1. 1. What is the value of 44
+ 4 × 4 − 4? D
44
+ 4 × 4 − 4
256 + 4 × 4 − 4
256 + 16 − 4
272 − 4
268
 PEMDAS/GEMDAS
o Parenthesis/grouping symbol
o Exponent
o Multiplication
o Division
o Addition
o Subtraction
2. Which of the following is/are true? C
(a) Is false because it is a geometric sequence with common ratio of 2
(b) Is true since it follows the Fibonacci sequence in which you have two add the two
consecutive term to get the next term ...3,5,(3+5=)8,(5+8=)13, (13+8=)21...
Thus, 34 is the next term which is 13+21.
(c) Is true since it is an arithmetic sequence with common difference of 4.
(d) Is false because the common ratio of the given geometric sequence is 0.2
3. Two rectangles measuring 6 cm × 7 cm and 8 cm × 9 cm
overlap as shown. The region shaded black has an area of 62
cm2
. What is the area of the gray region? C
Since the area of 8cm by 9cm rectangle is 72cm2
, the
area of the white rectangle is 72cm2
-62cm2
=10cm2
.
Hence the area of the gray region is (7cm x 6m) –
10cm2
= 32cm2

2. 4. If the height of a triangle is divided by 4 and the base is multiplied by 4, what is the ratio
of the area of the new triangle to the area of the original triangle?A
Area of the original triangle =
𝑏ℎ
2
Area of the new triangle=
(4𝑏)(
ℎ
4
)
2
=
𝑏ℎ
2
AΔnew : AΔorig =
𝑏ℎ
2
:
𝑏ℎ
2
= 1:1
5. The base of a pyramid has edges. In terms of n, what is the difference between the
number of edges of the pyramid and the number of its faces? B
EDGES: A pyramid whose base has n edges also has n edges rising to its apex
and hence 2n edges in total.
FACES: It has n + 1 faces, including the base.
So the difference between the number of edges of the pyramid and the number
of its faces is
2𝑛 − (𝑛 + 1) = 𝑛 − 1
6. Paul is 32 years old. In 10 years' time, Paul's age will be the sum of the ages of his three
sons. What do the ages of each of Paul's three sons add up to at present? D
NOW IN 10 YEARS
Paul 32 32+10 = 42
Sum of the ages of three
sons
X x+(3*10) = x+30
x + 30 = 42
x = 42-30
x=12 years
7. The diagram shows a grid of 16 identical equilateral
triangles. How many different rhombuses are there made up
of two adjacent small triangles? C
A rhombus formed from a pair of adjacent triangles is
in one of three orientations:
+ +
6 + 6 + 6 = 18

3. 8. Consider the set of numbers {1, 2, 2, . . . , 5, 5, 5, 5, 5}, where the number n appears n-
times for 1 ≤ 𝑛 ≤ 10. What is the absolute value of the difference between the mode and
the median of the set? A
{1,2,2,3,3,3,4,4,4,4,5,5,5,5,5}
Median: the element at the middle of the set
Mode: most frequently occurring element in the set
|mode-median| = |5-4| = 1
9. In triangle ABC, <CAB=84˚; D is a point on AB such that<CDB = 3 × <ACD and DC =
DB. What is the size of <BCD? E
Let <ACD = xo
<CDB = 3xo
Then, from the straight line ADB, <ADC = (180-3x)o
Consider triangle ADC with angle sum 180o
:
84° + 𝑥 + (180 − 3𝑥) = 180°
2𝑥 = 84
𝑥 = 42°
Hence, <BCD = <DBC =
1
2
(180 − 3𝑥) =
1
2
(180 − 3(42)) =
1
2
(54) = 27°
10. If two printers can print five pages in four minutes, how many printers are needed to
print 20 pages on 16 minutes? B
𝐽
𝑊𝑇
=
𝐽2
𝐽2 𝑊2
5
(2)(4)
=
20
(𝑥)(16)
5
8
=
20
16𝑥
80𝑥 = 160
𝑥 = 2 𝑝𝑟𝑖𝑛𝑡𝑒𝑟𝑠

