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Divisibility Rule
Number Rule Example
2 If last digit is 0,2,4,6, or 8 22, 30, 50, 68, 1024
3 If sum of digits is divisible by 3 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6
is divisible by 3)
4 If number created by the last 2
digits is divisible by 4
864 is divisible by 4 since 64 is divisible by 4
5 If last digit is 0 or 5 5, 10, 15, 20, 25, 30, 35, 2335
6 If divisible by 2 & 3 522 is divisible by 6 since it is divisible by 2 & 3
9 If sum of digits is divisible by 9 621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9
is divisible by 9)
10 If last digit is 0 10, 20, 30, 40, 50, 5550
Logarithms
Definition: ๐ = ๐๐๐ ๐ ๐ โ ๐ = ๐ ๐
Example: ๐๐๐ ๐ ๐๐ = ๐ โ ๐๐ = ๐ ๐
Properties
๐๐๐ ๐ ๐ = ๐ ๐๐๐ ๐ ๐ = ๐
๐๐๐ ๐ ๐ ๐
= ๐ ๐๐๐๐ ๐ ๐
= ๐
๐๐๐ ๐ ๐ ๐
= ๐๐๐๐ ๐ ๐
๐๐๐ ๐ ๐ ร ๐ = ๐๐๐ ๐ ๐ + ๐๐๐ ๐ ๐
๐๐๐ ๐
๐
๐
= ๐๐๐ ๐ ๐ โ ๐๐๐ ๐ ๐
Special Exponents
Reciprocal ๐
๐ ๐
= ๐โ๐
Power 0 ๐ ๐
= ๐
Power 1 ๐ ๐
= ๐
Operations Involving Exponents
Multiplication ๐ ๐
ร ๐ ๐
= ๐(๐+๐)
Division ๐ ๐
รท ๐ ๐
= ๐(๐โ๐)
Power (๐ ๐
) ๐
= ๐ ๐ร๐
Roots
๐
๐
๐ = ๐ ๐๐
= ( ๐
๐
) ๐
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Progressions
Arithmetic Progression: ๐ ๐ก๐
term of an
Arithmetic Progression
๐ ๐ = ๐ ๐ + ๐ โ ๐ ๐
Sum of n terms of an arithmetic expression
๐ ๐ =
๐ ๐ ๐ + ๐ ๐
๐
=
๐ ๐๐ ๐ + ๐ โ ๐ ๐
๐
The first term is ๐1, the common difference
is ๐, and the number of terms is ๐.
Geometric Progression: ๐ ๐ก๐
term of a
Geometric Progression
๐ ๐ = ๐ ๐ ๐ ๐โ๐
Sum of n terms of a geometric progression:
๐ ๐ =
๐(๐ ๐+๐
โ ๐)
๐ โ ๐
The first term is ๐1, the common ratio is ๐, and
the number of terms is ๐.
Infinite Geometric Progression
Sum of all terms in an infinite geometric series =
๐ ๐
(๐โ๐)
๐ค๐๐๐๐ โ ๐ < ๐ < ๐
Roots of a Quadratic Equation
A quadratic equation of type ๐๐ฑ ๐
+ ๐๐ฑ + ๐ has two solutions, called roots. These two solutions
may or may not be distinct. The roots are given by the quadratic formula:
โbยฑ b2โ4ac
2a
where
the ยฑ sign indicates that both
โbโ b2โ4ac
2a
and
โb+ b2โ4ac
2a
are solutions
Common Factoring Formulas
1. ๐ ๐
โ ๐ ๐
= ๐ โ ๐ ร ๐ + ๐
2. ๐ ๐
+ ๐๐๐ + ๐ ๐
= (๐ + ๐) ๐
3. ๐ ๐
โ ๐๐๐ + ๐ ๐
= (๐ โ ๐) ๐
4. ๐ ๐
+ ๐๐๐ ๐
+ ๐๐ ๐
๐ + ๐ ๐
= (๐ + ๐) ๐
5. ๐ ๐
โ ๐๐๐ ๐
+ ๐๐ ๐
๐ โ ๐ ๐
= (๐ โ ๐) ๐
Binomial Theorem
The coefficient of ๐ฅ(๐โ๐)
๐ฆ ๐
in (๐ฅ + ๐ฆ) ๐
is:
๐ถ๐
๐
=
๐!
