Y11: My Booklet of Half-term Fun
Name: ________________________________________________
The SMC is on the first Thursday when we get back. All of the questions are taken from the second half of SMC papers
– which is where the really fun and interesting questions are!
Have a go at these in the booklet. In the grid below you can record your progress and make a note of things you might
want to ask your teacher about afterwards. Especially good efforts should be shown to teachers for validation! There
is room on the side for working. You can find answers and solutions online on nlcsmaths.weebly.com/challenges, as
well as a link to a set of past papers from 2007 to 2011.
Number
p. 2
Done
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Algebra
p. 7
Done
 ?
Shapes
p. 12
Done
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Circles
p. 19
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Triangles
p. 27
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1. Number
1.
Timmy Riddle was selling toffee apples at the school fête. When I asked him
what they cost he said "One toffee apple costs the smallest amount that
cannot be paid exactly using four or fewer standard British coins".
I bought as many toffee apples as I could get for £1. How much change did I
receive?
A 1p
2.
C 18p
D 22p
E 24p
Eight unit cubes are arranged to form an imaginary 2 by 2 by 2 cube. What is
the largest number of unit cubes one can remove from this arrangement if
the resulting shape has to have the same surface area as the original?
A 0
3.
B 5p
B 1
C 2
D 3
E 4
A teacher gave a test to 20 students. Marks on the test ranged from 0 to 10
inclusive. The average of the first twelve papers marked was 6.5. What can
you conclude from this about the eventual average M for the whole group?
A 0.325 ≤ M ≤ 6.5
B 3.25 ≤ M ≤ 6.5
C
3.9 ≤ M ≤ 6.5
D 3.9 ≤ M ≤ 7.9
E
6.5 ≤ M ≤ 7.9
4.
If n is some integer, 1 ≤ n ≤ 9, what is the value of (0.n) / (0. n )?
.
A 1/10
B 9/10
C 1
D 10/9
2
E
it depends on n
5.
What is the value of
1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))))?
A 210 +1
B 211 −1
C 211 +1
D
212 −1
E 212 + 1
6.
I am trying to do a rectangular jigsaw puzzle. The puzzle was made by
starting with a rectangular picture and then cutting it into 1000 pieces by
sawing along the lines of a (wiggly!) rectangular grid. I start by separating out
all the edge and corner pieces.
Which of the following could not possibly be the number of corner and edge
pieces of such a jigsaw?
A
126
B
136
C
216
D
316 E
A-D are all possible
7.
Observe that 18 = 42 + 12 + 12 + 02.
How many of the first fifteen positive integers can be written as the sum of
the squares of four integers?
A 11
8.
B 12
C 13
D 14
E 15
Pascal, Newton, Galileo and Fermat all took the same test. The average
score of all four candidates was 16; Pascal and Newton had an average of
16, Pascal and Fermat had an average of 13, while Newton and Fermat had
an average of 18.
What was Galileo's score?
A 14
B 15
C 16
D 17
3
E 18
9.
Mary's height increased by 30% between her 5th birthday and her 10th
birthday. It
increased by 20% between her 10th birthday and her 15th birthday.
By how much did her height increase between her 5th birthday and her 15th
birthday?
A 50%
B 52%
C 54%
D
56%
E 60%
10.
The probability of a single ticket winning the jackpot in the National Lottery is
6
5
4
3
2
1





.
49 48 47 46 45 44
If I buy one ticket every week, approximately how often might I expect to win
the jackpot?
A once every hundred years
thousand years
B once every twenty
C once every hundred thousand years
D once every quarter of a million years
E once every million
years
11.
Damien wishes to find out if 457 is a prime number. In order to do this he
needs to check whether it is exactly divisible by some prime numbers. What
is the smallest number of possible prime number divisors that Damien needs
to check before he can be sure that 457 is a prime number?
A 8
12.
