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A. UniformDefinition 1 :
Two plane are congruent if all pair of sides on it planes have the same length and the pair
of corresponding angles have the same measure.
Definition 2 :
Two plane are uniform if and only if the pair of corresponding angles have the same
measure and the pair of corresponding sides have the same ratio.
B. Plane>>n-sides and circle
The characteristic of n-sides:
1. The sum of angles on n-side is (n-2) x1800.2. Special for regular n-sides,
a. The measure of angles on each of angle point in it circle isb. The measure of cental angle on each unit of triangles is
The characteristic of triangle:
1. The sum of two sides from a triangle is always longer then the third side.2. The some of angles in a triangle is 1800.3. The outside angles of a triangle equal with the sum of two inside angles.4. On triangleABC, the shortest sides are faced with a smaller angle.
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For equilateral triangle with the length a, then the area of triangle as follow :
The characteristic of circle:
1. Inscribe angle which face the same arc has the same measure of angle.2. Circle angle which face the same arc with inscribed angle, the measure is two times of
inscribed angle.
3. The angles that face of chord rectangle are supplementary.
4. If lineAB touch a circle at point C, then each chord CD pass point Cwe get = and = .
5. IfPis a common point inside a circle, then satisfying :
atau
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6. IfPis a common point outside a circle, then satisfying :
Specially ifPEis tangent line, then 2 = . = .
The important characteristics of circle and triangle:
1. The radius of inside circle of triangle :
2. The radius of outside circle of triangle :
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C. The Altitude, Median and Angle Bisector of Triangle
Definition:
>>CD is called altitude that appropriate with baseAB since CD is perpendicular withAB.
>>CEis called median that appropriate with base AB since AEand EB have the same
length.
>>CFis called angle bisector that appropriate with angle point Csince CFdevides angle
ABCbe two anglesACFand CFB that have the same measure.
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D. SpaceVolume and Area of Cylinder, Cone and Ball
E. Pythagoras Theorem
Pythagoras TheoremIn ABC. Right-angled at A, the following
formula is- always valid.
BC2
= AC2
+ AB2
a2
= b2
+ c2
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F. Trigonometry
The value of trigonometric with spesific angle:
Example :
Problem : The height of a tree is 5 m. The distance between the top of it and the tip of
its shadow is 13 m. The height of a tree is 5 m. The distance between the
top of it and the tip of its shadow is 13 m.
Solution :
132
= 52
+ L2(Pythagoras Theorem)
L2
= 132
- 52
L2
= 144
L = 12 m
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1. Look at the figure beside. Given the length of KL = 15 cm, NO = 5 cm, andLN = 8 cm,the length of NP is
a. 28b. 18c. 16d. 26
2. What is the perimeter of the figure shown?a. 16
b. 14c. 10d. 18
3. Each of the figure below shows a square with side 1 and inscribed circles. In which of thefigures do the circles have the greatest total area.
a. Figure 1b. Figure 2c. Figure 3d. Figure 2 and 3 have the same areae.
They all have the same area
4. Look at the figure beside. The mea sure ofisa. 400
b. 450c. 500d. 600e. 800
2
Figure 1 Figure 2 Figure 3
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5. A cube with a volume 64 is cut horizontally. The two halves are then glued togetheragain to form a beam. How much surface are a the new space?
a. 128b. 114c. 96d. 56e. 48
6. If the sum of the squares of all the length of the rectangle is 100,then the length of thediagonal of the rectangle is...
a. 2
5
b. 213c. 43d. 52e. 10
7. In triangle ABC, known AB= 4cm and BC= 6cm. If the length of the high line drawnfrom point C is 6 cm, then the length of the altitude drawn from point A is...a. 4
b. 5c. 6d. 7e. could not be determined
8. A car tire has an outer diameter 50 cm. If the radius is reduced to a half cm, what %increases in many rounds per meter?