๐! (๐ โ ๐)!
Applies for any real or complex numbers x and y, and any non-negative integer n.
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Summary of counting methods
Order matters Order doesnโt matter
With
Replacement If ๐ objects are taken from a set of ๐
objects, in a specific order with
replacement, how many different
samples are possible? ๐ ๐
N/A
Without
Replacement
Permutation Rule: If ๐ objects are
taken from a set of ๐ objects without
replacement, in a specific order, how
many different samples are
possible?
๐๐
๐
=
๐!
(๐ โ ๐)!
Combination Rule: If ๐ objects are
taken from a set of ๐ objects without
replacement and disregarding order,
how many different samples are
possible?
๐ถ๐
๐
=
๐!
๐!(๐โ๐)!
Probability
The probability of an event A, ๐ ๐ด is defined as
๐ ๐ด =
๐๐ข๐๐๐๐ ๐๐ ๐๐ข๐ก๐๐๐๐๐ ๐ก๐๐๐ก ๐๐๐๐ข๐ ๐๐ ๐๐ฃ๐๐๐ก ๐ด
๐๐๐ก๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐๐๐๐๐๐ฆ ๐๐ข๐ก๐๐๐๐๐
Independent Events: If A and B are independent events, then the probability of A happening
and the probability of B happening is:
๐ ๐ด ๐๐๐ ๐ต = ๐(๐ด) ร ๐(๐ต)
Dependent Events: If A and B are dependent events, then the probability of A happening and
the probability of B happening, given A, is:
๐ ๐ด ๐๐๐ ๐ต = ๐(๐ด) ร ๐(๐ต | ๐ด)
Conditional Probability: The probability of an event occurring given that another event has
already occurred e.g. what is the probability that B will occur after A
๐ ๐ต ๐ด =
๐(๐ด ๐๐๐ ๐ต)
๐(๐ด)
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Geometry Fundamentals
The sum of angles around a point will always
be 360 degrees.
In the adjacent figure
๐ + ๐ + ๐ + ๐ = ๐๐๐ยฐ
Vertical angles are equal to each other.
In the adjacent
figure ๐ = ๐ and
๐ = ๐
The sum of angles on a straight line is 180ห.
In the adjacent
figure sum of
angles and b is
180 i.e. ๐ + ๐ =
๐๐๐ยฐ
When a line intersects a pair of parallel lines,
the corresponding angles are formed are
equal to each other
In the figure above
๐ = ๐
When a line intersects a pair of parallel lines,
the alternate interior and exterior angles
formed are equal to each other
In the adjacent
figure alternate
interior angles
a=b and
alternate exterior
angles c=d
Any point on the perpendicular bisector of a
line is equidistant from both ends of the line.
In the figure, k
is the
perpendicular
bisector of
segment AB
and c=d, e=f
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Triangle Properties
1. Sum of all internal angles of a
triangle is 180 degrees i.e.
๐ + ๐ + ๐ = ๐๐๐ยฐ
2. Sum of any two sides of a triangle
is greater than the third i.e.
๐ + ๐ > ๐ or ๐ + ๐ > ๐ or ๐ + ๐ > ๐
3. The largest interior angle is
opposite the largest side; the
smallest interior angle is opposite
the smallest side i.e. if ๐ฉ > ๐ช โ
๐ > ๐
4. The exterior angle is supplemental
to the adjoining interior angle i.e.