B 9
C 10
D 11
E 12
The ratio of Jon's age to Jan's age is 3 : 1. Three years ago the ratio was 4 :
1.
In how many years time will the ratio be 2 : 1?
A 3
B 6
C 9
D 12
4
E 15
13.
How many two-digit numbers N have the property that the sum of N and the
number formed by reversing the digits of N is a square?
A 2
B 5
C 6
D
7 E
8
14.
In a sale, a shopkeeper reduced the advertised selling price of a dress by
20%. This
resulted in a profit of 4% over the cost price of the dress.
What percentage profit would the shopkeeper have made if the dress had
been sold at the original selling price?
A 16%
B 20%
C 24%
D
25%
E 30%
15.
In 1954, a total of 6 527 mm of rain fell at Sprinkling Tarn and this set a UK
record
for annual rainfall. The tarn has a surface area of 23 450 m2.
Roughly how many million litres of water fell on Sprinkling Tarn in 1954?
A 15
B 150
D
15 000
E 150 000
16.
Eight identical regular octagons are placed edge to edge in a ring in such a
way that a symmetrical star shape is formed by the interior edges. If each
octagon has sides of length 1, what is the area of the star?
D
A 5 + 10 2
B 8 2
16 – 4 2
E 8+4 2
C 1 500
C 9+4 2
5
17.
The number N is exactly divisible by 7. It has 4008 digits. Reading from left
to right, the first 2003 digits are all 2s, the next digit is n and the last 2004
digits are all 8s.
What is the value of n?
A 4
B 5
D
2 or 9
E 1 or 8
18.
What is the sum of the values of n for which both n and
A −8
19.
B −4
C 0 or 3
C 0
n2 – 9
are integers?
n –1
D 4
E 8
The factorial of n, written n!, is defined by n! = 1 × 2 × 3 ×…× (n − 2) × (n − 1)
× n.
For how many positive integer values of k less than 50 is it impossible to find
a value of n such that n! ends in exactly k zeros?
A 0
20.
B 5
C 8
D 9
Let N be a positive integer less than 102002. When the digit 1 is placed after
the last digit of N, the number formed is three times the number formed when
the digit 1 is placed in front of the first digit of N.
How many different values of N are there?
D
E 10
A 1
B 42
667
E 2002
C 333
6
2. Algebra
1.
A teacher gave a test to 20 students. Marks on the test ranged from 0 to 10
inclusive. The average of the first twelve papers marked was 6.5. What can
you conclude from this about the eventual average M for the whole group?
A 0.325 ≤ M ≤ 6.5
B 3.25 ≤ M ≤ 6.5
C
3.9 ≤ M ≤ 6.5
D 3.9 ≤ M ≤ 7.9
E
6.5 ≤ M ≤ 7.9
2.
Pascal, Newton, Galileo and Fermat all took the same test. The average
score of all four candidates was 16; Pascal and Newton had an average of
16, Pascal and Fermat had an average of 13, while Newton and Fermat had
an average of 18.
What was Galileo's score?
A 14
3.
B 15
C 16
D 17
E 18
A jogger runs a certain distance at V ms-1, and then walks half that distance
at U ms-1.
If the total time for the two stages is T seconds, what is the total distance
travelled (in metres)?
A
D
4.
3TUV
U  2V
TUV
2U + V
B
3TUV
2U  V
E
2TUV
2U  V
C
3T
U  2V
Given that a and b are integers greater than zero, which of the following
equations could be true?
A a – b = a b
B a + b = a b
D a+b=a–b
C a–b=a  b
E
a+ b = a + b
7
5.
The difference between two numbers is one quarter of their sum.
What is the ratio of the smaller number to the larger number?
A 3:8
B 1:2
C 5:7
D
1:4
E 3:5
6.
Consider the arithmetic sequences 1998, 2005, 2012,…and 1996, 2005,
2014,…. Which is the next number after 2005 that appears in both
sequences?
A 2054
B 2059
C 2061
D
2063
E 2068
7.
