Maths School work

8/11/2019 Maths School work

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Pg-298 =164,5-sd pg-

2)Novas--probs,P&C,graphs,3)Grubbers--functions,axioms,sets Geometry covers examples on triangle similarity & congruency, so its worth goingthrough.

1. When you multiply or divide both sides of an equation witha negative number, the direction of the inequality symbol must change. That IS, if

x >y, then -x < -y.

2. To find the total angle of ___gon = (n-2) x 180 where n= no of sides

3. If you have been given a total fig(U) which is divided into 2 parts(A & B), some elements of

a part may be repeated in the other(R), and some elements may not feature at all in these

parts(N) hence,U-A-B+R-N = 0

4. Percentage point= percentage difference.

5. Probability = P(AUB)=P(A)+P(B)-P(A*B)

P(AUBUC) = P(A)+P(B)+P(C)-P(A&B)-P(B&C)-P(C&A)+P(A&B&C)

6. interquartile range means difference between third quartile and first quartile

7. Important Notes on Standard Deviation

1) Adding or subtracting a constant from each element in the set has no effect on standard

deviation. Remains the same

2) Multiplying the elements of a set with an absolute value greater than 1 increases the standard

deviation

3) Dividing the elements of a set with an absolute value greater than 1 decreases the std deviation

4)Changing the signs of the element of a set or multiplying by -1 has no effect on std deviation. Remains

the same

8. In a cyclic quadrilateral opp sides total is always 180 degrees

9. Incirle/Inradius for Angle bisector

10. a^2 + a^3 is not equal to a^5

11. Million has six zeros 1000000 ,billion has 9 zeros 1000000000

12. 1mile=5,280 feet13. Parenthesis, exponentiation, negation, multiplication, division, addition, subtraction

14. On the GRE the square root sign ALWAYS refers to the positive root. Which means sqrt(4) is only

+2 but x^2 = 4 then answer can be +2 or -2. 15. 0 degree is not defined.16.

no .of derangements = n!(1/0! - 1/1! + 1/2! - 1/3! ..... (-1)^n/n!)

For n=4

no .of derangements = 4!(1/0! - 1/1! +1/2! - 1/3! +1/4!)

= 24(1 - 1 + 1/2 - 1/6 + 1/24)


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= 24(9/24)

9

therefore probability = 9/24 = 3/8

17. Every value of a discrete random variable X, say X=a has a probability denoted by P(a).A

histogram (table) showing all of values of X and their P(X) is called probability of X.The

mean of the random variable of X is the sum of the products XP(X) and all the values of

X.

18. When the termprofit is used in a question, it refers togross profit, which is the sales revenue minusthe cost of production. The profit does not involve any other amounts unless they are explicitly given.

19. if an answer to a question requires converting one unit of measurement to another, then therelationship between the units is given in the question, unless the relationship is a common

20. 25 is a multiple of only six integers: 1, 5, 25, and their negatives.

21. Since decimals donot terminate or repeat they are irrational nos.

22. |a+b|


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33. True discount = (total amount after x yrs initial amount) for SI

34. length of diagonal of isosceles trapezium is (ab+c^2 )^(1/2)in which a & b are two opposite parallel sides and c is the 2 parallelsides

35. How to check if a no is rational or irrational- 1. Reduce the no to

numerator/denominator form such that is not be further reduced then if the

denominator is one these- Terminate: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 100 thenonly it terminates

36.

37. To find out no of integers coming between A and B nos inclusive, use following

formula B - A+1.

38. If a percentage is given then take it always with respect to 1, for example 3% = 0.03

(of 1)

39. To get a larger standard deviation increase the spread by taking out the mean and

get lower standard deviation by eliminating the extreme values

40.

41.

Lateral Surface Area

Definition of Lateral Surface Area

Lateral surface area in a solid is the sum of the surface areas of all its faces excluding the base

of the solid.


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More about Lateral Surface Area

Lateral surface area of a cube = 4b2, where bis the base of a cube

Lateral surface area of a sphere is 4r2, where ris the radius of the sphere.

Lateral surface area of a cone = r l, where rand lare the radius and slant height of the

cone

Lateral surface area of a cylinder = 2rh

, whereris the radius and

his the height of the

cylinder

Lateral surface area of the right triangular pyramid = 3 area of lateral faces

Lateral surface area of a pentagonal prism = 5 area of each rectangle

42. Square root refers to only positive square root.


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43.

44. Range in statistics is given by: max term in the set least term in the set

45.

46.


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47.

48. Pythagorean triplets:


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49. Types of triangle and their sides:

50.

51.


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52.

53.

54.

55.


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56.

57.

58.

59. Any given amount is a larger percentage of a smaller number than it is of a bigger

number.(DI)

60.


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61.

62. DECIMALS


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63.

64. For function y = f(x):

65. GRAPH shifts(to shift graph:):

66.


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67.

