Oxford University Maths Test Online Preparation Course Session 1 WELCOME

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Oxford University Maths TestOn-line Preparation
CourseSession 1WELCOME!
The Further Mathematics Network
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Oxford University Maths TestOn-line Preparation
Course
The Further Mathematics Network
Thursdays 6.30 pm to 7.30 pm
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Further Mathematics Network Online Sessions
Participants Rules
The Further Mathematics Network

• All Further Mathematics Network online sessions
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The Further Mathematics Network
Oxford Maths Entrance Test Syllabus
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The Further Mathematics Network
Oxford Maths Entrance Test Syllabus
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The Further Mathematics Network
Oxford Maths Entrance Test Syllabus
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The Further Mathematics Network
Worked Solutions to Oxford Maths Entrance
Specimen Test 1
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Question 1 (A)
Curves intersect at (1, 1) and (2, 4)
Area bounded by curves y x2 and y x 2 is
(c)
Specimen Test 1
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Question 1 (B)
We need to picture the curve!
Find end-points and any stationary points over
0 x 2
f(x) 2x3 9x2 12x 3
? f(0) 3 and f(2) 7
f(x) 6x2 18x 12
0 for stationary points
? 6(x2 3x 2) 0
? 6(x 1)(x 2) 0
? x 1 or x 2
f(1) 8 and f(2) 7
(b)
Now join the dots!
Curve fits points one way
Smallest value is f(0) 3
Specimen Test 1
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Sketch line 3x 4y 50 and plot the point!
Question 1 (C)
Note that gradient of line
Gradient of line through (0, 0) and (3,4)
(a)
The two lines are perpendicular!
(9, 12)

• Reflection of (3, 4) in line 3x 4y 50
• lies on line through
• (0, 0) and (3,4)

(6, 8)
(3, 4)
Image (9, 12)
Specimen Test 1
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f(x) x3 30x2 108x 104
Question 1 (D)
By the factor theorem f(a) 0 ? (x a)
is a factor
By inspection f(2) 0 ? (x 2) is a
factor
? f(x) (x 2)(x2 28x 52)
(d)
Now try to factorise the quadratic factor
(x2 28x 52) (x 2)(x 26)
Hence
f(x) (x 2)(x 2)(x 26)
(x 2)2(x 26)
There is a repeated root.
Specimen Test 1
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Question 1 (E)
6 7 42
A counter example is used to show a conjecture
is false.
Consider each statement in turn
(a) The product of two odd integers is odd.
This conjecture is true so there are no counter
examples.
(b) If the product of two integers is not a
multiple of 4 then the integers are not
consecutive.
42 is not a multiple of 4,but 6 and 7 are
consecutive. Hence 6 x 7 42 is indeed a
counter example, showing that the conjecture is
false.
(b)
Specimen Test 1
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Question 1 (E)
6 7 42
A counter example is used to show a conjecture
is false.
Consider each statement in turn
(c) If the product of two integers is a
multiple of 4 then the integers are not
consecutive.
42 is not a multiple of 4, therefore we cannot
tell whether the statement is true or false.
(d) Any even integer can be written as the
product of two even integers.
This conjecture is false but our statement is
not a counter example, we would need to show all
factorisations of 42 contain an odd number.
Specimen Test 1
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Question 1 (F)
2 cos2 x 5 sin x 4
Solve as a quadratic equation in sin x, over
range 0 x 2p
2 (1 sin2 x) 5 sin x 4
? 2 sin2 x 5 sin x 2 0
? (2 sin x 1)(sin x 2) 0
? sin x ½ or sin x 2
(a)
Specimen Test 1
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Question 1 (G)
x2 3x 2 gt 0 and x2 x lt 2
Solve each quadratic inequality in turn
x2 3x 2 gt 0
? (x 1)(x 2) gt 0
? x lt 2 or x gt 1
x2 x 2 lt 0
? (x 1)(x 2) gt 0
? 2 lt x lt 1
The values of x which satisfy both inequalities
are given by 1 lt x lt 1
(b)
Specimen Test 1
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Question 1 (H)
log10 2 0.3010 (to 4 dp) and 100.2 lt 2
Manipulating the first expression
log10 2 0.3010
? 100 log10 2 100 0.3010
? log10 2100 30.10
? 2100 1030.10
? 2100 100.10 1030
(c)
Since 100.10 lt 100.2 lt 2 2100 begins with 1
and is 30 1 31 digits long
Specimen Test 1
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Question 1 (I)
The coefficient of xk in the expansion is given
by
0 k 10
To find the value of k which maximises ck, we
examine
Hence the terms ck are growing provided
(b)
? 10 k gt 2(k 1)
? 8 gt 3k
? k 0, 1 or 2
? c3 is largest coefficient
Specimen Test 1
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Question 1 (J)
Which of the four graphs represents this equation?
