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ALGEBRA 1

Composition Analysis: Ax + By = Cz
WORK PROBLEMS

LOGARITHM

x = logb N → N =bx
Properties

log(xy) = log x + log
x
y

y log

Rate of doing work = 1/ time
Rate x time = 1 (for a complete job)
Combined rate = sum of individual
rates Man-hours (is always assumed
constant)
(Wor ker s1)(time1) (Wor ker s2 )(time2 )
=
quantity
...
work1
quantity
...
work2

= log x − log y

ALGEBRA 2

log xn = nlog x
logb x =

UNIFORM MOTION PROBLEMS
log
x logb

S =Vt

loga a =1
REMAINDER AND FACTOR THEOREMS

Traveling with the wind or downstream:

Given:

Vtotal = V1 +V2

f (x)
(x − r)

Traveling against the wind or upstream:

Vtotal = V1 −V2

Remainder Theorem: Remainder = f(r)
Factor Theorem: Remainder = zero
QUADRATIC EQUATIONS
Ax2 + Bx +C = 0
− B ± B 2 − 4AC
Root = 2A
Sum of the roots = - B/A
Products of roots = C/A
MIXTURE PROBLEMS
Quantity Analysis: A + B = C

DIGIT AND NUMBER PROBLEMS

100h +10t +u →
where:

2-digit number

h = hundred’s digit
t = ten’s
digit u =
unit’s digit

CLOCK PROBLEMS

an = a m r n−m

where:
x = distance traveled by the
minute hand in minutes x/12 = distance
traveled by the hour
hand in
minutes

PROGRESSION PROBLEMS

nth term

r=

a 2 a3
=
a1 a2

S=

a ( r n −1)
1
→ r >1
r −1

Sum of ALL
terms, r >1

a1 (1 −r n )
S=
→ r <1
1 −r

Sum of ALL
terms, r < 1

S=

a1 = first term an = nth term

a1
1 −r

ratio

→ r <1 & n = ∞

Sum of ALL
terms,
r<1,n=∞

am = any term before an d =
common difference
= sum of all “n” terms

S

HARMONIC PROGRESSION (HP)
-

ARITHMETIC PROGRESSION (AP)
difference of any 2 no
...


COIN PROBLEMS

d = a − a = a − a ,
...


ALGEBRA 3
Fundamental Principle:
“If one event can occur in m different ways, and
after it has occurred in any one of these ways, a
second event can occur in n different ways, and
then the number of ways the two events can occur
in succession is mn different ways”

PERMUTATION

Properties of a binomial expansion: (x +
y)n

Permutation of n objects taken r at a time nPr
1
...
The powers of x decreases by 1 in the
successive terms while the powers of y
increases by 1 in the successive terms
...
The sum of the powers in each term is always
equal to “n”

= n!

4
...
objects
are alike

P=

is equal to “n+1”

of the terms having a coefficient of 1
...


Permutation of n objects arrange in a circle

P = (n−1)!

r th term = nCr-1 (x)n-r+1 (y)r-1
term involving yr in the expansion (x +
y)n

COMBINATION

y r term = nCr (x)n-r (y)r

Combination of n objects taken r at a time

sum of coefficients of (x + y)n

nCr =

Sum = (coeff
...
of y) n

−nr!)!r!

sum of coefficients of (x + k)n

Sum = (coeff
...
The sets are drawn as
circles
...


PLANE
TRIGONOMETRY

FULL OR PERIGON

Measurement
θ = 0°
0° < θ < 90°
θ = 90°
90° < θ < 180°
θ =180°
180° < θ < 360°
θ = 360°

Pentagram – golden triangle (isosceles)

36 °

72° 72 °

TRIGONOMETRIC IDENTITIES

sin 2 A+ cos2 A = 1 1+

a

= =

cot2 A = csc2 A
1+ tan2 A = sec 2 A sin(A± B) = sin
AcosB ± cos Asin B cos(A± B) =

