This document contains instructions and questions for a Grade 9 mathematics exam. It is divided into three levels of difficulty - easy, average, and difficult - with point values assigned to each level. The exam contains 30 multiple choice and open-ended questions testing a range of math skills like solving quadratic equations, working with ratios, geometry, trigonometry, and word problems involving measurement. Students have 60 minutes to complete all questions, with partial credit awarded based on difficulty level.

1. EXAMINATION IN
GRADE 9
MATHEMATICS
JULY 10, 2023
REPUBLIC OF THE PHILIPPINES
DEPARTMENT OF EDUCATION
REGION IV-A CALABARZON
DIVISION OF CAVITE
DISTRICT OF MARAGONDON
BUCAL NATIONAL INTEGRATED SCHOOL

2. INSTRUCTIONS
 This exam contains 30 questions and divided into 3
levels – easy, average, and difficult. Each level has a
corresponding point – 1 point for easy, 2 points for
average, and 3 points for difficult. This exam is an
open ended. You have 60 minutes or 1 hour to
answer all the questions. When the time is up, you
should pass your paper immediately and considered
all unanswered questions as wrong or zero point.

3. EASY [1 point]
LEVEL

4. Question #1:
What do you call a second-degree
polynomial equation that can be
written in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0,
where a, b and c are real numbers
and 𝑎 ≠ 0?

5. Question #2:
The dimensions of a rectangle with an
area of 56 meters2 are (3x - 1) meters
by (x + 4) meters. What quadratic
equation in standard form represents
the situation?

6. Question #3:
In the quadratic equation
3x2 + 7x – 4 = 0, which is
the quadratic term?

7. Question #4:
How many real roots does
the quadratic equation 𝑥2 +
8𝑥 + 12 = 0 have?

8. Question #5:
Solve x (x– 5) = 36.

9. Question #6:
What are the dimensions of
the largest rectangular field
that can be enclosed by 80 m
of fencing wire?

10. Question #7:
If y varies directly as x and
y = 24 when x = 6, find the
constant of variation.

11. Question #8:
What is the result after
simplifying
2 3 + 4 3 - 5 3 ?

12. Question #9:
In rectangle KAYE, YO =
18 cm. Find the length of
diagonal AE.

13. Question #10:
What is the perimeter of a 30-
60-90 triangle whose shorter
leg is 5 inches long?

14. AVERAGE [2 points]
LEVEL

15. Question #11:
Find the roots of the
equation (x + 5)2 + (x – 2)2 =
37.

16. Question #12:
Express y = 3x2 – 4x + 1 in
the form y = a (x - h)2 + k form
and give the values of h and
k.

17. Question #13:
The maximum load m of a beam varies directly as
the breadth b and the square of the depth d and
inversely as the length l. If a beam 3.6 m long, 1.2
m wide, and 2.4 m deep can safely bear a load up
to 909 kg, find the maximum safe load for a beam
of the same material which is 3 m long, 6 m wide,
and 1.8 m deep.

18. Question #14:
Luis walks 5 kilometers due
east and 8 kilometers due
north. How far is he from the
starting point?

19. Question #15:
Given the trapezoid EFGH
with median IJ. If IJ = x, HG =
8 and EF = 12, what is the
value of x?

20. Question #16:
The ratio of the volumes of
two similar rectangular
prisms is 125:64. What is the
ratio of their base areas?

21. Question #17:
ABCD is a parallelogram. If AB is
8 cm long, BC is 5 cm and their
included angle measures 100°,
how long is diagonal AC?

22. Question #18:
In right triangle PQR, PQ =
12 cm and QR = 5 cm.
What is cos R?

23. Question #19:
A balloon is 50 m high. Its angle of
elevation from observer A is 45° and from
observer B it is 30°. What is the
maximum distance between the two
observers? Express your answer to the
nearest meter.

24. Question #20:
The expression 2(sin 30°) –
tan 45° is equal to
_____________.

25. DIFFICULT [3 points]
LEVEL

26. Question #21:
A farmer is tilling a square field with
an area of 900 m2. After 3 hrs, he
tilled
2
3
of the given area. If the area of
the square field measures 180 m2,
find the length of its side.

27. Question #22:
The perimeter of a parallelogram is 34
cm. If a diagonal is 1 cm less than its
length and 8 cm more than its width,
what are the dimensions of this
parallelogram?

28. Question #23:
Solve for the
indicated distance
across the lake.

29. Question #24:
Two points P and Q on the same side of a river are
12 m apart. A tree on the opposite side of the river
is directly opposite a point between P and Q. The
lines of sight of a tree across the river make angles
of 78° and 57°, respectively, with the line joining P
and Q. Find the distance from point Q to the tree.
Express your answer to the nearest meter.

30. Question #25:
Two ships leave the same port at the
same time. One ship sail on a course of
110° at 32 mi/h. The other sails on a
course of 230° at 40 mi/h. Find the
distance between them after 2 hours.
Express your answer to the nearest mile.

31. Question #26:
Triangle BCA is right-angled at C.
If c = 23 and b = 17, find ∠A, ∠B
and a. Express your answers up
to two decimal places.

32. Question #27:
According to a lighting specialist for an art gallery,
for best illumination of a piece of art, it is
recommended that a ceiling mounted light be 1.8
m from the piece of art and that the angle of
depression of the art piece be 38°. How far from
the wall should the light be placed so that the
recommendations of the specialist are met? Give
your answer to the nearest tenth of a meter.

33. Question #28:
Find the exact value of
(sec 30°) (cos 30°) – (tan
60°) (cot 60°)

34. Question #29:
After heavy winds brought by “Hanging Habagat”
damaged a house, workers placed an 8-m brace
against its side at a 45° angle. Then, at the same
spot on the ground, they placed a second longer
brace to make a 30° angle with the side of the
house. How long is the longer brace? Round your
answer to the nearest tenth of a meter.

35. Question #30:
An observer in a lighthouse 48.8 m above sea
level saw two vessels moving directly towards
the lighthouse. He observed that the angles
of depression are 42° and 35°. Find the
distance between the two vessels, assuming
that they are coming from the same side of
the tower.