x2y2(x y) 1
Specimen Test 1
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Question 1 (J)
x2y2(x y) 1
Neither x nor y can equal 0, so that eliminates
(a) and (b)
Specimen Test 1
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Question 1 (J)
x2y2(x y) 1
Since x2y2 is always positive, so x y gt 0
This eliminates (d), hence the correct graph is
(c)
(c)
Specimen Test 1
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Question 2
(i)
Expand brackets for
f(x) (x 1)(x2 (cos ? sin ?)x cos ?
sin ?)
x(x2 (cos ? sin ?)x cos ? sin ?) (x2
(cos ? sin ?)x cos ? sin ?)
x3 (cos ? sin ?)x2 (cos ? sin ?)x x2
(cos ? sin ?)x cos ? sin ?)
x3 (1 cos ? sin ?)x2 (cos ? sin ?
cos ? sin ?)x cos ? sin ?
Specimen Test 1
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Question 2
(i)
Further factorising of f(x) gives
f(x) (x 1)(x2 (cos ? sin ?)x cos ?
sin ?)
(x 1)(x cos ?)(x sin ?)
Hence roots of f(x) are
x 1, x cos ?, x sin ?
(ii)
When x ?p roots of f(x) are
x 1, x cos (?p ) ½, x sin (?p ) ½v3
Specimen Test 1
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Question 2
(iii)
Roots of f(x) are
x 1, x cos ?, x sin ?
Two roots are equal if
(A)
cos ? 1 ? ? 0 ,
(B)
sin ? 1 ? ? ½p , or
(C)
sin ? cos ? ? tan ? 1 ? ? ¼p or ?
1¼p
(iv)
Since 1 cos ? 1 and 1 sin ? 1
The greatest difference between two roots is 2
when x 1 and cos ? 1 ? ? p
or x 1 and sin ? 1 ? ? 1½p
Specimen Test 1
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Question 2
(iv)
If ? p , sin ? sin p 0 and cos ? cos
p 1
? x3 (1 1 0)x2 (10 1 0)x 10
0
? x3 x 0
If ? 1½p , sin ? sin 1½p 1 and cos ?
cos 1½p 0
? x3 (1 0 1)x2 (01 0 1)x 01
0
? x3 x 0
Specimen Test 1
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The Further Mathematics Network
Worked Solutions to Oxford Maths Entrance
Specimen Test 2
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Question 1 (A)
Points P (2, 3) and Q (8, 3)
Let R be the point which divides PQ in the ratio
12 with R (x, y)
P
Then x 2 ?(8 2) 4 y 3 ?(3 3)
1
R
Hence point R is R (4, 1)
Q
(d)
Specimen Test 2
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Question 1 (B)
Transform y f(x) to y f(x 1)
The two operations are
Translate 1 unit to the left f(x 1) blue
curve
followed by
Reflect in the x-axis f(x 1) green curve
(a)
Specimen Test 2
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Consider each statement in turn.
