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DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160606
Last Minutes Review [1]
Mathematics Last Minutes Review
Form 5 Final Examination Date: 16 June 2016 (Thursday) Time: 09:00-11:15 (Paper 1)
11:45-13:00 (Paper 2) Venue: School Hall Chapters in Form 5 Chapter 1: Basic Properties of Circles Chapter 2: Tangents to Circles Chapter 3: Inequalities Chapter 4: Linear Programming Chapter 5: Applications of Trigonometry in 2-D Problems Chapter 6: Applications of Trigonometry in 3-D Problems Chapter 7: Equations of Circles Chapter 8: Locus Chapter 9: Measures of Dispersion Chapter 10: Permutation and Combination Chapter 11: More about Probability Chapters in Form 4 Chapter 1: Quadratic Equations in One Unknown (I) Chapter 2: Quadratic Equations in One Unknown (II) Chapter 3: Functions and Graphs Chapter 4: Equations of Straight Lines Chapter 5: More about Polynomials Chapter 6: Exponential Functions Chapter 7: Logarithmic Functions Chapter 8: More about Equations Chapter 9: Variations Chapter 10: More about Trigonometry Final Reminder Step by step (Do not jump!) Draw the graph and look! Get the keywords! Check answers (try to put them back!) Check units! Bring Calculator!!! Try your best! Never give up!
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151008
Last Minutes Review (F5 Chapters 1 and 2) [1]
Mathematics Last Minutes Review (Form 5 Chapters 1 and 2)
Revision – Circle properties
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151008
Last Minutes Review (F5 Chapters 1 and 2) [2]
Revision – Tangent properties
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151116
Last Minutes Review (F5 Chapters 3 and 4) [1]
Mathematics Last Minutes Review (Form 5 Chapters 3 and 4)
Revision – Inequalities ba cbca ba bcac (if 0c ) ba bcac (if 0c , change sign!)
ba c
b
c
a (if 0c ) ba
c
b
c
a (if 0c , change sign!)
ba ba
11 (reciprocal, change sign!)
Any number a 02 a
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 151116
Last Minutes Review (F5 Chapters 3 and 4) [2]
If 0a and , we have
02 cbxax 0 xxa x or x
02 cbxax 0 xxa x or x
02 cbxax 0 xxa x
02 cbxax 0 xxa x
No x-intercepts, all above x-axis: 042 acb (No roots)
Vertex
a
acb
a
b
4
4,
2
2
Steps of 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 3 points solid line vs. dotted line) 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)
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160121
Last Minutes Review (F5 Chapters 5 and 6) [1]
Mathematics Last Minutes Review (Form 5 Chapters 5 and 6)
Revision – Trigonometry functions
For a right-angled triangle
c
asin
c
bcos
b
atan
Pythagoras’ theorem: 222 cba
Identities: 1cossin 22
cos
sintan
For any triangle:
Area: bhArea2
1
Area: CabArea sin2
1
Area: csbsassArea
where 2
cbas
(Heron’s formula, FMLA03)
Sine law: C
c
B
b
A
a
sinsinsin
Cosine law: Cabbac cos2222 (FMLA02)
ab
bacC
2cos
222
(No FMLA)
a c
b
a c
b A
B
C
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160121
Last Minutes Review (F5 Chapters 5 and 6) [2]
Revision – 3-D problem
Direction: True bearing: 045 200
(Always from N, clockwise) Compass bearing: N45E S70W
(From N or S to E or W) Draw triangles (or other side) from top view front view side view
Projection of a point to a plane K is vertically below H △OHK is a right-angled triangle
Angles definitions:
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160407
Last Minutes Review (F5 Chapters 7 and 8) [1]
Mathematics Last Minutes Review (Form 5 Chapters 7 and 8 )
Revision – Cartesian coordinates
For any two points 11 , yxA and 22 , yxB ,
Mid-point formula:
2,
22121 yyxx
Section formula (r : s):
sr
rysy
sr
rxsx 2121 ,
Distance formula: 212
212 yyxxAB
Distance of 11 , yxA to line ky : ky 1 or 1yk
Distance of 11 , yxA to line hx : hx 1 or 1xh
For y -intercept, put 0x . For x -intercept, put 0y .