COSINE LAW

cosAcosB sin Asin B tan(A± B) =
tan A± tanB
1tan

b
c
sin A
sinB
sinC

a2 = b2 + c2 – 2 b c cos A b2 = a2 + c2
– 2 a c cos B c2 = a2 + b2 – 2 a b cos

AtanB

cot(A± B) =

C

cot AcotB 1 cot A± cotB
sin 2A = 2sin AcosB cos2A
= cos2 A−sin 2 A

AREAS OF TRIANGLES AND
QUADRILATERALS

tan2A =

TRIANGLES

2tan
2
A 1− tan A
cot2 A−1

1
...
Given two sides and included angle

Area = absinq

3
...
Quadrilateral circumscribing in a circle

Area = s(s − a)(s −b)(s − c)
Area = rs

Area = abcd

s=a+b+c
2
4
...
Triangle circumscribing a circle

Area=rs
6
...
Given diagonals and included angle

1

Area =

d1d2 sinq

2
2
...
Cyclic quadrilateral – is a quadrilateral inscribed in a
circle

Area = (s − a)(s −b)(s −c)(s − d)
s = a +b + c + d
2
(ab +cd)(ac +bd)(ad +bc)
r=
4(Area)

d1d 2= ac+bd →Ptolemy’s Theorem

q = (n − 2)(180°)

SIMILAR TRIANGLES

A1
C

A
2

2

B

n

2

2

H

Value of each exterior angle
A2

a

b

c

h

SOLID GEOMETRY

a =180°−q =

360°
n

Sum of exterior angles:
POLYGONS
3 sides – Triangle
4 sides –
Quadrilateral/Tetragon/Quadrangle
5 sides – Pentagon
6 sides – Hexagon
7 sides – Heptagon/Septagon
8 sides – Octagon
9 sides – Nonagon/Enneagon
10 sides – Decagon
11 sides – Undecagon
12 sides – Dodecagon
15 sides – Quidecagon/ Pentadecagon
16 sides – Hexadecagon
20 sides – Icosagon
1000 sides – Chillagon
Let: n = number of sides
θ = interior angle
α = exterior angle
Sum of interior angles:

S = n α = 360°

Number of diagonal lines (N):

N=

n

(n − 3)

2
Area of a regular polygon inscribed in a circle of
radius r

Area = 1 nr2 sin
360
2
n
Area of a regular polygon circumscribing a
circle of radius r

Area = nr2 tan

n

S = n θ = (n – 2) 180°
Value of each interior angle

180

Area of a regular polygon having each side
measuring x unit length

Area = 1 nx2 cot

180

4

1

A=

n

d1d2sinq

2

PLANE GEOMETRIC FIGURES
RHOMBUS

CIRCLES

pd 2 2 A =
=pr

1

A=

4

d1d2 = ah

2

Circumference =pd = 2pr

A = a2 sina

Sector of a Circle
SOLIDS WITH PLANE SURFACE

A = rs = r2q

Lateral Area = (No
...
The bounding
planes are referred to as the faces and the intersections of
the faces are called the edges
...


PRISM

V = Bh
A(lateral) = PL
A(surface) = A(lateral) + 2B
where: P = perimeter of the base L
= slant height
B = base area

Truncated Prism

V=B
number∑heightsof heights

PYRAMID

V

Bh

A(lateral) = ∑ Afaces
A(surface) =A(lateral) +B
Frustum of a Pyramid

V=

h

(A1 + A2 + A1A2 )

3
A1 = area of the lower base
A2 = area of the upper base

PRISMATOID

h
V = 6(A1 + A2 +4Am)
Where: x = length of one edge
Am = area of the middle section

SOLIDS WITH CURVED SURFACES

REGULAR POLYHEDRON

CYLINDER

a solid bounded by planes whose faces are congruent
regular polygons
...

B
...

D
...


Tetrahedron
Hexahedron (Cube)
Octahedron
Dodecahedron
Icosahedron

A(lateral) = PkL = 2 π r h
A(surface) = A(lateral) + 2B
Pk = perimeter of right section
K = area of the right section
B = base area
L= slant height
CONE

V

=Bh

A(lateral) =prL

V = 3 (3r − h)
FRUSTUM OF A CONE

V = h (A1 + A2 + A1 A2
3

p
V=

+h2 )

2

+3b2 + h2 )

(3a

p

h

V=

(3a
6

SPHERE

=pr3

V

2

6

A(lateral) =p(R + r)L
SPHERES AND ITS FAMILIES

h

SPHERICAL WEDGE
is that portion of a sphere bounded by a lune and the planes
of the half circles of the lune
...


