CEER 2012 Math Lecture

The lecture shall begin
shortly…

5M sa math lecture
1. MAKINIG
2. MAG-BEHAVE
3. MAGTANONG
4. MAGSAGOT
5. MAG-ENJOY 

Bago ang lahat....

(a) 0
(b) 1
(c) 5
(d) 7

QUESTION 1

Determine the domain and range of

(a)
(b)
(c)
(d)

QUESTION 1 Solution:

The domain of y excludes values of x that will make the
denominator zero. Thus, the domain is
To solve for the range, we first solve for x in terms of y:
x 5
y y x 1 x 5 Therefore, the
x 7 range is
xy 7y x 5
xy x 7y 5
x y 1 7y 5
7y 5
x
y 1

QUESTION 2

If and
find

(a) (c)

(b) (d)


Recall: for functions F and G:

and


QUESTION 2 Alternative Solution:
SUBSTITUTE a value of x and test which choice will give
the same value.
Para madali, let x = 0.


QUESTION!
Which will give a value of 3 at x = 0?

(a) (c)

(b) (d)

Astig, ‘di ba? 

QUESTION 3

Which of the following is a linear function?

(a)
(b)
(c)
(d)


Recall that a linear function is a polynomial function
wherein the highest power of the independent
variable is 1.
Is LINEAR, so the answer is (a)
(a)
Is QUADRATIC because of
(b) the terms 3x2

WAIT! This is
(c) also linear!
cannot be a linear function since x and are
(d) in the denominator

The answers are BOTH (a) & (c)! Weh, ‘di nga?!

QUESTION 4

What is the equation of the linear function y
whose graph passes through the point (2, 4) and
has the given slope m = 5/7?

(a) (c)

(b) (d)


We use the slope-intercept form

STRATEGY: Substitute x = 2, y = 4 and m = 5/7 then
solve for b.
Hence, the equation of the line is

or


QUESTION 4 Alternative Solution:

Check the choices! Which among the choices… CLUE: 5 ang
nasa unahan
1. Has slope 5/7? ng x at 7
ang nasa
denominator
2. Has a value y = 4 when x = 2?

QUESTION 5

Determine the distance from the point ( 2, 9) to
the line 3x + 4y = 2.


"No choice" Solution: We have NO CHOICE but use the
following formula for the distance D of a point (x0, y0)
from a line with equation Ax + By + C = 0:

Before doing anything, rewrite 3x + 4y = 2 as
3x + 4y 2 = 0
Then, substitute the values
A = 3, B = 4, C = 2, x0 = 2, and y0 = 9.

QUESTION 6

If ax2 + bx + c = 0, where a, b and c are real
numbers and a ≠ 0, which of the following
statements is true about the discriminant D?

(a) If D < 0, the two roots are real and equal.
(b) If D < 0, the two roots are imaginary and unequal.
(c) If D > 0, the two roots are real and unequal.

(d) If D < 0, the two roots are imaginary and equal.


Recall: the solutions or ROOTS of the quadratic equation
ax2 + bx + c = 0, where a, b and c are real numbers and
a ≠ 0, can be solved using the QUADRATIC FORMULA:

The DISCRIMINANT D of ax2 + bx + c = 0 is the value
INSIDE THE SQUARE ROOT; i.e.,


The DISCRIMINANT D determines the type or
NATURE of solutions or roots a quadratic equation
with real coefficients has.

As an ASIDE…
Some "UPCAT-level" problems that can be solved
using the discriminant:

QUESTION 7

Determine the radius of the circle whose
equation is

(a) 2 y
(b) 3
(c) 4 r

(d) 5
x

QUESTION 7 Solution

The CENTER-RADIUS FORM of the equation of a
circle with radius r and center at (h, k) is

To write x2 + y2 􀀀 8x + 6y = 0 in center-radius
form, complete the square:
The radius is

QUESTION 8

Find the quotient of

QUESTION 10

What is x in the equation ?

(a) 5
(b) 3
(c) 3
(d) 2

QUESTION 11

Evaluate

(a) 3/2
(b) 2/3
(c) 3
(d) 6

QUESTION 11 Solution

By definition, the LOGARITHM of a positive number x to
the base b, denoted by logb x, is the POWER y of b
equal to x; i.e.,

Example: log3 9 = 2 since 32 = 9. Simple, ‘di ba?

