any of its vertices. So we have
√
π √
3 3 2
2π
1 2
=
r ;
P = 6 × r sin
= 3 3r.
A = 3 × r sin
2
3
4
3
Hence
√
3 3r2 /4
5
r
A
√
=
= =
P
4
4π
3 3r
and the answer is (a).
J. The area QP R is largest when Q and R are symmetrically placed about P , for if (say) P Q
were longer than P R then Q could be moved so as to gain more area than would be lost by the
corresponding move of R. This means that the angles QP O and RP O are both θ/2; the angle QOR
is 2θ as the angle subtended by QR at the centre O is twice that subtended at the circumference P .
Q
P
Ȑ2
Ȑ2
O
2Θ
R
We then see that the area of P QR equals
1
1
2 × × 12 × sin QOP + × 12 × 2θ = sin (π − θ) + θ = sin θ + θ
2 2
triangles
sector
and the answer is (b).
3
Sample Solutions for Extract from 2012 Mathematics Admissions Test
2. (i) Note that
f 2 g(x) = f (f (g(x))) = f (f (2x)) = f (2x + 1) = 2x + 2;
gf(x) = g(x + 1) = 2(x + 1) = 2x + 2.
(ii) Using the identity f 2 g = gf we see that
gf 2 g = g(f 2 g) = g(gf) = g 2 f
and also that
gf 2 g = gf (f g) = f 2 g(fg) = f 2 gf g
and finally that
f 2 gf g = f 2 (gf ) g = f 2 f 2 g g = f 4 g 2 .
These four, f 4 g 2 , f 2 gf g, gf 2 g, g 2 f are the only sequences that lead to 4x + 4.
(iii) Note that
f k (x) = x + k;
gf k (x) = 2 (x + k) = 2x + 2k;
f j gf k (x) = (2x + 2k) + j = 2x + 2k + j;
gf j gf k (x) = 2 (2x + 2k + j) = 4x + 4k + 2j;
f i gf j gf k (x) = (4x + 4k + 2j) + i = 4x + 4k + 2j + i.
(iv) We need to consider the ways we can have 4k + 2j + i = 4m where i, j, k 0. Clearly k can take
the values 0 k m and then we need
2j + i = 4 (m − k) .
Then j can take values 0 j 2 (m − k) — that is 2m − 2k + 1 choices for j (given k). The choice
of j then determines i. So the number of possible combinations is
m m m
(2m − 2k + 1) = (2m + 1)
1 −2
k
k=0
k=0
k=0
1
= (2m + 1) (m + 1) − 2 × m(m + 1)
2
= (m + 1) (m + 1)
= (m + 1)2 .
4
Sample Solutions for Extract from 2012 Mathematics Admissions Test
′
2
3. (i) For there to be two distinct turning points the derivative f (x) = 3x + 2ax + b must have two
distinct real roots. This is determined by the discriminant
(2a)2 − 4 × 3 × b = 4 a2 − 3b .
Thus y = f (x) has two distinct turning points if, and only if, a2 − 3b > 0.
(ii) The x-coordinates of the turning points are given by
√
√
a ± a2 − 3b
−2a ± 4a2 − 12b
=−
.
x=
2×3
3
If we call these x1 and x2 with x1 < x2 then
√
√
1
−a + a2 − 3b + a + a2 − 3b
x2 − x1 =
3
2√ 2
=
a − 3b.
3
(iii) The equation of the translated graph has a repeated root at x = 0 and another root at t (say),
where t is a positive real number. It is thus the case that
g(x) = x2 (x − t) = x3 − tx2 ,
and
g ′ (x) = 3x2 − 2tx = x(3x − 2t).
There are then turning points at x = 0 and x = 23 t. Thus, using the result from (ii), we see that
2√ 2
2
t=
a − 3b
3
3
and hence t =
√
a2 − 3b, as required.