a. 1,8 %b. 2 %c. 2,2 %d. 2,4 %e. 2,6 %
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9. In square ABCD, point X is located on the line DC so that DX: XC = 5:2 and Y islocated on the line BC so that BY: YC = 3:4.The ratio of area of the triangle AXC and
ABYare...
a. 2 : 7b. 2 : 3c. 3 : 4d. 4 : 9e. 9 : 16
10. Given a circle of radius r units. Then made a circle of radius r units with a central pointon a point of the circle first. Furthermore, also made a circle of radius r, with center point
located on the intersection of two circles before. The area of slices of third circle is...
a. 122 3
b. 2( 3)c. 1
122(3 23)
d. 1122(2 33)
e. Not answer11. Suppose a, b, and c are the length of the sides of a triangle, with a, b and c are
consecutive natural numbers with arithmetic mean 6. If we make altitude towards the
side with length b, then the length ofthe altitude is...
a. 66b. 46c. 26d. 4
2
e. 2212. A cube has length 4 m. Block all cube with color red and then cut into 64 cubes with
sides of 1 m. How many cubes which have a face exactly that can be colored red?
a. 6b. 16c. 24d. 36e. 64
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a.b.c.d.e.
13. Triangles ABC and ABD are isosceles triangle with AB=AC=BD and BD cut AC atpoint E. If BD is perpendicular to AC, then the sum of the angles C and D is...
a. 1150b. 1200c. 1300d. 1350e. could not be determined
14. Consider a right triangle ABC with hypotenuse AB=c and the formula is exemplified bythe high h, the triangle ABC the following:
Which is the correct formulation?
a. 1)b. 2)c. 3)d. 1) and 2)e. 2) and 3)
15. The radius (inner circle) right triangle is 6cm. What is the area of triangle?
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a.b.c.d.e.
a.b.c.d.e.
16. In the picture below the triangle ABC, BC and DE are parallel. If angle BAC=x0, angleBCA=(x+300) and the angle BDE=4x0, how large angle abc?
a. 150b. 450c. 600d. 900e. 1200
17. In the circle below, AB is the diameter of the circle. The lines CD and EF isperpendicular to line AB. If the length of AP, PQ, and QB, respectively, 5, 7and 9, what
is the sum of the lengths of CP and EQ?
18. In right triangle ABC below, AC= 4 and BC=3 with square CDEF. The length of EF is...
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.
19. A circle of radius r. AB and CD is the diameter of the circle that perpendicular toCM=0,2. If MN//AB and NG//CD, then the length of MG is...
a. 0,8rb. 0,4rc. 0,2rd. 0,r + 0,2e. r
20. A circle of radius 4 in an equilateral triangle. If the area outside the circle and inside thetriangle is Value of x + y is...
a. 64b. 60c. 48d. 30e. 24
ANSWER KEYS ::
1. C2. A3. E4. D5. A6. D7. A8. B9. B10.A
11.C12.C13.D14.D15.16.E17.B18.A19.E20.A
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Chapter 4
COMBINATORIC, OPPORTUNITY, STATISTICS
4.1PermutationDef. of Permutation:
The ways to arrange r element of a set that consist of n element with observe the
serial is:
, = =! !
where ! = 1 3 2 1.Ifanobject canbe selectedmorethan oncethen it is calledpermutationwithrepetitionwith
the formula:
, = =
The ways to arrange n element where each ofkelement appears 1,2, , times is:
= !1! 2! !
4.2CombinationDef. of Combination:
The ways to arrange relement of a set that consist ofn element without observe the
serial is:
, = = !
! !
If an object can be selected more than once then it is called combination with
repetition where:
+ 1, = +1 = + 1!! 1!
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4.3OpportunityDefinition 1:
Let A is an event in sample space S, then the opportunities forA is:
= ()()
Definition 2:
Let S be a finite sample space with events A and B, then the conditional probability of
event A on the condition B, write as (/), that is:
/
=(/)()
4.4Coefficient of BinomialThis principle is used to determine the cardinality of the sets that should not be
combined with each other separated.