๐ + ๐ = ๐๐๐ยฐ. Since ๐ + ๐ + ๐ =
๐๐๐ยฐ it follows that ๐ = ๐ + ๐
5. The internal bisector of an angle
bisects the opposite side in the
ratio of the other two sides. In the
adjoining figure
BD
DC
=
AB
BC
Pythagoras Theorem
๐ ๐
= ๐ ๐
+ ๐ ๐
Commonly Used Pythagorean Triples
Height, Base Hypotenuse
3,4 or 4,3 5
6,8 or 8,6 10
5, 12 or 12,5 13
7, 24 or 24,7 25
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Special Right Triangles
The lengths of the
sides of a 45๏ฐ- 45๏ฐ-
90๏ฐ triangle are in
the ratio of ๐: ๐: ๐
The lengths of the sides
of a 30๏ฐ- 60๏ฐ- 90๏ฐ
triangle are in the ratio
of ๐: ๐: ๐
Test of Acute and Obtuse Triangles
If ๐ ๐
< ๐ ๐
+ ๐ ๐
then it is an acute-angled
triangle, i.e. the angle facing side c is an acute
angle.
If ๐ ๐
> ๐ ๐
+ ๐ ๐
then it is an obtuse-angled
triangle, i.e. the angle facing side c is an
obtuse angle.
Polygons
Sum of interior angles of a quadrilateral is 360ห
๐ + ๐ + ๐ + ๐ = 360ยฐ
Sum of interior angles of a pentagon is 540ห
๐ + ๐ + ๐ + ๐ + ๐ = 540ยฐ
If n is the number of sides of the polygon then, sum of interior angles = (n - 2)180ยฐ
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Properties of a Circle
The angle at the centre of a circle is twice any
angle at the circumference subtended by the
same arc.
In the adjacent figure
angle ๐ = ๐๐
Every angle subtended at the circumference
by the diameter of a circle is a right angle
(90ห).
In a cyclic quadrilateral, the opposite angles are
supplementary i.e. they add up to 180ห.
In the figure
above angle
๐ + ๐ =
๐๐๐ยฐ and
๐ + ๐ = ๐๐๐ยฐ
The angles at the circumference subtended
by the same arc are equal.
In the
adjacent
figure angle
๐ = ๐
A chord is a straight line joining 2 points on the
circumference of a circle.
A radius that
is
perpendicular
to a chord
bisects the
chord into
two equal
parts and
vice versa.
In the adjacent figure PW=PZ
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Area and Perimeter of Common Geometrical Figures
Rectangle
Area ๐จ = ๐ ร ๐
Perimeter ๐ท = ๐ ร (๐ + ๐)
Square
Area ๐จ = ๐ ๐
Perimeter ๐ท = ๐๐
Triangle
Area ๐จ =
๐
๐
ร ๐ ร ๐
Circle
Area ๐จ = ๐ ๐ ๐
Perimeter ๐ท = ๐๐ ๐
Equilateral Triangle
Area ๐จ =
๐
๐
ร ๐ ๐
Perimeter ๐ท = ๐๐
Altitude ๐ = ๐
๐
๐
Trapezoid
Area ๐จ =
๐
๐
๐ (๐ ๐ + ๐ ๐)
Parallelogram
Area ๐จ = ๐ ร ๐
Perimeter ๐ท = ๐ ร (๐ + ๐)
Ring
Area ๐จ = ๐ (๐น ๐
โ ๐ ๐
)
Sector
๐ in degrees
Area ๐จ = ๐ ๐ ๐
ร
๐
๐๐๐
Length of Arc ๐ณ = ๐ ๐ซ ร
๐
๐๐๐
Perimeter ๐ท = ๐ณ + ๐๐
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Volume and Surface Area of 3 Dimensional Figures
Cube
Volume ๐ = ๐ ร ๐ ร ๐
Surface Area
๐จ = ๐(๐๐ + ๐๐ + ๐๐)
Sphere
Volume ๐ =
๐
๐
๐ ๐ ๐
Surface Area ๐จ = ๐๐ ๐ ๐
Right Circular Cylinder
Volume ๐ = ๐ ๐ ๐
๐ก
Surface Area ๐จ = ๐๐ ๐ซ๐ก
Pyramid
Volume ๐ =
๐
๐
๐ฉ๐
Surface Area = ๐จ = ๐ฉ +
๐๐
๐
B is the area of the base
๐ is the slant height
Right Circular Cone
Volume ๐ =
๐
๐
๐ ๐ ๐
๐ก
Surface Area ๐จ = ๐ ๐(๐ + ๐)
Frustum
Volume
๐ =
๐
๐
๐ ๐ก(๐ซ ๐
+ ๐ซ๐ + ๐ ๐
)
Coordinate Geometry
Line: Equation of a line ๐ = ๐๐ + ๐
In the adjoining
figure, c is the
intercept of the
line on Y-axis i.e.
c=2, m is the
slope
Slope of a line ๐ =
๐ ๐โ๐ ๐
๐ ๐โ๐ ๐
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The slopes of parallel lines are equal
In the
adjoining
figure, the two
lines are
parallel to
each other i.e.