Horizontal line: ky Vertical line: hx
Slope of a line (2 points): 12
12
xx
yym
Angle of inclination: tanm
Two-point form: 12
12
1
1
xx
yy
xx
yy
Point-slope form: mxx
yy
1
1
Intercept-slope form: cmxy
Relationship between two lines: parallel ( 21 mm )
perpendicular ( 121 mm )
overlapping (same slope 21 mm & y-intercept 21 cc )
intersecting ( 21 yy find the point)
Check whether a point yx, is on a line:
Put yx, into the line equation and see whether LHS = RHS
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160407
Last Minutes Review (F5 Chapters 7 and 8) [2]
Revision – Equation of circle
Given Centre kh, and radius r :
222 rkyhx
Given 022 FEyDxyx :
Centre =
2,
2
ED
Radius = FED
22
22
Given 3 points:
Let 022 FEyDxyx Sub 3 points 3 equations 3 unknowns
Relationship between a circle (with centre O , radius r ) and a point A : Inside circle: rOA On the circle: rOA Outside circle: rOA
Relationship between a circle C and a straight line L :
)2(:
)1(: 22
cmxyL
FEyDxyxyC
Sub (2) into (1) Quadratic equation in x 0 2 intersections acb 42 0 1 intersection (line is tangent to circle) 0 0 intersections
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 A circle
Fixed distance to a line 2 parallel lines Fixed distance to a line segment 2 semi-circles & 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 A parabola
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160520
Last Minutes Review (F5 Chapters 9, 10 and 11) [1]
Mathematics Last Minutes Review (Form 5 Chapters 9, 10 and 11)
Revision – Statistics
Mean: n
xxxx n
21 or n
nn
fff
xfxfxfx
21
2211 (Calculator: SD Shift 2 1)
Median: Middle of the sorted data (odd vs. even) Mode: Number with highest frequency
Upper class boundary = 2
1 (upper limit + lower limit of next higher class)
Lower class boundary = 2
1 (lower limit + upper limit of next lower class)
Class size (Class width, Class length) = upper class boundary – lower class boundary
Class mid-point (Class mark) = 2
1 (lower class limit + upper class limit)
Range = Highest value – Lowest value (for single data)
= Highest class boundary – Lower class boundary (for grouped data)
Upper Quartile (UQ, Q3) = median of upper half data (odd vs. even) Lower Quartile (LQ, Q1) = median of lower half data (odd vs. even) Inter-Quartile Range (IQR) = UQ – LQ
Box and Whisker Diagram: Min, LQ, median, UQ, Max (Note: every part is 25%)
Standard Deviation
n
xxxxxx n22
22
1
(Calculator SD Shift 2 2)
Variance = 2
Standard Score
xxz
Exactly 50% of data more than (or less than) x About 68% of data lying between x and x ( 1 z ) About 95% of data lying between 2x and 2x ( 2 z ) About 99.7% of data lying between 3x and 3x ( 3 z )
Change of data: cXX Mean: cxx S.D.:
kXX Mean: xkx S.D.: k
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160520
Last Minutes Review (F5 Chapters 9, 10 and 11) [2]
Revision – Permutation and Combination Counting: Use when the cases are happened together (AND case)
Use when the cases are happened either way (OR case)
Factorial: 1221! nnnn
With positions: )!(
!
rn
nP n
r
Select a group: !)!(
!
rrn
nC n
r
Revision – Probability
Probability: outcomepossiblen
EforoutcomenEP
_
__)( , 1)(0 EP
Complementary: 1)'()( EPEP where 'E is the complementary of E
Addition: A and B are mutually exclusive, )()()( BPAPBorAP
A and B are not mutually exclusive, )()()()( BandAPBPAPBorAP
Multiplication: A and B are independent, )()()( BPAPBandAP
Conditional: )(
)()|(
AP
BandAPABP (Note: still probability of B!)