CHALLENGE: What is the value of ?

QUESTION 12

Solve for all possible values of x in the equation

(a) 3 and 2
(b) 2 and 3
(c) 6 and 9
(d) 9 and 6


A property of logarithm is that

Shortest solution:
SUBSTITUTE the
choices to the
original equation!

QUESTION 13

Solve for q in the equation

(a) (c)

(b) (d)


NOSE
BLEEEED!

QUESTION 14

Faye is 5 greater than twice the age of Luigi. 5
years from now, Faye will be twice as old as
Luigi. How old is Faye 3 years ago?

(a) 41 (c) 39

(b) 38 (d) 37


AGE PROBLEM:

Let x = Luigi’s age
2x+5 = Faye’s age
Age 5 years
Age now
from now

Luigi x x+5

(2x + 5) + 5 =
Faye 2x + 5
2x + 10

QUESTION 15

Paolo can finish compiling the books in library in 25
minutes. Kevin can finish it in 25 minutes while
Carmela took her 50 minutes. How many minutes
will it take them if they were to compile the books
altogether?

(a) 10 (c) 20

(b) 25 (d) 33


WORK PROBLEM:
Let x = no. of min they can finish the job together
No. of Rate per EQUATION:
minutes minute

Paolo 25 1/25

Kevin 25 1/25

Carmela 50 1/50

Together x 1/x

QUESTION 16

There are 570 students in a school. If the ratio of
female to male is 7:12, how many male students
are there?

(a) 300 (c) 380

(b) 370 (d) 390


570 students in the ratio 7:12

MALES FEMALES

One block =

QUESTION 17

When each side of a square lot was decreased by
3m, the area of the lot was decreased by 105 sq.
m. What was the length of each side of the original
lot?

(a) 18 (c) 20

(b) 19 (d) 21


Let x = length of the side of the square

EQUATION:
Length
of a Area
side

Original x x2

New x 3 (x 3)2

QUESTION 18

The difference of 2/3 of an even integer and one-
half of the next consecutive even integers is equal
to 5. What is the odd integer between these two
even integers?

(a) 26 (c) 36

(b) 27 (d) 37


Let x = 1st even integer
x + 2 = 1st even integer

EQUATION: The ODD
integer in
between is
the one
AFTER 36,
which is 37 

QUESTION 19

Find the average of all numbers from 1 to 100 that
end in 8.

(a) 53 (c) 51

(b) 52 (d) 45


The average looks like this:

The numerator is actually a sum of an ARITHMETIC
PROGRESSION with first term a1 = 8 and tenth term
a10 = 98, given by
The average is
then 530/10 = 53

As an ASIDE…
FACT: The average of the first n terms of an
arithmetic progression is just actually the
AVERAGE of the FIRST AND LAST TERM!

QUESTION 20

A tank is 7/8 filled with oil. After 75 liters of oil are
drawn out, the tank is still half-full. How many
liters can the tank hold?

(a) 200 (c) 240

(b) 220 (d) 260

CAPACITY
QUESTION 20 Solution = 25(8) = 200 L

25 L
25 L
75 L
drawn out 25 L
25 L
25 L
25 L
25 L
25 L
7/8 full 1/2 full

QUESTION 21

Two new aquariums are being set up. Each one
starts with 150 quarts of water. The first fills at the
rate of 15 quarts per minute. The second one fills
at the rate of 20 quarts per minute. When would
the first tank contain 6/7 as much as the second
tank?
(a) After 7 min (c) After 9 min

(b) After 8 min (d) After 10 min


Let x = no. of minutes

EQUATION:

QUESTION 22

In a classroom, chairs are arranged so that each
row has the same number. If Ana sits 4th from the
front and 6th from the back, 7th from the left and
3rd from the right. How many chairs are there?
(a) 49

(b) 64

(c) 81
(d) 100


FRONT

LEFT anna RIGHT

NO. OF CHAIRS:
9 X 9 = 81 BACK

QUESTION 23

A circle with radius of 5 m and a square of 10 m are
arranged so that a vertex of the square is at the
center of the circle. What is the area common to
the figures?