Theorem:
Ifx andyare variables and n is natural numbers, then apply:
( + ) =
=0
If( + )be expanded then apply some of the following properties: There are( + 1)tribes, the number ofranksof xand yin eachtribeis n. The rankofxdownone byonefrom nto 0, while therankofyup oneby onefrom
0ton.
The coefficientofthe tribeswhichis equidistant fromandare the same.
Example and Solution:
1. Anita has 6-digits, that is digit 1, 2, 3, 4, 5, and 6.How many ways set the 6-digit numbers into different 6-digit?
Answer:
Remember that the numbers 123456 is different with 213465.
So, the ways to set the sixth digit is the order of preparation. In the place of hundred
thousand there are 6-digit can be chosen. Since one digit has been used, so in the
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place of ten thousands there are only 5-digit can be chosen. Thus, the next until the
place of ones.
So, there are 6 5 4 3 2 1 = 720 ways.
2. On mathematics Olympiad had selected 6 finalists. From 6 finalists will be selected 3finalists to be first winner, second winner, and third winner. So, how many ways can
be used?
a. 120b. 60c. 40d. 24e. 20Answer:
Suppose that the six finalists are A,B, C, D, E, and F. Notice that if A be the first
winner, B be the second winner, and C be the third winner are different with if A be
the second winner, B be the first winner, and C be the third winner. So the ways to
choose three winners depending on their sequence. This result is a permutation case.
Then as a finalist is not possible to obtain two degrees, so this is particularly the case
of permutations without repetition.
63 =6!
6 3! =6!
3!= 6 5 4 = 120
The correct answer is : 120 (A)
3. There are 15 participant of Olympiad development, but the participant that will berepresent Indonesia to mathematics Olympiad only 5 participants. So, how many
ways to selected 5 participants from 15 participants?
Answer:
This problem is a combination problem without observe the repetition:
155 =15!
5! 15 5! =15 14 13 12 11
5 4 3 2 1 = 3003
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Problem:
I. Choose the right answer of a, b, c, d, or e from the question below!1. In mathematics contest with ten problems, a students gains 5 points for a correct
answer and lose 2 points for an incorrect answer. If Olivia answered every problem
and her score was 29, how many correct answers did she have?
a. 5
b. 6
c. 7
d. 8
e. 9
2. Carlos Montado was born on Saturday, November 9, 2002, on what day of the weekwill Carlos be 706 days only?
a. Mondayb. Wednesdayc. Fridayd. Saturdaye. Sunday
3. The following data are the result of a mathematics test.Score 5 6 7 8 9 10
Frequency 3 5 8 5 2 1
The average score of the test is...
a. 6.88b. 7.00c. 7.04d. 7.50e. 8.00
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4. The scores achieved by 10 students on mathematics test are as follows:6,p-1, 5, 9, 5,p, 8, 8, 10, 9.
If the average of the scores is 7.3, what is the value ofp?
a. 6b. 7c. 8d. 9e. 10
5. The average of science test scores of students in a class is 78. Six other students jointhe class. Each has a score of 72. What is the average of science test scores?
a. 74.50b. 75.08c. 76.62d. 77.71e. 78.00
6. Given that the data of the weights of the girls in class IXA are:62 53 48 50 52 45 50 60
43 54 56 50 51 60 55 45
The median is...
a. 54b. 53.5c. 51.5d. 50e. 55
7. Given two dice, the example points below are members of the sample space,EXCEPT..
a. (2, 5)b. (4, 4)c. (1, 3)d. (5, 7)e. (5, 1)
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8. Umi had a coin and two dice. She rolled them together. What was the total of thesample points?
a. 72b. 36c. 12d. 8e. 6
9. The probability that the day will rain in November is 715
. How many days will not rain
in November?
a. 14b.