68.16 is + 4 times greater + than + 4 i.e. 16/4 = 4 times greater 'than'

read it as : 16 is + 12 greater + by + 4 i.e. 4+12 = 16 ...'by'

69. There are 40% as many men as women means -> 40% = m/w

63. A divisor must always be greater than the remainder.

64. Revenue = profit + C.p


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65. Average concept.

66. For any evenly spaced set, the median equals mean.

67. Definition of a mutually exclusive event

If event A happens, then event B cannot, or vice-versa. The two events "it rained on

Tuesday" and "it did not rain on Tuesday" are mutually exclusive events. When calculating

the probabilities for exclusive events you add the probabilities.

Independent events

The outcome of event A, has no effect on the outcome of event B. Such as "It rained on

Tuesday" and "My chair broke at work". When calculating the probabilities for independent

events you multiply the probabilities. You are effectively saying what is the chance of both

events happening bearing in mind that the two were unrelated.

68. A square is one type of rectangle. As the definition of rectangle says that all sides are at right

angle to one another.


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69. 25 is a multiple of only six integers: 1, 5, 25, and their negatives.

70.

71.


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72.

73.

74. This means that in the standard form(Parabola),y= ax2+ bx+ c, the expression gives

thex-coordinate of the vertex.

75. Inscribed angle subtended by same chord is always same:


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76. A list can have repetitions while a set cannot.

77.

For such a problem where number of data points are odd the median is not taken in either lower

or upper half for forming quartiles.

78.


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78.

79.

80.


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81.


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82.Area of a triangle can range from 0 to the area when 2 sides are perpendicular to each other.


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83.

84.

85. When x is divided by 3, the remainder is 1. When x is divided by 7, the remainder is 2. How many

positive integers less than 100 could be values for x?

4

86.


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cons a rectangle and three straight lines drawn over it.. with the straight lineintersecting each other inside the rectngle.. what is the maximum number of portionsthe rectangle can be cut into, using these three st lines..

Ans- (n(n+1)/2)+1

87.

88.


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89.

90.


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Surface area of cylinder is 2pi r h + 2pi r^2


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GEOMETRY :-


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Third side length rule:-


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----


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Question 15

If all the members of the Auto Mechanics Association who held advanced degrees and all

the members of the Auto Mechanics Association who had at least 5 children voted for a

measure, how many more votes were needed to gain a majority?

-173

-344-556

-636

-It cannot be determined from the information given.

15. E There is not enough information given to answer this question since all, some, or

none of

the members who hold advanced degrees could also have 5 or more children.


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3. C Although AirlineA has fewer airplanes than does Airline B, you have no information

about each airlines customers; choice (A) is incorrect. While the percent of airplanes

purchased by Airline B is twice as large as that purchased by AirlineA, the actual number of

airplanes purchased by Airline B is approximately four times as large as those for AirlineA,

not twice as large. So, choice (B) is incorrect. To evaluate choice (C), you need to add thepercents in each column year by year until you get to 50%. The airplane with the median

age for AirlineA was purchased in 1997. For Airline B, the median is between 1995 and

1996, making the median age for the airplanes in Airline Bs inventory older. This validates

choice (C), the only correct

answer.


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4. C, D, and E

Ignore the population growth: The question is asked inper capita terms, so its asking

about the population as a whole. Coal production was 25 in 1979, when, according to the

graph, coal represented 27% of energy; since 25 is 27% of 92.6, the total energy was 92.6

in 1979, and double that, or 185.2, in 2004. The lower end of the range is oil, at about 17%,

and 17% of 185.2 is about 31.5; the upper end of the range is coal, at about 34%, or about

63. Choices (C),(D), and (E) fall within the range.


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11. B, C, E, and F

One standard deviation captures 68% of the data points, or 34% in each direction from the

mean. For this question, that means that 68% of the months of the year will have average

high temperatures within one standard deviation of the annual average high temperature.

12 68% = 8.16, so 8 months will be within one standard deviation. Standard deviation

assumes a normal (balanced on the high and low ends) distribution, so the 2 months with

the greatest average monthly high temperature and the 2 with the lowest average monthly

high temperature will fall outside of one standard deviation. To answer the questions, use

POE to eliminate the 2 most extreme months on each end. (Note that you dont have to

calculate anything to find the correct answers.) It is clear that January has the lowest

average high temperature, so eliminate choice (A). February probably comes next, but it

doesnt matter since neither it nor December is an answer choice. November is definitely

not one of the two months

with the lowest average high temperatures, so choice (F) is in. July has the highest average

high temperature, so eliminate choice (D). Whether June or August is second doesnt matter


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for us because neither is an answer choice, but that means March, May, September, and

November all make the cut.

Question 11

If 4 a 9, and 3 b 2, then what is the greatest possible value of a b ?