Three consecutive even numbers are such that the sum of four times the
smallest
and twice the largest exceeds three times the second by 2006.
What is the sum of the digits of the smallest number?
A 8
8.
B 11
C 14
D 17
E 20
Heather and Rachel each has some pennies. Heather has more than
Rachel. In fact, the number of pennies that Heather has is the square of the
number that Rachel has. The total number of pennies they have between
them makes a whole number of pounds.
What is the smallest this total could be?
A £1
B £6
D
£99
E £101
9.
What is the greatest number of the following five statements about numbers
a, b
which can be true at the same time?
1 1
a2 > b2
a<b
a<0
b<0
a<b
A 1
B 2
C £57
C 3
D 4
8
E 5
10. Given that x  y1 , where x and y are unequal and non-zero, which of the
following is
 x  1  y  1 


x 
y ?
always equal to 
A y2 − x2
B x2 − y2
C 2y
D 2x
E 0
11. For how many integer values of n does the equation x2 + nx − 16 = 0 have
integer solutions?
A 2
12.
B 3
C 4
D 5
E 6
B x8 + x6 + x4 + x2 + 1 C
x8 + 1
( x – 1)( x 4 + 1)( x 2 + 1)( x + 1) equals
A x8 − 1
D x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1
13.
E
x8 − x6 + x4 − 1
In triangle PQR, angle P = 90°, PR = 15 cm and QR = 17 cm. Circular arcs
are drawn with centres at P, Q and R, and each arc touches the other two
arcs.
What is the radius of the arc with centre R?
Q
17 cm
P
A 10 cm
B 10.5 cm
R
15 cm
C 11 cm
D 11.5 cm
E 12
cm
14.
The following equation is true for all a, b and c:
a3 + b3 + c3 = (a + b + c)3 − 3 (a + b + c) (ab + bc + ca) + kabc
What is the value of k?
A −6
B −3
C 0
D 3
9
E 6
15.
1.
The ratio of Jon's age to Jan's age is 3 : 1. Three years ago the ratio was 4 :
In how many years time will the ratio be 2 : 1?
A 3
16.
B 6
C 9
D 12
E 15
Five peaches, three oranges and two melons cost £3.18. Four peaches,
eight oranges and three melons cost £4.49.
How much more expensive is a peach than an orange?
A 8p
B 7p
D
5p E
more information needed
17.
Which positive integer n satisfies the equation
3
4
5
n3 – 5 n3 – 4 n3 – 3
+
+

...
+
+
+
= 60?
n3 n3 n3
n3
n3
n3
A 5
2006
18.
C 6p
B 11
x
Given that y 
x
x
xy
C 31
D 60
E
, for which of the following values of x is y not a real
number?
A −6
19.
B −3
C 1
A function f has the property that f(n + 3) =
D 3
E 6
f (n )  1
for all positive integers n.
f (n )  1
Given that f (2002) is non-zero, what is the value of f (2002) × f (2008)?
A 1
needed
B −1
C 2
D −2
E more information
10
20. X is a positive integer in which each digit is 1; that is, X is of the form
11111… .
Given that every digit of the integer pX2 + qX + r (where p, q and r are fixed integer
coefficients and p > 0) is also 1, irrespective of the number of digits in X, which of
the following is a possible value of q?
A –2
B –1
C 0
D 1
11
E 2
3. Shapes
1.
A cube ABCDEFGH has ABCD as square base, with E, F, G, H above A, B,
C, D
respectively.
What is the cosine of the angle ∠CAG ?
A 1/√3
√3/2
2.
B √2/3
C 1/√2
D √(2/3)
E
A square has the same perimeter as a 4cm by 2cm rectangle.
What is the area of the square (in cm2)?
A 4
3.
B 8
C 9
D 10
E 12
ABCDEFGH is a regular octagon. P is the point inside the octagon such that
triangle ABP is equilateral.
What is the size of angle APC?