4. 11. A square is cut into two rectangles, as shown, so that the sum
of the lengths of the perimeters of these two rectangles is 30
cm. What is the length of a side of the square? B
Let x cm be the width of rectangle 1 and y cm be the
width of rectangle 2.
Then, both rectangles have length of x + y cm.
Perimeter of rec 1: 2𝐿 + 2𝑊 = 2(𝑥 + 𝑦) + 2𝑥 = 4𝑥 + 2𝑦
Perimeter of rec 2: 2𝐿 + 2𝑊 = 2(𝑥 + 𝑦) + 2𝑦 = 2𝑥 + 4𝑦
(4𝑥 + 2𝑦) + (2𝑥 + 4𝑦) = 30
6𝑥 + 6𝑦 = 30
𝑥 + 𝑦 = 5𝑐𝑚
(𝑥 + 𝑦)𝑐𝑚 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑠𝑞𝑢𝑎𝑟𝑒.
12. An athletics club has junior (i.e. boy or girl) members and adult members. The ratio of
girls to boys to adults is 3 : 4 : 9 and there are 16 more adult members than junior
members. In total, how many members does the club have?
Nine-sixteenths of the total club members are adults and seven-sixteenths are
junior. So two-sixteenths of the total, the difference between the number of adults
and juniors, is sixteen. Thus, one-sixteenth of the total membership is 8.
The total membership therefore is 16 × 8 = 128.
Alternative solution:
Let x be the total number of members.
Let y be the common ratio of the partitive proportion.
3𝑦 + 4𝑦 + 9𝑦 = 𝑥
9𝑦 = 7𝑦 + 16
2𝑦 = 16
𝑦 = 8
Substitute the value of y to solve for x:
3(8) + 4(8) + 9(8) = 𝑥
24 + 32 + 72 = 𝑥
𝑥 = 128

5. 13. What is the integer x so that
𝑥
9
lies between
71
7
and
113
11
?
We require
𝑥
9
>
71
7
7𝑥 > 639
𝑥 > 91
2
7
We also require
𝑥
9
<
113
11
11𝑥 < 1017
𝑥 < 92
5
11
Since 𝑥 is an integer, 𝑥 = 92.
14. What is the value of √3102 − 2013?
√3102 − 2013 = 1089
= √9 × 121
= √32 × 112
= √(3 × 11)2
= √332
= 33
15. Simplify
5+√3
2−√3
.
5 + √3
2 − √3
=
5 + √3
2 − √3
×
2 + √3
2 + √3
=
13 + 7√3
4 − 3
= 13 + 7√3
16. What percentage of ¼ is 1/5?
Let 𝑥 be the percentage.
1
4
𝑥 =
1
5
5𝑥 = 4
𝑥 =
4
5
𝑥 = 0.80
0.80 × 100 = 80%

6. 17. Calculate the value of x in the diagram shown.
2𝑥° + (𝑥 + 32)° + 40° = 180°
3𝑥 + 72 = 180
3𝑥 = 180 − 72
3𝑥 = 108
𝑥 = 36°
18. Divide 5𝑥3
− 14𝑥 + 3 by 𝑥 − 2. What is the remainder?
Solving by substitution:
Let 𝑥 = 2
5𝑥3
− 14𝑥 + 3
5(2)3
− 14(2) + 3
5(8) − 28 + 3
40 − 28 + 3
12 + 3
15
Alternative Solution: Performing division.

7. 19. Not all characters in the Woodentops series tell the truth. When Mr Plod asked them,
“How many people are there in the Woodentops family?”, four of them replied as
follows:
Jenny: “An even number.”
Willie: “An odd number.”
Sam: “A prime number.”
Mrs Scrubitt: “A number which is the product of two integers greater than one.”
How many of these four were telling the truth?
The number of people in the Woodentops family is a positive integer
which is greater than one.
Every such integer is either even or odd but not both. So precisely, one of
Jenny and Willie is telling the truth, but we don’t know which.
Also, every integer greater than one is either a prime number or the
product of two integers greater than one (composite number), but not both. So
precisely, one of Sam and Mrs. Scrubitt is telling the truth, but again we don’t
know which.
It follows that exactly two of them were telling the truth, though we don’t
know which two.
20. If 𝑐 =
𝑏−2𝑎
𝑎𝑏
, then a=?
𝑐 =
𝑏 − 2𝑎
𝑎𝑏
𝑎𝑏𝑐 = 𝑏 − 2𝑎
𝑏 = 𝑎𝑏𝑐 + 2𝑎
𝑏 = 𝑎(𝑏𝑐 + 2)
𝑏
(𝑏𝑐 + 2)
=
𝑎(𝑏𝑐 + 2)
(𝑏𝑐 + 2)
𝒂 =
𝒃
(𝒃𝒄 + 𝟐)

