Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

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

---