Question 1 (C)
(a)
(b)
(c) Let
then
(d) Let
then
Since c lt 1 and d gt 1
(d)
is largest in value
Specimen Test 2
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2x 3y 23, x 2 3y, 3y 1 4x
Question 1 (D)
Draw each boundary line then shade out unwanted
region
(d)
2x 3y 23 x 2 3y 3y 1 4x
Hence the largest value of x which satisfies all
3 inequalities is 7
(b)
Specimen Test 2
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Question 1 (E)
Solve cos (sin x) ½ for 0 x 2p
First solve cos ? ½
Since
have no solutions
(a)
Specimen Test 2
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Question 1 (F)
Parabola y x2 2ax 1
First complete the square
y (x a)2 a2 1
Hence turning point is at
(a, 1 a2)
Distance d from the origin satisfies
d2 a2 (1 a2)2
a2 1 2a2 a4
a4 a2 1
(a2 ½)2 ¾
Thus turning point is closest to the origin when
(d)
Specimen Test 2
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2 6 5 2
Question 1 (G)
Notice that 26 13 2 65 13 5
52 13 4
If we now extend the number by adding 33 652s
2 6 5 2 6 5 2 6 5 2 6 5 2 6 5 2 6 5 2
We now have a 100 digit number with similar
properties.
Starting with the digit 9 gives a similar pattern
9 1 3 9 1 3 9 1 3 9 1 3 9 1 3 9 1 3 9
where 91 13 7 13 13 1 39
13 3
Thus the 100-digit number begins and ends in a 9.
(d)
Specimen Test 2
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Question 1 (H)
(x2 1)10 2x x2 2
Complete the square on the RHS
2x x2 2 x2 2x 2
(x 1)2 1
? (x2 1)10 (x 1)2 1
LHS is positive for all values of x
RHS is negative for all values of x
Therefore there is no real number x which
satisfies the equation.
(b)
Specimen Test 2
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Question 1 (I)
23 8, 25 32, 32 9, 33 27
Rearranging these in ascending order gives
23 8, 32 9, 33 27, 25 32
? 23 lt 32 lt 33 lt 25
Now take logarithms to the base 2 throughout
? log2 23 lt log2 32 lt log2 33 lt log2 25
? 3 lt 2 log2 3 and 3 log2 3 lt 5
? 3/2 lt log2 3 and log2 3 lt 5/3
Hence
(b)
Specimen Test 2
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Question 1 (J)
y x2, y x2 2x, y x2 2x 2
Sketch the graphs by completing the square
y x2 y x2 2x (x 1)2 1 y
x2 2x 2 (x 1)2 1
Counting regions, the plane is divided into 7
regions.
Alternatively count regions as you sketch first
one, then the second, then the third.
(d)
Specimen Test 2
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Question 2
x4 Ax2 B (x2 ax b)(x2 ax b)
for all x
(i)
Comparing constants
x4 Ax2 B (x2 ax b)(x2 ax b)
? B b2
Comparing coefficients of x2
x4 Ax2 B (x2 ax b)(x2 ax b)
? A 2b a2
Specimen Test 2
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Question 2
x4 Ax2 B x4 20x2 16 ? A 20 and B
16
(ii)
? 16 b2 .. 1
and 20 2b a2 .. 2
From 1
16 b2 ? b 4
If b 4 then from 2
20 8 a2 ? a2 28
? a v28 2v7
? x4 20x2 16 (x2 2v7x 4)(x2 2v7x
4) ... 3
If b 4 then from 2
20 8 a2 ? a2 12
? a v12 2v3
? x4 20x2 16 (x2 2v3x 4)(x2 2v3x
4) ... 4
Specimen Test 2
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Question 2
From 3 x4 20x2 16 0
(iii)
? (x2 2v7x 4)(x2 2v7x 4) 0
? (x2 2v7x 4) 0 or (x2 2v7x 4)
0
Completing the square
(x v7)2 7 4 0 ? (x v7)2 3
? x v7 v3
? x v7 v3
(x v7)2 7 4 0 ? (x v7)2 3
? x v7 v3
? x v7 v3
Specimen Test 2
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Question 2
From 4 x4 20x2 16 0
(iii)
? (x2 2v3x 4)(x2 2v3x 4) 0
? (x2 2v3x 4) 0 or (x2 2v3x 4)
0
Completing the square
(x v3)2 3 4 0 ? (x v3)2 7
? x v3 v7
? x v3 v7
(x v3)2 3 4 0 ? (x v3)2 7
? x v3 v7
? x v3 v7
Specimen Test 2