8. 21. Find a quadratic function whose roots are the square of the roots of x2
- 6x + 9 = 0
(𝑥 2
− 6𝑥 + 9) = 0
(𝑥 − 3)(𝑥 − 3) = 0
Root = 3
Square of the root = 9
𝑓(𝑥) = (𝑥 − 9)(𝑥 − 9)
𝑓(𝑥) = 𝑥2
− 18𝑥 + 81
22. Compute for sin 7π/12
sin (A + B) = sinAcosB + cosAsinB
sin 7π/12 = sin (π/4 + π/3)
sin π/4cos π/3 + cos π/4 sin π/3 =
√2
2
(1/2) +
√2
2
(
√3
2
) =
√𝟐+√𝟔
𝟒
23. Factor: (
4𝑥2
𝑦2
) − (9𝑎 − 𝑏)2
Difference of two squares:
(𝑎2
− 𝑏2) = (𝑎 + 𝑏)(𝑎 − 𝑏)
(
4𝑥2
𝑦2
) − (9𝑎 − 𝑏)2
(
2𝑥
𝑦
)
2
− (9𝑎 − 𝑏)2
[(
2𝑥
𝑦
) + (9𝑎 − 𝑏)] [(
2𝑥
𝑦
) − (9𝑎 − 𝑏)]
(
𝟐𝒙
𝒚
+ 𝟗𝒂 − 𝒃) (
𝟐𝒙
𝒚
− 𝟗𝒂 + 𝒃)
24. Find the equation of the tangent line to the circle 𝑥2
+ 𝑦2
= 289 𝑎𝑡 (8, 15).
a.) 15x–8y= 0 c.) 15x+8y-240= 0 e.) NOTA
b.) 8x–15y+161= 0 d.) 8x+15y-289= 0
Equation of circle with center at origin (0,0): 𝑥2
+ 𝑦2
= 𝑟2
Equation of circle with center at a point (h, k): (𝑥 − ℎ)2
+ (𝑦 − 𝑘)2
= 𝑟2
where r is the radius (distance from the center to a point on the circle).

9. In finding the equation of a line, we need:
(i) Two points; or
(ii) Slope and a point
For this question, we’ll use (ii).
Slope of radius: P(0,0) and (8,15)
m=
𝑦2−𝑦1
𝑥2−𝑥1
=
15−0
8−0
=
15
8
Note: The line tangent to a circle is perpendicular to the radius of the circle.
When two lines are perpendicular, their slopes are negative reciprocal of each
other.
M tangent line=
−8
15
, P (8, 15)
We now have the slope (−
8
15
)and a point(8,15) on the tangent line.
𝑦 − 𝑦1= 𝑚(𝑥 − 𝑥1)
𝑦 − 15 =
−8
15
(𝑥 − 8)
15𝑦 − 225 = −8𝑥 + 64
𝟖𝒙 + 𝟏𝟓𝒚 − 𝟐𝟖𝟗 = 𝟎
25. Which of the following graphs does not represent a function?
Use the vertical line test. The graph represents a function if the vertical line
intersects the graph at exactly one point.