Set language:
AB is the event that both A and B occurs AB is the event that A or B (or both) occurs AB means A is a subset of B {x}B means x is an element of B
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [1]
Mathematics Last Minutes Review (Form 4 Term 2)
Revision – Quadratic expressions
02 cbxax a
acbbx
2
42
02 cbxax 0 nmxlkx
0 lkx or 0 nmx k
lx or
m
nx
222 2 bababa ,
222 2 bababa (Not just 222 baba !!!)
bababa 22
2233 babababa
2233 babababa
Special cases: 02 bxax 0 baxx 0 x or 0 bax (Not cancelling x !) 42 x 2 x (Not just 2x , Not 4x !!!)
For problems, Let x be … set equation The answer is … (with units)
Given 02 cbxax Discriminant ( acb 42 ): 042 acb 2 real roots 042 acb 1 repeated real root 042 acb 0 real roots (2 complex roots)
Given and are the roots ( x - intercepts), we have 0 xx Given 02 cbxax , if and are the roots, we have
a
b (sum of roots),
a
c (product of roots)
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [2]
Special cases:
11
111
222 22222
222
42 2222
If SUM and PRODUCT , then the equation is
0)()(2 PRODUCTxSUMx
Revision – Complex numbers
i411616 (Putting 1i for simplicity)
ii , 12 i , ii 3 , 14 i , ii 5 , … (so 14 ni , ii n 14 , 124 ni , ii n 34 )
Note: 42 x 142 x
222 2 ix
ix 2 (Not ix 4 , Not just ix 2 )
idicbia dicbia
idbca
dicbia 2bdibciadiac
ibcadbdac
bdbciadiac
1
dic
bia
dic
dic
dic
bia
22
222
2
dc
ibcaddbacidcdicdic
bdibciadiac
dicdic
dicbia
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [3]
Revision – Functions and graphs
Domain: x
xf1
, then 0x 1 xxf , then 1x
Function: cbxaxxf 2 Then, cbaf 222 2
Graph of cbxaxy 2 : yc intercept (Put 0x then cy ) Roots x intercept (Put 0y , find x , FMLA)
0a , 0a
acb 42 : 0 2 roots (2 x intercepts); 0 1 root (1 x intercept, just touch);
0 0 roots (0 x intercepts, not touch)
Vertex
a
acb
a
b
4
4,
2
2
Axis of symmetry: a
bx
2
Minimum / Maximum value:
a
acb
4
42
For b, vertex on right ba, different sign vertex on left ba, same sign
Graph of khxay 2 : Vertex kh, (Not h !!!) Axis of symmetry: hx (Not just h !!!) Minimum / Maximum value: k
Completing square: 5122 2 xxy
562 2 xxy Note: Take 3b
59962 2 xxy Note: 22 2/2/ bb
518962 2 xxy Note: Only 3 terms, last term out!
2332 2 xy Note: Just 2/b
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [4]
Revision – Straight lines
Horizontal line: ky
Vertical line: hx
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 ,
Slope of a line: 12
12
xx
yym
( tan )
Two-point form: 12
12
1
1
xx
yy
xx
yy
Point-slope form: mxx
yy
1
1
Intercept-slope form: cmxy (Intercept form: 1interceptintercept
y
y
x
x)
General form: 0 CByAx x-intercept = A
C (Put y = 0)
y-intercept = B
C (Put x = 0)
B
Cx
B
Ay Slope =
B
A
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, see below)
Intersection of 2 lines
)2(0
)1(0
rqypx
cbyax
Sub (1) into (2) Get both x and y for the coordinates!
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [5]
Revision – Polynomials
For polynomial dcxbxaxxf 23 , Degree = 3, Coefficient of 2x is b , Constant Term is d
For xf is divided by kx , Remainder is kf .
For xf is divided by nmx , Remainder is
m
nf .
0kf kx is a factor of 0xf .
0
m
nf nmx is a factor of 0xf .