The area common to the figures is
10 m equal to ¼ the area of the circle:

10 m

5m

5m

QUESTION 24

How many liters of 20% chemical solution must be
mixed with 30 liters of 60% solution to get a 50%
mixture?
(a) 5 L

(b) 10 L
(c) 15 L
(d) 20 L


Let x = no. of L of 20% chemical sol’n
%
Amount of
Vol (L) concen-
chemical
tration

Sol 1 x 20% 0.2x

Sol 2 30 60% 30(0.6) = 18

mixture (x + 30) 50% 0.5(x + 30)


EQUATION:

QUESTION 25

A URent-A-Car rents an intermediate-size car at a
daily rate of 349.50 Php plus 1.00 Php per km. a
business person is not to exceed a daily rental
budget of 800.00 Php. What mileage will allow the
business person to stay within the budget?

(a) 300 (c) 400
(b) 350 (d) 450


Let x = mileage
EQUATION:

Rules of Counting
The Fundamental Principle of Counting:
If an operation can be performed in n1 ways, and
for each of these a second operation can be
performed in n2 ways,then the two operations
can be performed in n1n2 ways.
Extension: The Multiplication Rule
If an operation can be performed in n1 ways, and
for each of these a second operation can be
performed in n2 ways, a third operation in n3
ways,…, and a kth operation in nk ways, then the k
operations can be performed in n1n2n3…nk ways.

Permutations
Rules of Counting
PERMUTATION – based on arrangement of
objects, with order being considered
Permutation of n objects:
n(n – 1)(n – 2)… (3)(2)(1) = n! (n factorial)

Permutation of n objects taken r at a time:
n!
n Pr
n r !
Permutation of n objects with repetition:
n!
n1 ! n2 !...nk !

Combinations Rules of Counting
Combination – based on arrangement of objects,
without considering order

Combination of n objects taken r at a time:

n n!
n Cr
r r! n r !

13,983,816
possible combinations

QUESTION 26

How many 3-digit numbers can be formed from the
digits 1, 2, 3, 4, 5, and 6 , if each digit can be used
only once?

(a) 100 (c) 120

(b) 110 (d) 130


6 5 4
1st digit: 2nd digit: 3rd digit:
6 choices 5 choices 4 choices

By the Multiplication Rule:

QUESTION 27

The basketball girls are having competition for
inter-colleges. There are 15 players but the coaches
can choose only five. How many ways can five
players be chosen from the 15 that are present?
(a) 3,103 (c) 3,000

(b) 2,503 (d) 3,003


Since order is NOT important in choosing the
five players out of 15, we use the
Combination rule with n = 15 and r = 5:

QUESTION 28

A coach must choose first five players from a team
of 12 players. How many different ways can the
coach choose the first five?

(a) 790 (c) 800

(b) 792 (d) 752


Same as no. 27 

QUESTION 30

What is the perimeter of the triangle defined by
the points (2 , 1), (4 , 5) and (2 , 5)?


We can use the DISTANCE FORMULA to compute the
perimeter of a triangle in the Cartesian plane
(i.e., the sum of the lengths of the sides of the
triangle).

Kaya lang, ‘di ko na realize na easy lang ang case
sa problem kasi RIGHT TRIANGLE na! (See the board
for the solution) :p

QUESTION 32

If arcs AB and CD measure 4s - 9o and s + 3o
respectively and angle X is 24o, find the value of s.
See figure below.

(a) 6 (c) 18

(b) 12 (d) 20


GEOMETRY FACT:

QUESTION 33

How many possible chords can you form given 20
points lying on a circle?

(a) 380 (c) 382

(b) 190 (d) 191


The number of chords can be obtained using
the Combination Rule with n = 20, r = 2

QUESTION 34

Which of the following sets of numbers cannot be
the measurements of the sides of a triangle?
(a) 1, 2, 2
(b)
(c) 3, 4, 5
(d) 1, 2, 3


Use the TRIANGLE INEQUALITY:
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.