10. 26. A rectangle has area 20 cm2
. Reducing the ‘length’ by 2 ½ cm and increasing the width’
by 3 cm changes the rectangle into a square. What is the side length of the square?
Let s be the side of the square.
Then the rectangle has length (𝑠 +
5
2
)cm and width (𝑠 − 3)cm.
The area of the rectangle based on the given information is
(𝑠 +
5
2
) (𝑠 − 3) = 20
(
2𝑠 + 5
2
) (𝑠 − 3) = 20
(2𝑠 + 5)(𝑠 − 3)
2
= 20
(2𝑠 + 5)(𝑠 − 3) = 40
(2𝑠2
− 𝑠 − 15) = 40
2𝑠2
− 𝑠 − 15 − 40 = 0
2𝑠2
− 𝑠 − 55 = 0
(2𝑠 − 11)(𝑠 + 5) = 0
𝑠 =
11
2
𝑐𝑚 = 5
1
2
𝑐𝑚 𝑠 = −5𝑐𝑚
Since the measurement of a square cannot have a negative value,
𝑠 = −5𝑐𝑚 is an extraneous root.
Therefore, the length of each side of the square is 5
1
2
𝑐𝑚.
27. The distance between the points 𝑃1(−4, 2)𝑎𝑛𝑑 𝑃2(3, −1)
Distance formula between two points P1 and P2
𝐷 = √(𝑥2 − 𝑥1)2 + (𝑦2−𝑦1)2
𝐷 = √(3 − (−4))
2
+ ((−1) − 2)
2
𝐷 = √(3 + 4)2 + (−1 − 2)2
𝐷 = √(7)2 + (−3)2
𝐷 = √49 + 9
𝑫 = √𝟓𝟖
28. Which of the following is divisible by 6?
Divisibility rules:
 By 2 – even number
 By 3 – sum of the digits is divisible by 3
 By 4 – last two digits is divisible by 4 or last two digits is 00
 By 5 – last digit is 5
 By 6 – divisible by 2 and 3

11.  By 8 – last three digits is divisible by 8 or last three digits is 000
 By 9 – the sum of the digits is divisible by 9
 By 10 – the last digit is 0.
For this number, we’ll use the divisibility rule for 6:
a. 1 000 000 – 1= 999 999 (divisible by 3 but not by 2)
b. 1 000 000 – 2= 999 998 (divisible by 2 but not by 3)
c. 1 000 000 – 3= 999 997 (not divisible by 3 and 2)
d. 1 000 000 – 4= 999 996 (divisible by 2 and 3)
e. 1 000 000 – 5= 999 995 (not divisible by 2 and 3)
29. How many quadrilaterals are there is this diagram, which
is constructed using 6 straight lines?
Quadrilaterals: 9
30. Which of the following has the least value?
a. 10
− 01
= 1 − 0 = 1
b. 21
− 12
= 2 − 1 = 1
c. 32
− 23
= 9 − 8 = 1
d. 43
− 34
= 64 − 81 = −17
e. 𝟓 𝟒
− 𝟒 𝟓
= 𝟔𝟐𝟓 − 𝟏𝟏𝟖𝟒 = −𝟓𝟓𝟗
31. Jane has 20 identical cards in the shape of an isosceles right-angled triangle. She uses
the cards to make the five shapes below. Which of the shapes has the shortest perimeter?
Special right triangle:
 Isosceles Right triangle (45° − 90° − 45)
𝑐 = 𝑎√2
𝑎 =
𝑐√2
2
 30° − 60° − 90°
𝑐 = 2𝑎
𝑏 = 𝑎√3

12. Let be the representation of the card.
a. 𝑃 = 𝑠 + 𝑠 + 𝑠 = (𝑎 + 𝑎) + (𝑎 + 𝑎) + (𝑎√2 + 𝑎√2 = 4𝑎 + 2𝑎√2
b. 𝑃 = 𝑠 + 𝑠 + 𝑠 + 𝑠 = (𝑎 + 𝑎) + (𝑎 + 𝑎) + (𝑎√2) + (𝑎√2 =
4𝑎 + 2𝑎√2
c. 𝑃 = 𝑠 + 𝑠 + 𝑠 + 𝑠 = 𝑎 + 𝑎 + 𝑎 + 𝑎 = 4𝑎
d. 𝑃 = 𝑠 + 𝑠 + 𝑠 + 𝑠 = (𝑎 + 𝑎) + (𝑎 + 𝑎) + 𝑎 + 𝑎 = 6𝑎
e. 𝑃 = 𝑠 + 𝑠 + 𝑠 + 𝑠 = 𝑎 + (𝑎 + 𝑎 + 𝑎) + 𝑎√2 + 𝑎√2 =
4𝑎 + 2𝑎√2

13. 32. For which of the following numbers is the sum of all its factors not equal to a square
number?
a. 3
b. 22
c. 66
d. 70
e. 40
33. The sum one + four = seventy becomes correct if we replace each word by the number of
letters in it to give 3+ 4 =7. Using the same convention, which of these words could be
substituted for x to make the sum
three + five = x true?
a. eight
b. nine
c. twelve
d. seventeen
e. eighteen
34. Which of the expressions below is equivalent to (𝑎 ÷ (𝑏 ÷ 𝑐)) ÷ ((𝑎 ÷ 𝑏) ÷ 𝑐)
a. a2
b. b2
c.
1
𝑎𝑏𝑐
d. 1
e. c2