A 90°
135°
4.
B 112.5°
C 117.5°
D 120°
E
The smaller circle has radius 10 units; AB is a diameter. The larger circle has
centre A, radius 12 units and cuts the smaller circle at C.
What is the length of the chord CB?
C
A
A 8
B 10
B
D 10√2
C 12
12
E 16
5.
The size of each exterior angle of a regular polygon is one quarter of the size
of an interior angle.
How many sides does the polygon have?
A 6
6.
B 8
C 9
D 10
E 12
A square piece of wood, of side 8 cm, is painted black and fixed to a table.
An equal square, painted white, is placed on the table alongside the black
square and has a point P marked one quarter of the way along a diagonal,
as shown. Whilst keeping the same orientation on the table and always
remaining in contact with the black square, the white square now slides once
around the black square.
Through what distance does P move?
P
A 32 cm
B 48 cm
C 64 cm
D
72 cm
E 80 cm
7.
A trapezium has parallel sides of length a and b, and height h. Sides a and b
are both decreased by 10% and the height h is increased by 10%.
What is the percentage change in the area of the trapezium?
a
h
b
A 10% decrease
B 1% decrease
D 10% increase
C
no change
E 30% increase
13
8.
ABCDEF is a regular hexagon of area 60.
What is the area of the kite-shaped figure ABCE?
A
B
F
C
E
A 20 3
9.
D
B 40
C 49
D 50
E 51
The diagram shows two concentric circles of radii r and 2r respectively.
What is the ratio of the total shaded area to the total unshaded area?
120°
A 5:7
10.
B 7:5
C 1:1
D 2:3
E 3:2
The diagram shows a 2 × 2 square and a 3 × 1 rectangle. One vertex of the
square lies on a side of the rectangle. The sides of the rectangle are parallel
to the diagonals of the square.
What is the area of the shaded triangle?
A 1
2
B 1
C 3
2
D 2
14
E 5
2
11.
A triangle is cut from the corner of a rectangle. The resulting pentagon has
sides of
length 8, 10, 13, 15 and 20 units, though not necessarily in that order.
What is the area of the pentagon?
A 252.5
282.5
12.
B 260
C 270
D 275.5
E
The point O is the centre of both circles and the shaded area is one-sixth of
the area of the outer circle.
What is the value of x?
A 60
B 64
O
13.
C 72
x
1
D 80
E 84
3
How many hexagons can be found in the diagram below if each side of a
hexagon must consist of all or part of one of the straight lines in the diagram?
A 4
B 8
C 12
D 16
15
E 20
14.
The base of a pyramid has n edges.
What is the difference between the number of edges the pyramid has and
the number of faces the pyramid has?
A n−2
B n−1
C n
D n+1
E n+
2
15.
PQRS is a square with U and V the mid-points of the sides PS and SR
respectively.
Line segments PV and UR meet at T.
What fraction of the area of the square PQRS is the area of the quadrilateral
PQRT?
P
Q
U
T
S
5
B 8
1
A 2
16.
R
V
3
D 4
2
C 3
5
E 9
In triangle PQR, S and T are the midpoints of PR and PQ respectively; QS is
perpendicular to RT; QS = 8; RT = 12.
What is the area of triangle PQR?
Q
T
P
A 24
R
S
B 32
C 48
D 64
16
E 96
17.
The sum of the lengths of the 12 edges of a cuboid is x cm. The distance
from one
corner of the cuboid to the furthest corner is y cm.
What, in cm2, is the total surface area of the cuboid?
x 2 – 2y 2
2
A
xy
6
D
18.
B x2 + y 2
E
C
x 2 – 4y 2
4
x 2 – 16 y 2
16
A paperweight is made from a glass cube of side 2 units by first shearing off
the eight
tetrahedral corners which touch at the midpoints of the edges of the cube.
The remaining inner core of the cube is discarded and replaced by a sphere.
The eight corner pieces are now stuck onto the sphere so that they have the
same positions relative to each other as they did originally.
