mathematics last minutes review - webpersonal.tsss.edu.hk/roh/f5/mathematics last minutes note (form...

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!


Last Minutes Review (F5 Chapters 1 and 2) [1]

Mathematics Last Minutes Review (Form 5 Chapters 1 and 2)

Revision – Circle properties



Revision – Tangent properties




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



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)




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



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:



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



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


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


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


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)



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



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



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!



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



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:



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!



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


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

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