๐ ๐ = ๐ ๐
The slopes of perpendicular lines are opposite
reciprocals of one another.
In the
adjoining the
two lines are
perpendicular
to each other
i.e. ๐ ๐ =
โ๐
๐ ๐
Midpoint between two points (๐ ๐, ๐ ๐) and
(๐ ๐, ๐ ๐) in a x-y plane:
๐ ๐, ๐ ๐ =
๐ ๐ โ ๐ ๐
๐
,
๐ ๐ โ ๐ ๐
๐
Distance between two points (๐ ๐, ๐ ๐) and
(๐ ๐, ๐ ๐) in a x-y plane:
๐ = (๐ ๐ โ ๐ ๐) ๐ + (๐ ๐ โ ๐ ๐) ๐
Horizontal and Vertical line
Equation of line parallel to x-axis (line p) with
intercept on y-axis at (0, b) is ๐ = ๐. Equation
of line parallel to y-axis (line q) with intercept
on x-axis at (a,0) is ๐ = ๐
Equation of a circle with center (h,k) and
radius r
(๐ โ ๐) ๐
+ (๐ โ ๐) ๐
= ๐ ๐
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Trigonometry Basics
Definition: Right Triangle definition for angle ฮธ
such that 0 < ฮธ < 90ยฐ
๐ ๐๐๐ =
๐๐๐๐๐๐ก
๐๐ฆ๐๐๐ก๐๐๐ข๐ ๐
, ๐๐ ๐๐ =
1
๐ ๐๐๐
๐๐๐ ๐ =
๐๐๐ ๐
๐๐ฆ๐๐๐ก๐๐๐ข๐ ๐
, ๐ ๐๐๐ =
1
๐๐๐ ๐
๐ก๐๐๐ =
๐๐๐๐๐๐ก
๐๐๐ ๐
, ๐๐๐ก๐ =
1
๐ก๐๐๐
Pythagorean Relationships
๐ ๐๐2
๐ + ๐๐๐ 2
๐ = 1
๐ก๐๐2
๐ + 1 = ๐ ๐๐2
๐
๐๐๐ก2
๐ + 1 = ๐๐ ๐2
๐
Tan and Cot
๐ก๐๐๐ =
๐ ๐๐๐
๐๐๐ ๐
,
๐๐๐ก๐ =
๐๐๐ ๐
๐ ๐๐๐
Half Angle Formulas
๐ ๐๐2
๐ =
1
2
(1 โ cos 2๐ )
๐๐๐ 2
๐ =
1
2
1 + cos 2๐
๐ก๐๐2
๐ =
(1 โ cos 2๐ )
(1 + cos 2๐ )
Even/Odd
sin โ๐ = โ๐ ๐๐๐
cos โ๐ = ๐๐๐ ๐
tan โ๐ = โ๐ก๐๐๐
sin 2๐ = 2๐ ๐๐๐๐๐๐ ๐, cos 2๐ = ๐๐๐ 2
๐ โ ๐ ๐๐2
๐
Periodic Formula: Where n is an integer
sin ๐ + 2๐๐ = sin ๐ โ sin ๐ + 180ยฐ = ๐ ๐๐๐
cosโก(๐ + 2๐๐) = cos ๐ โ cosโก(ฮธ + 180ยฐ) = ๐๐๐ ๐
tan ๐ + ๐๐ = tan ๐ โ tan ฮธ + 90ยฐ = tanฮธ
0ยฐ 30ยฐ 45ยฐ 60ยฐ 90ยฐ
Sin 0 1
2
1
2
3
2
1
Cos 1 3
2
1
2
1
2
0
Tan 0 1
3
1 3 โ