What is the diameter of the sphere?
A √8 – 1
1
C 3 (6 + √3)
B √8 + 1
4
D 3 √3
E 2√3
19. A solid red plastic cube, volume 1cm3, is painted white on its outside.
The cube is cut by a plane passing through the midpoints of various edges, as
shown.
What, in cm2, is the total red area exposed by the cut?
A
B
F
C
E
D
A
D
3 3
2
3 E
B 2
C
9 2
5
3( 3 + 2 )
4
17
20.
AA’ and BB’ are arcs of concentric circles with centre O and with radii a and
b respectively. Let ∠A' OA = x°. The length of the arc AA′ is equal to the total
distance from A to A′ via the arc BB′.
Find the value of x to the nearest integer.
A'
B'
x°
D
O
B
A
C
125
A 115
B
120
135
E
it depends on a and b
18
4. Circles
1.
A circular disc of diameter d rolls without slipping around the inside of a ring
of internal
diameter 3d, as shown in the diagram.
By the time that the centre of the inner disc returns to its original position for
the first time, how many times will the inner disc have turned about its
centre?
A 1
2.
B π
C 3
D 2π
E 2
On 2 July 2002, Steve Fossett completed the first solo balloon
circumnavigation of the
world after 13 1 days.
2
Assuming the balloon travelled along a circle of diameter 12 750 km, roughly
what was the average speed of the balloon in km/h?
A 12
3.
B 40
C 75
D 120
E 300
The diagram shows seven circles of equal radius which fit snugly in the
larger circle.
What is the ratio of the unshaded area to the shaded area?
A 7:1
B 7:2
C 2√3 : 1
D 9:2
19
E 1:1
4.
The 80 spokes of the giant wheel The London Eye are made from 4 miles of
cable.
Roughly what is the circumference of the wheel in metres?
A 50
5.
B 100
C 500
D 750
E 900
A roll of adhesive tape is wound round a central cylindrical core of radius 3
cm. The outer radius of a roll containing 20 m of tape is 4 cm.
Approximately, what is the outer radius of a roll containing 80 m of tape?
A 5 cm
B 5·5 cm
C 6 cm
D
7 cm
E 12 cm
6.
In the diagram, O is the centre of the circle, AOB = α and COD = β.
What is the size of AXB in terms of α and β?
A
.O
D
B
C
X
A 1 α – 1β
2
2
C
D
B 90  – 1 α – 1 β
2
2
α−β
180° − α − β
E more information needed
20
7.
Three circles touch, as shown in the diagram. The2 two
 1 larger circles both
have radius 1 and the smaller circle has radius
.
What is the perimeter of the shaded region?
A
8.
π
( 2  1)
4
B
π
( 2  1)
2
C
π
2
D
π
( 2  1)
4
E
2
A sculpture consists of a row of 2 metre rods each placed with one end
resting on horizontal ground and the other end resting against a vertical wall.
The diagram shows how the rods BT, CU, DV, … look from above. The
bases of the rods B, C, D, … lie on a straight line on the ground at 45° to the
wall. The top ends of the rods T, U, V… lie on part of a curve on the wall.
What curve is it?
T U V
B
C
D
A a straight line
B a parabola
D
a sine curve
E a quartic curve
9.
All six vertices of hexagon UVWXYZ lie on the circumference of a circle;
ZUV = 88° and XYZ = 158°.
C a circle
What is the size of VWX?
D
π
A 92°
B 114°
C 120°
132°
E it is impossible to determine
21
10.
The trunk of a monkey-puzzle tree has diameter 40 cm. As a protection from
fire, the
trunk of the tree has a bark which makes up 19% of its volume.
On average, roughly how thick is the bark of the trunk?
A 0.4 cm
B 1.2 cm
C 2 cm
D
2.8 cm
E 4 cm
11.