14. 35. The diagrams show squares placed inside two identical semicircles
In the lower diagram the two squares are identical. What is the ratio
of the areas of the two shaded regions?
a. 1: 2
b. 2 : 3
c. 3 : 4
d. 4 : 5
e. 5 : 6
36. Which is the smallest positive integer for which all these are true?
(i) It is odd.
(ii) It is not prime.
(iii) The next largest odd integer is not prime.
a. 9
b. 15
c. 21
d. 25
e. 33
37. An equilateral triangle is placed inside a larger equilateral triangle so that the diagram has
three lines of symmetry. What is the value of x?
a. 110
b. 120
c. 130
d. 140

15. e. 150
38. The numbers x and y satisfy the equations x(y + 2) = 100 and y(x + 2) = 60. What is the
value of x − y?
a. 60
b. 50
c. 40
d. 30
e. 20
39. Zac halves a certain number and then adds 8 to the result. He finds that he obtains the
same answer if he doubles his original number and then subtracts 8 from the result. What
is Zac’s original number?
a. 8
2
3
b. 9
1
3
c. 9
2
3
d. 10
1
3
e. 10
2
3

16. 40. The diagram shows a circle with centre O and a triangleOPQ. Side PQ is a tangent to the
circle. The area of the circle is equal to the area of the triangle.
What is the ratio of the length of PQ to the circumference of the circle?
a. 1 : 1
b. 2 : 3
c. 2 : π
d. 3 : 2
e. π : 2
41. A machine cracks open 180 000 eggs per hour. How many eggs is that per second?
a. 5
b. 50
c. 500
d. 5000
e. 50 000
42. How many weeks are there in 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 minutes?
a. 1
b. 2
c. 3
d. 4
e. 5
43. Tuwing bakasyon, ginagamit ni Leah ang 50% ng kanyang oras sa pagbabasa ng mga
nobela. Sa kabuuan, kay a niy ang matapos ang isang nobela ng pitong oras. Kung si a ay
natutulog ng walong oras araw-araw, ilang nobela ang kay a niy ang tapusin sa loob ng
dalawang linggo?

17. a. 14
b. 16
c. 18
d. 20
44. Using the table below, fin d the value of 92.7
30.1
= 1.116
30.2
= 1.246
30.3
= 1.390
30.4
= 1.552
30.5
= 1.732
30.6
= 1.933
30.7
= 2.158
30.8
= 2.408
30.9
= 2.688

18. a. 377.098
b. 277.098
c. 377.136
d. 277.136
45. The numbers 2, 3, 12, 14, 15, 20, 21 may be divided into two sets so that the product of
the numbers in each set is the same. What is this product?
a. 420
b. 1260
c. 2520
d. 6720
e. 6 350 400
46. Which of these is the largest number?
a. 2 + 0 + 1 + 3
b. 2 × 0 + 1 + 3
c. 2 + 0 × 1 + 3
d. 2 + 0 + 1 × 3
e. 2 × 0 × 1 × 3
47. What is the value of
1
2−3
−
4
5−6
−
7
8−9
?
a. -10
b. 10
c. -11
d. 11
e. NOTA
48. What is the value of 11
+ 22
+ 33
+ 44
- (14
+ 23
+ 32 + 41
)?
266
49. Mike drank 60% of his glass of milk. Afterwards, 80 ml of milk remained in the glass.
What volume of milk was initially in the glass? 200m
50. The numbers x and y satisfy the equations x(y + 2) = 100 and y(x + 2) = 60. What is the
value of x − y?

19. a. 60
b. 50
c. 40
d. 30
e. 20
51. If E is the midpoint of AC and BD and 2AB = 2CD = AC = BD, what is the m∠AED?
a. 60°
b. 120°
c. 150°
d. Cannot be determined
e. NOTA
52. If
(a−b)
(c−b)
= −28, what is the value of
(a−c)
(b−c)
?
a. 29 c.) 28 e.) NOTA
b. 30 d.) 27
53. If x + 1 is a factor of f(x) = 2x3
– 7x2
– 5x + k , find the value of K.
a.) 3 c.) -2 e.) NOTA
b.) 4 d.) 5
E
D
C
A
B