(iv) The area of the region R is given by
t
t
−
g(x) dx = −
x2 (x − t) dx
0
0
t
4
x3
x
− t
=−
4
3 0
t4
=
12
2
(a2 − 3b)
=
,
12
which is rational when the coefficients a, b are rational.
√
4
2
(v) Yes, it
is
possible
for
R
to
be
rational
when
a
and
b
are
both
irrational.
For
example,
let
a
=
2
√
and b = 2.
5
Sample Solutions for Extract from 2012 Mathematics Admissions Test
2
4. (i) We need the gradient of the line segment joining (x, x ) to C = (0, 2). This is, for x = 0, given
by
x2 − 2
x2 − 2
=
.
x−0
x
(ii) Let the coordinates of B be (x1 , x21 ). Then the gradient of the parabola (and also of the circle)
at B is equal to 2x1 . The tangent to these curves at B is perpendicular to the line segment joining
(x1 , x21 ) to (0, 2), so that
x21 − 2
× 2x1 = −1.
x1
Solving this, and taking the appropriate solution, gives x1 = 32 and x21 = 32 .
(iii) The square of the radius CB is
2 2
3
7
3
= ,
+ 2−
2
2
4
so the area of the circle is
7π
.
4
The angle at the centre subtended by the minor arc AB is


1
1
−1  2 
−1
√ ,
= 2 cos
2 cos
7
7
4
from which we see the sector’s area equals
1 7
1 2
r θ = × × 2 cos−1
2
2 4
1
√
7
7
= cos−1
4
1
√
7
(iv) Consider the equation
x2 − a
× 2x = −1.
x
This can be rearranged to give x2 = a − 12 . For two distinct real solutions we require a > 12 .
(v) If 0 < a 1
2
then, from (iv), the circle is resting on the vertex of the parabola. In this case a = r.
On the other hand, if a >
1
2
then there are two distinct points points of contact, and we have
r2 = x2 + (x2 − a)2 .
Now, using the fact that x2 = a − 12 , we obtain
1
1
1
2
+ =⇒ a = r2 + .
r = a−
2
4
4
6
Sample Solutions for Extract from 2012 Mathematics Admissions Test
5. (i) From the recursive definition
P0 = F,
P1 = P0 L P0 R = F L F R
P2 = P1 L P1 R = F L F R L F L F R R.
(ii) Say there are fn commands F in Pn . As Pn+1 = Pn L Pn R then fn+1 = 2fn . As f0 = 1 then
fn = 2n .
(iii) Let ln denote the total number of commands in Pn . As Pn+1 = Pn L Pn R then
ln+1 = ln + 1 + ln + 1 = 2ln + 2.
If we set mn = ln + 2 (following the hint) then we see
mn+1 = ln+1 + 2 = 2ln + 4 = 2 (ln + 2) = 2mn .
As m0 = l0 + 2 = 3 then mn = 3 × 2n and
ln = mn − 2 = 3 × 2n − 2.
(iv) The robot again faces along the positive x-axis after each Pn because each Pn contains as many
Ls as it does Rs.
(v) P4 is sketched below:
4
2
-4
2
-2
4
-2
-4
(vi) After performing Pn the robot sits at (xn , yn ) facing "east". Turning left it now faces "north"
so what otherwise would have led to a movement of (xn , yn ) instead achieves (−yn , xn ). So from the
recursion Pn+1 = Pn L Pn R we can see that
(xn+1 , yn+1 ) = (xn , yn ) + (−yn , xn ) = (xn − yn , xn + yn ) .
Note then that
(xn+2 , yn+2 ) = (xn+1 − yn+1 , xn+1 + yn+1 ) = (−2yn , 2xn) ;
(xn+4 , yn+4 ) = (−2yn+2 , 2xn+2 ) = (−4xn , −4yn ) ;
(xn+8 , yn+8 ) = (−4xn+4 , −4yn+4 ) = (16xn , 16yn ) .
Hence
(x8 , y8 ) = (16, 0)
and
7
(x8k , y8k ) = 16k , 0 .