A sculpture is made up of 12 wooden cylinders, each of height 2cm. They
are glued together as shown. The diameter of the top cylinder is 2cm and
each of the other cylinders has a diameter 2cm more than the one
immediately above it. The exhibit stands with its base on a marble table.
What, in cm2, is the total surface area of the sculpture, excluding the base?
A 456π
B 356π
C 256π
D
156π
E 144π
12.
A circle is inscribed in an equilateral triangle. Small circles are then inscribed
in each
corner as shown.
What is the ratio of the area of a small circle to that of the large circle?
A 1:3
B 1 : 4.5
C 1 : 33
D 1:6
9
22
E 1:
13.
The curvy shape ABC shown here is called a Reuleaux triangle (after the
French engineer Franz Reuleaux (1829 - 1905)). Its perimeter consists of
three equal arcs AB, BC, CA each with the same radius and centred at the
opposite vertex. In the Reuleaux triangle shown, each arc has radius 3cm.
What is the area (in cm2) of the inscribed circle?
A
C
A 6π(2 − √3)
14.
B
B 9π / 4
C 2π(3 − √3)
E 9π
Circles with radii r and R (where r < R) touch each other and also touch two
perpendicular lines as shown.
What is the value of R/r?
D
D 3π / 4
A 5 2
B 5.75
6 E
32 2
C
40
23
15.
The area of each large semicircle is 2.
What is the difference between the black and grey shaded areas?
1
2
A 0
B
C 1+ 2 2
D
5
E
9
23 – 16 2
16.
What is the radius of the shaded semicircle?
2
1
1
A
2 1
1
B
C 32 2
2
D
1
E
2
17.
The figure shows two parallel lines,  1 , and  2 . Line  1 is a tangent to
2 2
circles C1 and C3, line  2 is a tangent to circles C2 and C3 and the three circles
touch as shown. Circles C1 and C2 have radius s and t respectively.
What is the radius of circle C3?
1
2
1
2
C2
C1
C3
24
A 2 s2  t 2
B s+t
E
18.
C 2 st
D
4st
s+t
more information needed
In the diagram AB, CB, and XY are tangents to the circle with centre O and
ABC = 48°.
What is the size of XOY?
C
X
B
48°
O
Y
A
A 42°
B 69°
C 66°
D 48°
19. A company logo has a centrally-symmetric white cross of width
circle. The dark corner pieces have sides of length 1 as indicated.
E 84°
2 on a dark
What is the total area of the corners?
1
1
2
2
A π (2 – √2) +
C
2
2
B π
1
2
π(4 – √2) – 4√2
D
(π  2)
2
E
25
π (2 + 2)
– 2√2
2
20.
The diagram shows two concentric circles. The chord of the large circle is a
tangent to
the small circle and has length 2p.
What is the area of the shaded region?
A πp2
needed
B 2πp2
C 3πp2
D 4πp2
26
E
more information
5. Triangles
1.
How many differently shaped triangles exist in which no two sides are the
same length,
each side is of integral unit length and the perimeter of the triangle is less
than 13 units?
A 2
2.
B 3
C 4
D 5
E 6
A cube ABCDEFGH has ABCD as square base, with E, F, G, H above A, B,
C, D
respectively.
What is the cosine of the angle ∠CAG ?
A 1/√3
B √2/3
C 1/√2
D √(2/3)
E
√3/2
3.
Which of the following equations could be the equation of the "curve", part of
which is shown here?
y
x
A y = sin x B |y| = sin x
impossible
4.
D |y| = |sin x|
If cos θ =1/2, which of these cannot equal sin 2θ?
A sin θ
E
C y = |sin x|
B 1/2
C –√3/2
D √3/2
2 cos θ sin θ
27
E
A-D all
5.
Which of the following could be the graph of y = sin (x2)?
A
B
C
D
6.
E
P is a vertex of a cuboid and Q, R and S are three points on the edges as
shown. PQ = 2 cm, PR = 2 cm and PS = 1 cm.