(d) 1, 2, 3
1 + 2 = 3 – should be GREATER!

QUESTION 35

The figure shows a square inside a circle that is inside the
bigger square. If the diagonal of the bigger square is
units, what is the area of the shaded region?

As an ASIDE…

The Pythagorean Theorem and Special Right
Triangles


Note that:
•The side of the
larger square is 2
(special right
triangle)
2 •The side of the
square is the
diameter of the
circle, so the radius
of the circle is 1


Note that:
•The diagonal of the
smaller square is also 2.
s
•If s is the side of the
smaller square, then
2

•The area of the shaded
area is then

QUESTION 36

Which of the following statements is NOT true about the
figure? Parallel lines a and b are intersected by line x
forming the angles 1, 2, 3, 4, 5 and 6.

(a) Angles 1 and 6 are congruent.
(b) Angles 1 and 5 are
supplementary with each other.
(c) Angles 3 and 4 are congruent.
(d) Angles 2 and 4 are
supplementary with each other.


(a) Angles 1 and 6 are
congruent. (alt. ext.)
(b) Angles 1 and 5 are
supplementary with each
other. (ext.)
(c) Angles 3 and 4 are
congruent. (alt. int.)
(d) Angles 2 and 4 are NOT
supplementary with
each other – they are
CONGRUENT
(corresponding angles)

QUESTION 38

How many sides does a polygon have if the sum of
the measurements of the interior angles is 1980o?

(a) 11
(b) 12
(c) 13
(d) 14


The sum of the interior angles of a triangle is given by

http://www.mathopenref.com/polygoninteriorangles.html

QUESTION 39

An ore sample containing 300 milligrams of radioactive
material was discovered. It was known that the material has
a half-life of one day. Find the amount of radioactive
material in the sample at the beginning of the 5th day.

(a) 9.375 mg
(b) 18.75 mg
(c) 37.5 mg
(d) 75


This can be solved using a geometric progression with
first term a1 = 300 common ratio r = ½, and n = 5 days

QUESTION 40

A survey of 60 senior students was taken and the following
results were seen: 12 students applied for UST and UP only,
6 students applied for ADMU only, 29 students applied for
UST, 2 students applied for UST and ADMU only, 10 students
applied for UST, ADMU and UP, 33 students applied for UP
and only 1 applied for ADMU and UP only. How many of the
surveyed students did not apply in any of the three
universities (UP, UST, ADMU)?

(a) 0 (b) 8 (c) 14 (d) 20


Using Venn Diagram
12 – UP & UST only
UP UST 6 – ADMU only
12 2 – UST and ADMU only

10 10 – all three
1 2 1 – UP and ADMU only

6
ADMU


33 (12 + 10 + 1) = 10
UP UST 29 (12 + 10 + 2) = 5
12 Add all numbers in the
10 5
circles: 46
10
1 2
What’s outside:
6 60 46 = 14
ADMU
14

BRIEF TIPS AND
TRICKS
1. READ EACH QUESTION CAREFULLY.
2. Take each solution one step at a time. Some
seemingly difficult questions are really just a
series of easy questions.
3. Remember thy formulas and important facts
(especially in Geometry)
4. Answer the easy items first. If you can’t solve a
problem right away, SKIP it and proceed to the
next.

BRIEF TIPS AND
TRICKS
4. Try the PROCESS OF ELIMINATION. A little
guessing might work.
5. Employ the EASIEST way as possible (e.g.,
substitution, shortcuts, tricks, etc.)
6. Use your scratch paper wisely…
7. If you still have time, CHECK your answers,
ESPECIALLY your shaded ovals!
8. RELAX…. Don’t panic!

PRACTICE PROBLEMS!

1. If x + y = 4 and xy = 2, find the value of x2 + y2
2. If 1/3 of the liquid contents of a can evaporates
on the first day and 3/4 of the remaining
contents evaporates on the second day, what is
the fractional part of the original contents
remaining at the end of the second day?
3. What is the smallest three-digit number that
leaves a remainder of 1 when divided by 2, 3, or
5?
4. The average of 4 numbers is 12. What is the new
average if 10 is added to the numbers?
5.

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CEER 2012 Math Lecture

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