Long Division: Rational functions:
Dividend xf ( Divisor ) ( Quotient ) + ( Reminder )
2147820232018142 223 xxxxxx
H.C.F. and L.C.M.:
)2)(2)(1(3)(
)2)(1(2)( 2
xxxxg
xxxf
∴
)2)(1(H.C.F. xx
)2()2)(1(6
)2()2()1(32L.C.M.2
2
xxx
xxx
78 202
214
23478
20 87
6020
20 1820
6 2
20 181423
2
2
2
23
23
xx
x
x
xx
xx
xx
xxxx
)5)(5(
25
)5)(5(
102153
)5)(5(
)5(2
)5)(5(
)5(35
2
5
3
)5(3
6
)5(
3
153
6
5
32
xx
x
xx
xx
xx
x
xx
xxx
xxx
x
xxx
x
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [6]
Revision – Index and logarithms
nmnm aaa , nmn
mnm a
a
aaa
m
m
aa
1 ,
mm
aa
1
10 a
mmm abba , m
m
mmm
b
a
b
aba
, mnnm aa
aa 2
1
, 33
1
aa , nn aa 1
, n mn
m
aa
abba , b
a
b
a ,
aaa , baba 2 ba no calculation! (common factor)
110log , 1log aa , 01log a , 0log undefined!
NMMN aaa logloglog
NMN
Maaa logloglog
MkM a
ka loglog NMa log no calculation!
a
MMa log
loglog ,
xyya a
x log,
yxaa yx yxyx loglog
axkykay x logloglog (By taking log on both sides)
Graphs:
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [7]
Revision – More about Equations
Special cases: 02 bxax 0 baxx 0 x or 0 bax (Not cancelling x !) For 4x or 6x , let 2xu 42 xu ; 3xu 62 xu For x2 or x3 , let xu 2 xxu 4222 ; xu 3 xxu 9322
For xx
1
1
1
, same denominator add ( ) expand carefully break ( )
For 12 x , put on left only then square. MUST CHECK answers
For log, combine log then cancel. MUST CHECK answers
For problem, Let x be … set equation (t
ds or
s
dt ) The answer is …
Relationship between a circle and a straight line:
)2(
)1(0 22
cmxy
FEyDxyx
Sub (2) into (1) Quadratic equation in x 0 2 intersections 0 1 intersection (line is tangent to circle) 0 0 intersections
Revision – inequalities ba cbca Any number a 02 a ba bcac (if 0c ) ba bcac (if 0c , change sign!)
ba c
b
c
a (if 0c ) ba
c
b
c
a (if 0c , change sign!)
ba ba
11 (reciprocal, change sign!)
Revision – Rate and Ratio
xzxywycbazycb
xwba::::
::
::
Direct varies: kxy , where k is a non-zero constant
Inversely varies: x
ky , where k is a non-zero constant
Jointly varies: kxyz , where k is a non-zero constant
Partly varies: xkkz 21 , where 1k and 2k are non-zero constants
ykxkz 21 , where 1k and 2k are non-zero constants Check unit!
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 150609
Last Minutes Review (Term 2 Examination) [8]
Revision – Estimation & Percentage x is increased by r%, new value %1 rx ; x is decreased by r%, new value %1 rx
x is increased to y, % increase %100
x
xy; x is decreased to y, % increase %100
x
yx
% change %100
old
oldnew
% profit %100Cost
profit; Selling price profitCost %1
% loss %100Cost
loss; Selling price lossCost %1
% discount %100_
priceMarked
Discount; Selling price discountpriceMarked %1_
Amount for Simple Interest nrPP %
Amount for Compound Interest nrP %1 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
Identities: 1cossin 22 ,
cos
sintan
ff
/
360
180o
o ,
coff
/270
90o
o
tan
1tan,
sincos,
cossin,*
coff
coff
coff
Revision – Graph of the functions 0 90 180 270 360 sin 0 1 0 – 1 0 cos 1 0 – 1 0 1 tan 0 undef. 0 undef. 0
Properties: 1sin1 , 1cos1 tan has no max/min
sin and cos are periodic of 360 tan are periodic of 180
a c
b
DSE MATHEMATICS TAK SUN SECONDARY SCHOOL 160606
Last Minutes Review (Extra) [1]
Mathematics Last Minutes Review (Extra)
Revision – Estimation & Percentage Absolute error = difference between Actual and Measured values
Maximum error = Half of a “unit”
Upper limit = Measured value + Maximum error
Lower limit = Measured value – Maximum error
Relative error = valuemeasured
errormaximum
_
_ or
valueactual
errorabsolute
_
_
% error = Relative error 100% Revision – Areas and Volumes Circle Area 2r
Circumference r2
Sector Area o
2
360
r
Arc length o360
2 r