11.12You are being asked to subtract the a and b terms. Be careful that you dont justcombine the largest value of a with the largest value of b to get a b = 9 2 = 7, the wrong

answer. When you combine inequalities, you have to make four calculations to check the

four possibilities. Subtract the smallest values of a and b: 4 (3) = 1. Subtract the

largest values of a and b: 9 2 = 7. Subtract the smallest value of a and the largest value of

b: 4 2 = 6. Subtract the largest value of a and the smallest value of b: 9 (3) = 12. Of the

four possible values above, the greatest possible value is 12.

Question 14

If 12


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Thus, the total number of favorable rolls is 6 15 = 90. Because there are 216 possible

rolls, 90 of which are favorable, the probability of getting exactly two of the three dice to

show the same number is 90/216 = 5/12


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Question 1

Six state governors meet at an annual convention. They line up in random order to pose for

a photograph. If the governors of Alaska and Hawaii are among the six governors, how

many different ways can the governors line up for the picture so that these two governors

are adjacent?

510

120

240

720

1. Two members(A & B) are always together hence we need to find permutation of 5 that is

5x4x3x2x1 = 120 but as the 2 members can be arranged as AB or BA we need to multiply

120 with 2, hence the final ans 120x2 = 240.

Question 14Paul and Allen are choosing ties out of a selection of three distinct red ties, five distinct

green ties, and six distinct blue ties. If Paul and Allen each wear one tie, how many different

ways could they wear ties of the same color?

14. 56 This problem involves permutations, because the order matters since the ties are all

distinct. Start with Paul wearing a red tie. He has 3 red tie choices and Allen has 2

remaining red tie choices, giving them a product of 6 permutations. For green ties, they

have 5 4 = 20 permutations. For blue ties, they have 6 5 = 30 permutations. The grand

total is 56 permutations


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1. A Use Process of Elimination to solve this one. First, only coordinate pairs with a

positive -xvalue and a negativey-value will fall in the proper quadrant, so eliminate

choices (B), (C), and (E). The line that divides the correct quadrant into shaded and

unshaded regions has a slope of 1 because it goes through the origin and the point

(10, 10). On this line, the absolute value of thex-coordinate equals the absolute

value of they-coordinate. In the shaded region, then, |x| > |y|, so choice (D) can be

eliminated. Only choice (A) remains. Alternatively, realize that this figure is drawnaccurately, because of the placement of (10, 10), and plot all 5 points, eliminating

all of those that fall outside the shaded region.

if an integer n is chosen from the integers 1 through 72, what is the probability that n(n+1)(n+2) will bedivisible by 8?

a. 1/4b. 3/8

c. 1/2

d. 5/8e. 3/4

you take If any 3 consecutive numbers: n, n+1, n+2; We will have two cases, either nis odd or n is even

Case a: n is odd


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If n is odd, then n+2 will also be odd and n+1 will be evenAs both n and n+2 are odd, they wont countTherfore for n(n+1)(n+2) to be divisible by 8, n+1 must be a multiple of 8.

Between 1 & 72, no. of multiples of 8 = 9

(i.e., n+1 = 8,16,24,32,40,48,56,64,72

=> n = 7,15,23,31,39,47,55,63,71)

Case b: n is even

If n is even, n+1 is odd and n+2 is even

and one among n & n+2 will also be a multiple of 4.

example: 6,7,8 => 8 is a multiple of 4

20,21,22 => 20 is a multiple of 4

therfore, when we multiple n and n+2 we will automaticall get a multiple of 8.

no. of even number between 1 & 72 = 36

Total no. of possible choices for n = 72No. of choices for n to make n(n+1)(n+2) a multiple of 8 = 9+36 = 45

so, probability = 45/72 = 5/8


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9.N {12,18,2,6}

P {1,4,2,3}

If n and p are to be selected at random from sets N and P respectively, what is the probability that n/2p

will be a member of set P?

Correct Answer:3/8 OR 6/16

Explanation:

n can take four different values, each of which can be combined with the four possible values for p. This

gives us 16 possible pairs.

Now find out how many of these pairs can give a value for n/2p which is in set P.


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If n = 12 and p = 2 the expression gives 12/4 = 3 which is in set P. Similarly n=12 and p = 3 works out to

2, which is in the set.

With n = 18, p can be 3. With n = 2, p can be 1. With n = 6, p can be 1, or 3

This gives us 6 pairs out of 16 = 3/8

1.The distance from town A to town B is five miles. C is six miles from B. Which of the following could be

the distance from A to C?

Indicate ALL such distances.

A. 11

B. 7

C. 1

Correct Answer:ABC

Explanation:

Do not assume that AB and C are on a straight line. Make a diagram with A and B marked 5 miles apart.

Draw a circle centered on B, with radius 6. C could be anywhere on this circle. The minimum distance will

be 1, and maximum 11, but anywhere in between is possible.


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Four men and four women are to be seated around a circular table such that no two men are together, in

how many ways can they be seated ?

men can be seated in 'n' number of ways and the women then gets limited to just 'n-1' number of ways to be seated... here'n' is 4...so the answer is 4!*3!

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