2

A

V

=

V = 1 A(zone)r
3

° 90
SPHERICAL ZONE
is that portion of a spherical surface between two
parallel planes
...


A(surface) = A(zone) + A(lateralofcone)
SPHERICAL PYRAMID
is that portion of a sphere bounded by a spherical
3
polygon and the planes of its sides
...


ph 2

E = [(n-2)180°]
E = Sum of the angles E =
Spherical excess
n = Number of sides of the given spherical polygon

SOLIDS BY REVOLUTIONS

V12

AA12

2

3

V

TORUS (DOUGHNUT)
a solid formed by rotating a circle about an axis not
passing the circle
...

It is a special ellipsoid with

d = (x2 − x1)2 + (y2 − y1)2

c=a

V = pa2b

Slope of a line

PROLATE SPHEROID

m = tanq = yx22 −− xy11

a solid formed by rotating an ellipse about its major axis
...


x = x1rr12 ++rx2 r1

y=

y1rr12 ++ry22r1
V = pr2h

2

SIMILAR SOLIDS

V1
V2

A2

3

H
h

A1

r
H

h

Location of a midpoint
R

3

L

l

2

R
r

x1 +2 x2

3

l

2

L

2

x=
STRAIGHT LINES

y = y1 +
2

y2

d = Ax1 + By2 +1 B+2C

General Equation Ax + By + C

=0

±A

Point-slope form
Note: The denominator is given the sign of B

y – y1 = m(x – x1)
Two-point form

y − y1 = yx22 −−

Distance between two parallel lines d =

C1 −C2

xy11 (x − x1)

A2 + B2
Slope relations between parallel lines: m1 =
m2

Slope and y-intercept form

y = mx + b
Intercept form

x

Slope relations between perpendicular lines:
m1m2 = –1

y
+

a

=1
b

Slope of the line, Ax + By + C = 0 m

=−

Line 1 → Ax + By + C1 = 0 Line
2 → Ax + By + C2 = 0

Line 1 → Ax + By + C1 = 0
Line 2 → Bx – Ay + C2 = 0
PLANE AREAS BY COORDINATES

= 1 x1,x2,x3,
...
yn, y1

Angle between two lines
−1

m−1mm12
tan

1m+2
q=

Note: Angle θ is measured in a counterclockwise
direction
...


Distance of point (x1,y1) from the line
Ax + By + C = 0;

Note: The points must be arranged in a counter clockwise
order
...


SPACE COORDINATE SYSTEM
Length of radius vector r:

r = x2 + y2 + z2
Distance between two points P1(x1,y1,z1)
and
P2(x2,y2,z2)

Parabola

B2 - 4AC = 0

Ellipse

B2 - 4AC < 0, A ≠ C

A≠C
same sign
Sign of A
opp
...


ANALYTIC
GEOMETRY 2

Standard Equation:

CONIC SECTIONS
a two-dimensional curve produced by slicing a plane
through a three-dimensional right circular conical surface

(x – h)2 + (y – k)2 = r2
General Equation:

Ways of determining a Conic Section
1
...

3
...


x2 + y2 + Dx + Ey + F = 0

By Cutting Plane
Eccentricity
By Discrimination
By Equation

Center at (h,k):

General Equation of a Conic Section:
2

D

h =−

E

; k =−

2A

2

2A

Ax + Cy + Dx + Ey + F = 0 **
Radius of the circle:
Circle
Parabola
Ellipse
Hyperbola

Circle

Cutting plane

Eccentricity

Parallel to base

e→0

Parallel to element

e = 1
...
0

PARABOLA

Parallel to axis

e > 1
...


Discriminant

Equation**

B2 - 4AC < 0, A = C

A=C

r 2 = h2 + k 2 −

F or r = 1 D2 +E2 −4F
A

2

STANDARD EQUATIONS:
Opening to the right:
where: a = distance from focus to vertex
= distance from directrix to vertex

(y – k)2 = 4a(x – h)
Opening to the left:

AXIS HORIZONTAL:

Cy2 + Dx + Ey + F = 0
Coordinates of vertex (h,k):

(y – k)2 = –4a(x – h)
Opening upward:

(x – h) 2 = 4a(y – k)

k =−

Opening downward:

2EC

substitute k to solve for h
Length of Latus Rectum:

(x – h) 2 = –4a(y – k)
Latus Rectum (LR)
a chord drawn to the axis of symmetry of the curve
...


