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GENERALo Good Numbers to test a value of X [ -10, -1, -1/2, 0, , 1, 10]

o For a Triangle to find one length you must know at-least one another length

ARITHMETIC

Number Propertyo An odd integer divided by an odd integer can yield an only odd integer or a

non-integer.

o Any number with an integer sixth root will have an integer square root.

Given 3 consecutive positive integers, the product of the smallest number and the

largest number will be divisible by 2 three times (or 8) if the middle number is odd.

Any three consecutive integers include exactly one multiple of 3.

Every prime number greater than 3 must, by definition, be ODD (since the only evenprime number is 2).

In order for any number to be divisible by 6, it must be divisible by both 2 and 3.Thus, in order for y to be divisible by 6, y must be even (which is the same as beingdivisible by 2) and y must be a multiple of 3.

Exponentso X to the power y will always be a positive no. The only difference y=0,

then x= 1, y>0, then x>1, y0

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o Any number that is not a perfect square will automatically have an evennumber of factors.

o A perfect square will always be able to be expressed as the product of aneven number of prime factors because a perfect square is formed by takingsome integer, in this case 6, and squaring it. In this way, the prime factors of aperfect square will always appear in pairs, so there must be an even numberof them.

o p, the square root of the perfect square p2 will have an odd number of

factors ifp itself is a perfect square as well and an even number of factors ifp is not a perfect square.

o The greatest common factor (GCF) of two integers is the largest integer that

divides both of them evenly (i.e. leaving no remainder).

o In any set of four consecutive integers, two of the integers will be even and

two will be odd. In addition one of the two even integers will be a multiple of4, since every other even integer on the number line is a multiple of 4 (4 yes,6 no, 8 yes, etc...).

o For any two positive consecutive multiples of an integer n, n is the greatest

common factor of those multiples. E.g. for nos, 8 & 12, are multiples of 4(n),and then the GCD is 4(n).

Word Problems

o Two objects moving towards each other : Add both the rates to arrive at cumulative rate

Divide the total distance by the cumulative rate

Resultant is the time when both objects will meet each other.

Example - If Greg is moving 60 mph and Liz is moving 40 mph, thentogether they are traveling 100 miles per hour relative to one another.At that speed, they will collectively travel 350 miles in 3.5 hours,because T = (350 miles)/(100 mph) = 3.5.

o Two objects moving in the same direction at different speeds and a time /

distance lag:

Calculate the difference between the speeds

Divide the gap between distance by difference in speeds Resultant is the time when both objects will catch up.

Example - Car B is going 15 mph faster than Car A. That means it willcatch up to Car A at a rate of 15 mph. Car B is currently 45 milesbehind Car A, and R T = D, so (15 mph) T = 45 miles. Solving for Tgives us T = 3 hours.

GEOMETRY

Triangles:

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o If the sum of two angles of a triangle is equal to the third angle - the triangle

must be a right triangle.

o The right triangle with the largest area will be an isosceles right triangle

(where both the base and height are of equal length).

o Similar triangles are those in which all corresponding angles are equal and thelengths of corresponding sides are in proportion.

o If triangle DBC is inscribed in a semicircle, i.e the 3 points of the triangletouches 3 points within a circle and 1 side is the diameter then it must be aright triangle.

o The height of any equilateral triangle divides the triangle into two 30-60-90

triangles.

Parallelogram:

o Is a quadrilateral with opposite sides are parallel and equal in length.

o Opposite angles are equal in measure to each other. If a parallelogram has

one right angle, all of its angles are right angles.

o In a parallelogram opposite angles are equal and adjacent angles add up to

180) therefore the parallelogram is a rectangle.

o The diagonals of a parallelogram bisect each other.

o If the diagonals are equal--thus telling us that ABCD has right angle corners

(The only way for a quadrilateral to have equal diagonals is if its corners are90 degrees.)

Rectangles:o Opposite sides are equal in length + opposite sides are parallel + All

sides are right (i.e 90 degrees) Squares:

o All sides of a quadrilateral being equal AB=BC=DC=AD does not

necessarily mean its a square. It could also be a Rhombus.

o To prove that a quadrilateral is a square, you must prove that it is both a

rhombus (all sides are equal) and a rectangle (i.e one of the angles is 90

degrees / all angles are equal / that its diagonals are equal).

Rhombus:o The diagonals of a rhombus are perpendicular bisectors of one another.

o A quadrilateral with four equal sides is by definition a rhombus.

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o All sides of a quadrilateral being equal AB=BC=DC=AD does not

necessarily mean its a square. It could also be a Rhombus.

Circles:o Probability of a square within a circle -

Interestingly, the probabilitydoes not depend on x at all, meaning that this probability holds true for anysquare inscribed in a circle (regardless of size).

Polygon:o Sum of the interior angles of a polygon is computed according to the following

formula: 180(n 2), where n represents the number of sides in the polygon.

Sphere

o Volume is R2H.

Cylinder

o Volume = R2Ho Surface Area = 2r2 + 2rho The surface of a right circular cylinder can be broken into three pieces as

follows:o To understand why the curved side of the cylinder can be represented as a

rectangle, imagine taking a sheet of 8.5x11 paper and rolling the sidesforward until the edges meet in the front:

Cone

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o Volume =

Co-ordinate Geometry

Slope is positive => y is below the x-axis => y < 0

If slope of line A is the negative reciprocal (-1/x) of the slope of line B (x), we canconclude that line A is perpendicular to line B.

The formula for the distance between two points (x1, y1) and (x2, y2) is:

The equation of a circle given in the form indicates that the circle has

a radius ofr and that its center is at the origin (0,0) of the xy-coordinate system.

The slope of a line is calculated by using the formula (y2 y1) / (x2 x1)

Permutation & Combination

Arranging n things in which p are of one type, q of a second type and r of

third type. Formula !!!

!

rqp

n

Ex: In how many ways can you arrange the letters of word Banana?

Ans. !2!3

!6

Total 6 words, divide by repeats 3 a & 2n

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Circular Permutations: (n 1) !

When there are n people and all n people to be seated, then it is n!

When Order matters / Gives unique arrangements -( )!

!

rn

nP

r

n

=

o For e.g. A and B sitting on chair can be AB or BA so these are two

distinct arrangements.

o When n distinct items present and r have to be selected and then

arranged. E.g How many ways can you arrange 4 people in 5 chairs

= 5 P 4

o Keywords seating, sitting, sequence, order, alphabets, schedule,

ranking, itinerary, codes.

When Order Does not matter - ( ) !!!

rrn

nC

r

n

=

o For e.g. - Choosing A and B from a group of 3 or four alphabets. The

order does not matter. India playing a match against Australia is the

same as Australia playing against India.

o Keywords team, committee, balls, handshakes, matches, picking etc.

Operators - AND denotes Multiplication / OR denotes Addition

Special Cases

o 2 or more persons will not sit together

1) Calculate n!

2) Calculate (n-no. of people not sit together) !

3) Multiple 2) * No. of people not sitting together

4) Deduct 3) from 1)