15
c. 16d. 17e. 18
10.A letter is randomly taken from the word MATEMATIKA. The probability oftaking the letter T is...
a. 110
b. 15c. 3
10
d. 25
e. 310
11.Three coins are rolled at once. The probability of getting at least two heads is ...a. 1
12
b. 16
c. 14
d. 15
e. 12
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12.For A = 1, B = 2, ..., Z=26,Then defined that A : B = 2, A : B : C = 6, A : B : L = 36, and A : B : ... : Z = 26!.
What are the three letters that produce1001?
a. C, K, Wb. C, Q, Sc. G, I, Md. G, K, Me. I, K, M
13.How many pairs(m, n) of positive integers satisfying the equation 4 + 2 = 1a. 1b.
2c. 3d. 4e. Greater than 4
14.The pattern of ABBCCCDDEEEEEFFFABBCCCD....repeated to finite.What the letter that occupying 35 53 22 ?a. A
b. Bc. Cd. De. E
15.How many the number ofb so that there isapositive integerthat satisfy2 = 3?a. 1
b. 2c. 3d. 4e. Greater than 5
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16.On the selection of candidates for coach of the circus which was followed by fivecontestants, it is known that the winner gets 10 votes. if it is also known that no two
contestants who obtained the same number of votes, then the largest possible gains for
the contestant with the least votes is...sound
a. 3b. 4c. 5d. 6e. 7f. 8
17.Budi had 5 pieces of hundreds of thousands of rupiah and 10 coins fifties rupiah.If he wants to buy things that cost 500 rupiah, then the number of combinations of the
money he paid was..
a. 4b. 5c. 6d. 7e. 8
18.Every two different points on the plane are determining the position of a straight line.What is the number of straight lines determined by 12 points in the plane if there is no
three points on a line?
a. 11b. 66c. 121d. 11!e. 12!
19.How many phone numbers that consist of 7 digits can be made with 4-digit first, theremaining three digits must be different from each other and not a number 0, 3, or 5,
and the last digit is not the number 9?
a. 216b. 210c. 180d. 120e. 125
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20.Johan contracted to work a day. For each day that Johan worked, just received a Bdollars. For each day that Johan does not work, he shall indemnify C dollars. A day
after, Johan receives the results of D dollars. How many days does not work?
a. +
b. c. ++ d. e. not one of theoptionsabove
II.
Essay1. There are nine people shake their hands each other. The total number of their
handshakes is...
2. A car traveled on e mile in 1 minute, so for traveling on e mile in 40 seconds the carshould increase the speed...%
3. Know that 32 27 is an integer, so the smallest positive integer ofnis..4. There are 25 questions of math test. Each correct answer given a score of 4, while a
wrong answer was given a score of -1. Fahmi answers all questions and got score of
35. Determine the number of questions was answered correctly by Fahmi?
5. There are not all of students pass the test. Only 2/3of male students and 3/4 of femalestudents who pass the test. The number of male students and female students who
pass the test are the same. Then, determine the ratio of students who pass the test
from the number of students!
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Answer Keys:
I. MULTIPLE CHOICE1. C 11. E2. C 12. D3. C 13. D4. B 14. E5. C 15. E6. C 16. D7. D 17. C8. A 18. B9. C 19. C10.B 20. A
II. ESSAY1. The total number of their handshakes is the combination 2 of 9.So,
29 =9!
2!7!=
9 8
2= 36
2. Firstvelocity :160
Final velocity :140
Additional speed :140
160 =
1120
So, the velocity of car increase as much as :
=
1120
160
100% =1
2 100% = 50%
3.
32
27 =
42 2
32 3 = 12
2
3
So, n is integer if = 2 3 = 64. Let B be the number of questions were answered correctly and the S be the
number of questions were answered wrong.
4BS = 35
B + S = 25
From the both equation, we get: 5B = 60 B = 12
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5. Let L be the number of male students and P be the number of female students.2
3 = 3
4 = 9
8
The ratio of students who pass the test from the number of students is:
23 + 34 + =
34 + 34 9
8 + =
32
17
8 =
12
17