AXIS VERTICAL:

e=1

for a parabola

2

Ax + Dx + Ey + F = 0
Coordinates of vertex (h,k): h

=−

ELLIPSE
a locus of a moving point which moves so that the sum of its
distances from two fixed points called the foci is constant and
is equal to the length of its major axis
...


Coordinates of the center:

D

h=−
E

;k =−

2A

2C

If A > C, then: a2 = A; b2 = C If A < C,
then: a2 = C; b2 = A

d = distance from center to directrix a
= distance from center to vertex c =
distance from center to focus

STANDARD EQUATIONS
Transverse axis is horizontal

KEY FORMULAS FOR ELLIPSE
Length of major axis: 2a
Length of minor axis: 2b
Distance of focus to center:

c= a2 −b2

(x−h)2
2−

(y−k)2
b2
=1 a

Transverse axis is vertical:

(y − k)2

(x − h)2
− = 1 a2 b2

GENERAL EQUATION

Ax2 – Cy2 + Dx + Ey + F = 0
Length of latus rectum:

2b2

Coordinates of the center:

D
2 A; k =−

h =−
E
2 C

e = c = aa
d
POLAR COORDINATES SYSTEM

If C is negative, then: a2 = C, b2 = A
If A is negative, then: a2 = A, b2 = C
Equation of Asymptote:

x = r cos θ

y=r
sin θ

(y – k) = m(x – h)
Transverse

axis

is

horizontal:

m=± ba

Transverse

axis

vertical:

is

m=± a b

r = x2 + y2
tanq = x

y

KEY FORMULAS FOR HYPERBOLA
Length of transverse axis: 2a
Length of conjugate axis: 2b
Distance of focus to center:

c = a2 +b2

Important propositions
1
...


Length of latus rectum:

2b2
LR =
a
Eccentricity:

SPHERICAL
TRIGONOMETRY

2
...

3
...


a+b>c
4
...


0° < a + b + c < 360°
5
...


180° < A + B + C < 540°
6
...


A + B < 180° + C
SOLUTION TO RIGHT TRIANGLES

3
...


QUADRANTAL TRIANGLE
is a spherical triangle having a side equal to 90°
...


sinb
sinc
=
sin A sin B
sinC

Law of Cosines for sides:

cosa = cosbcosc + sinbsinccos A cosb
= cosacosc + sinasinccosB
cosc = cosacosb + sinasinbcosC
Law of Cosines for angles:

Napier’s Rules

cos A = −cos BcosC + sin Bsin C cosa cos
B = −cos AcosC + sin Asin C cosb
cosC = −cos Acos B + sin Asin Bcosc

1
...


Co-op
AREA OF SPHERICAL TRIANGLE
2
...


Tan-ad

p R 2E
A=

180°

Important Rules:
1
...

2
...


R = radius of the sphere
E = spherical excess in degrees,

E = A + B + C – 180°

TERRESTRIAL SPHERE
Radius of the Earth = 3959 statute miles
Prime meridian (Longitude = 0°)
Equator (Latitude = 0°)
Latitude = 0° to 90°
Longitude = 0° to +180° (eastward)
= 0° to –180° (westward)

d
dv
(uv) = u
dx
dv

v
−u
d u = dx 2 dx
dx v
v
d (u n ) = nun−1
du dx dx du d dx u
=
dx
2u

1 min
...
x→a g(x) x→a g'(x) x→a g"(x)
Shortcuts
Input equation in the calculator
TIP 1: if x → 1, substitute x = 0
...
)
Y’
MAX
0

Y”
(-) dec

Concavity
down

MIN

0

(+) inc

up

INFLECTION

-

No change

-

du

u 1−u 2 dx
d −1u) =
(csch

dx

−1

du
du
−1 u)

LIMITS
Indeterminate Forms

−1

du

u 1+u 2 dx

HIGHER DERIVATIVES nth
derivative of xn

d nn (xn) =
n! dx

∫[ f (u) + g(u)]du

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