mathematics last minutes note ii - tak sun …personal.tsss.edu.hk/roh/f6/mathematics last minutes...
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DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151112
Last Minutes Review II [1]
Mathematics Last Minutes Review II
Revision – Straight lines and locus Horizontal line: ky , Vertical line: hx
Intercept-slope form: cmxy Point-slope form: mxx
yy
1
1
Slope of a line (2 points): 12
12
xx
yy
Two-point form: 12
12
1
1
xx
yy
xx
yy
General form: 0 CByAx B
Cx
B
Ay x-intercept =
A
C (Put y = 0)
Slope = B
A y-intercept =
B
C (Put x = 0)
Relationship between two lines: overlapping (same slope 21 mm & y-intercept 21 cc )
parallel ( 21 mm )
perpendicular ( 121 mm )
intersecting ( 21 yy find the point) Linear programming:
1. Let x and y be the amount of those items Note: 0x , 0y , integer / number? Set up equations according to the questions
2. DRAW lines! (Get 2 points solid vs. dotted) SHADE area! (Try a point (0,0) shade / plot integral points)
3. Write down the Cost function of x and y Set Cost = 0 Plot the line move in parallel to the max/min
4. Get max/min point and find Cost (or get the point from all extremes)
Intersection of 2 lines
)2(0
)1(0
rqypx
cbyax Sub (1) into (2)
For any two points 11 , yxA and 22 , yxB , Distance formula: 212
212 yyxxAB
Mid-point formula:
2,
22121 yyxx
Section formula (r : s):
sr
rysy
sr
rxsx 2121 ,
Distance of A to line ky : ky 1 or 1yk Distance of A to line hx : hx 1 or 1xh
Finding locus: 1. Let yxP , be the movable point of the locus 2. Use distance formula, slope properties, … to set up a relationship 3. Express the equation as 022 FEyDxByAx
Description of locus: Fixed distance to a point Circle Fixed distance to a line 2 parallel lines Equal distance to parallel lines A line in the midway Equal distance to 2 points A perpendicular bisector Equal distance from 2 crossing lines A pair of angle bisectors Equal distance to a point and a line Parabola
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151112
Last Minutes Review II [2]
Revision – Circle properties
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151112
Last Minutes Review II [3]
Revision – Tangent properties
Revision – Equation of circle Given Centre kh, and radius r 222 rkyhx
Given 022 FEyDxyx Centre =
2,
2
ED, Radius = F
ED
22
22
Given 3 points Let 022 FEyDxyx Sub 3 points 3 equations 3 unknowns
Relationship between a circle and a straight line:
)2(
)1(22
cmxy
FEyDxyxy
Sub (2) into (1) Quadratic equation in x 0 2 intersections 0 1 intersection (line is tangent to circle) 0 0 intersections
Revision – Trigonometry functions For a right-angled triangle
c
asin ,
c
bcos ,
b
atan
For yxP , on a rectangular plane
r
ysin ,
r
xcos ,
x
ytan , 22 yxr
Properties: 1sin1 , 1cos1 tan has no max/min sin and cos are periodic of 360 tan are periodic of 180
Identities: 1cossin 22 ,
cos
sintan
ff
o
o
360
180,
coff
o
o
270
90
tan
1tan
sincos
cossin*
coff
coff
coff
a c
b
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151112
Last Minutes Review II [4]
Revision – Graph of the functions 0 90 180 270 360 sin 0 1 0 – 1 0 cos 1 0 – 1 0 1 tan 0 +/– 0 –/+ 0
Translation of yxP , Reflection of yxP , Rotation of yxP ,
Translation of xfy Reflection of xfy Enlargement/ Reduction of
Revision – 3-D problem Draw triangles from side and top views Find projection of a point to a plane
Triangle properties: Pythagoras’ theorem: 222 cba
Sine law: C
c
B
b
A
a
sinsinsin
Cosine law: Cabbac cos2222
Area: of △: Cabsin2
1
Or Heron’s formula Direction:
True bearing: 045 200 Compass bearing: N45E S70W
(x, y+k)
(x, y–k)
(x+h, y) (x–h, y)
(x, y)
(x, –y)
(x, y) (–x, y)
(–y, x)
(y, –x)
(x, y)
(–x, –y)
y–k = f(x)
y+k = f(x)
y = f(x–h) y = f(x+h)
y = f(x)
–y = f(x)
y = f(x) y = f(–x)
y/k = f(x)
yk = f(x)
y = f(x/k) y = f(xk)
xfy