What is the area, in cm2, of triangle QRS?
Q
P
R
S
A
7.
15 /4
B 5/2
C
6
D 2 2
E
10
L, M and N are midpoints of the sides of a skeleton cube, as shown.
What is the value of angle LMN?
N
M
L
A 90°
E
B 105°
C 120°
D 135°
150°
28
8.
The diagram shows a square and two equilateral triangles. All the sides have
length 1.
What is the length of XY?
1
A
3 1
X
Y
B
2
3
C
3
4
D
3
2
E
2 3
4
9.
It takes two weeks to clean the 3312 panes of glass in the 6000m2 glass roof
of the British Museum, a task performed once every two years.
Assuming that all the panes are equilateral triangles of the same size,
roughly how long is the side of each pane?
A 50 cm
B 1m
C 2m
D 3m
E 4m
10. In a triangle the perpendicular from a vertex to the opposite side is called an
altitude.
If h, h′, h″, denote the lengths of the three altitudes of a triangle, which of the
following ratios never occurs as the ratio h : h′ : h″?
A 2:3:4
E
B 2:3:5
C 2:4:5
D 3:4:5
3:4:6
11. Triangle PQR has a right angle at Q and PQ = QR. The line through Q which
divides the angle PQR in the ratio 1:2 meets PR at S.
What is the ratio RS:SP?
P
S
Q
R
A √2 :1
B √3 :1
D √5 :1
C 2:1
29
E 3:1
12.
If α < β, how many different values are there among the following
expressions?
sin α sin β
A 1
value of α
13.
sin α cos β
B 2
cos α sin β
C 3
cos α cos β
D 4
E It depends on the
The two triangles have equal areas and the four marked lengths are equal.
What is the value of x?
2x°
x°
A 30
needed
14.
B 45
C 60
D 75
E more information
Triangle ABC is isosceles with AB = AC, and D is the midpoint of AB.
If ∠BCD = ∠BAC = θ , then cos θ equals
A 3/4
B √7/(2√2)
C 1/√2
D
√7/4
E 1/(2√2)
15.
AA’ and BB’ are arcs of concentric circles with centre O and with radii a and
b respectively. Let ∠A' OA = x°. The length of the arc AA′ is equal to the total
distance from A to A′ via the arc BB′.
Find the value of x to the nearest integer.
A'
B'
x°
A 115
B
depends on a and b
O
B
A
120
C
125
D
30
135
E
it
∧
16.
∧
Triangle ABC has A B C = 90  and A C B = 30  .
If a point inside the triangle is chosen at random, what is the probability that it
is nearer to AB than it is to AC?
A
17.
3
2
B
1
2
C
1
D
3
1
3
E
1
4
The outer equilateral triangle has area 1. The points A, B, C are a quarter of
the way along the sides as shown.
What is the area of the equilateral triangle ABC?
A
C
B
A
18.
3
8
B
7
16
C
1
D
2
9
16
E
5
8
Which of the following expressions is identically equal to sin3 x + cos3 x?
A sin 3x + cos 3x
D (sin x + cos x)3
B 1
C (sin x + cos x) (1 − sin x cos x)
E (sin x + cos x) (2 sin x cos x + 1)
31
19.
A company logo has a centrally-symmetric white cross of width 2 on a dark
circle. The dark corner pieces have sides of length 1 as indicated.
What is the total area of the corners?
1
1
2
2
A π (2 – √2) +
2
2
B π
1
C
2
π(4 –
√2) – 4√2
D
20.
(π  2)
2
E
π (2 + 2)
– 2√2
2
The black triangle is drawn, and a square is drawn on each of its edges. The
three shaded triangles are then formed by drawing three lines which join
vertices of the squares and a square is now drawn on each of these three
lines. The total area of the original three squares is A1, and the total area of
the three new squares is A2.
Given that A2 = kA1 then
A k=1
B k=3
2
C k=2
D k=3
needed
32
E
more information