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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
...
If two angels of a spherical triangle are unequal,
the sides opposite are unequal, and the greater
side lies opposite the greater angle; and
conversely
...
The sum of two sides of a spherical triangle is
greater than the third side
...
The sum of the sides of a spherical triangle is less
than 360°
...
The sum of the angles of a spherical triangle is
greater that 180° and less than 540°
...
The sum of any two angles of a spherical
triangle is less than 180° plus the third angle
...
When the hypotenuse of a right spherical triangle
is greater than 90°, one leg is of the first quadrant
and the other of the second and conversely
...


SOLUTION TO OBLIQUE TRIANGLES
Law of Sines:

sina
=

NAPIER CIRCLE
Sometimes called Neper’s circle or Neper’s pentagon,
is a mnemonic aid to easily find all relations between
the angles and sides in a right spherical triangle
...
The sine of any middle part is equal to the
product of the cosines of the opposite parts
...
The sine of any middle part is equal to the
product of the tangent of the adjacent parts
...
In a right spherical triangle and oblique angle
and the side opposite are of the same
quadrant
...
When the hypotenuse of a right spherical
triangle is less than 90°, the two legs are of the
same quadrant and conversely
...
on great circle arc = 1 nautical mile
1 nautical mile = 6080 feet
= 1852 meters

du
+v dx
dx du

−c du d
= 2dx

c
dx u

u

1 statute mile = 5280 feet
= 1760 yards

d

u

1 statute mile = 8 furlongs
= 80 chains

du

du
(a ) = a ln a dx
dx d u
u

(e ) = e
dx

u

dx
du
log a e

d (ln a u) =
dx
u du
d

dx

dx

(lnu) = dx u d
du (sin u) =
cosu dx
dx d
du (cosu) =
−sin u dx dx
d
du
2
(tanu) = sec u dx dx

Derivatives

d

dC
=0
dx
d

du
(u +v) =
dx
dx

dv
+
dx

du
(cotu) = −csc u dx
dx d du (secu) =
secu tanu dx
dx
d
du (cscu)
=−cscucotu dx
dx d
−1 u) = 1
du
(sin
2

1−u 2 dx
d −1 u) = −1 du
(cos
dx
1−u2 dx
dx

DIFFERENTIAL
CALCULUS

dxd (tan−1 u) = 1+1u2 dudx dxd ( −1 u) =
1+−u12 dudx cot
d (sec−1 u) =
1
du dx
dx d (csc−1 u) =
−1
du dx
dx d du (sinh u) = coshu dx
du (coshu) = sinh u dx
(tanhu) = sech2u du dx
dx

u u 2 −1
u u 2 −1
dx d
dx d

d (cothu) = −csch2u du dx
(sechu) =−sechu tanhu dx
(cschu) =−cschucothu dx
=
1
du
(sinh
dx
u 2 +1 dx

dx d
dx d
dx d

d −1 u) =
(cosh

1

du

u2 −1 dx

dx

(tanh
dxd −1 u) = 1−1u 2 dudx
(sinh
dxd −1 u) = u 2−−11 dudx
d −1u) =
(sech
dx

L’Hospital’s Rule

Lim
f (x) = Lim
f '(x) = Lim
f "(x)
...
999999
TIP 2: if x → ∞, substitute x = 999999
TIP 3: if Trigonometric, convert to RADIANS then
do tips 1 & 2

MAXIMA AND MINIMA
Slope (pt

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