revision summary of dse mathematics (compulsory part)

Revision Summary of DSE Mathematics (Compulsory Part)

Number and Algebra

Page 1

Polynomials

Like terms and unlike terms

e.g. 2x and 6x are like terms and 2x + 6x can be simplified as (2 + 6)x = 8x 4pq

2 and –pq2 are like terms and 4pq

2 – pq2 can be simplified as (4 – 1)pq

2 = 3pq2

6ab and 5ac are unlike terms and they cannot be simplified Factorization

1. 2 2 ( )( )a b a b a b− ≡ + −

2. 2 2 22 ( )a ab b a b+ + ≡ +

3. 2 2 22 ( )a ab b a b− + ≡ −

4. 3 3 2 2( )( )a b a b a ab b+ ≡ + − +

5. 3 3 2 2( )( )a b a b a ab b− ≡ − + +

Remainder and Factor Theorems

Remainder theorem When a polynomial ( )f x is divided by a linear polynomial ax b+ where 0a ≠ , the remainder

is ( )b

fa

−.

Factor theorem

If ( )f x is a polynomial and ( ) 0b

fa

−= , then the polynomial ax b+ is a factor of the

polynomial ( )f x .

Conversely, if a polynomial ax b+ is a factor of polynomial ( )f x , then ( ) 0b

fa

−= .

Let 3 2( ) 4 3 18f x x x x= + − − .

(a) Find (2)f .

3 2(2) (2) 4(2) 3(2) 18 0f = + − − =

(b) Factorize ( )f x .

Since (2) 0f = , 2x − is a factor of ( )f x .

3 2 2 24 3 18 ( 2)( 6 9) ( 2)( 3)x x x x x x x x+ − − = − + + = − +

Let ( ) ( 3)(2 1) 3f x x x x= − + + + . Find the remainder when ( )f x is divided by 3x + .

The required remainder

[ ][ ]( 3) ( 3) 3 2( 3) 1 ( 3) 3 6( 5) 30f= − = − − − + + − + = − − =

Number and Algebra

Page 2

Solving Quadratic Equations in One Unknown

Factor method

e.g. Solve x2 + 2x – 3 = 0.

1 or 3

0 1or 03

0)1)(3(

0322

=−=

=−=+

=−+

=−+

x x

xx

xx

xx

Completing the square

Convert it to the form of 2 2( 2 )a x px p q+ + = , 2( )a x p q+ = , q

x pa

= − ±

Quadratic formula

e.g. Solve x2 – 7x + 9 = 0. Using the quadratic formula,

2 2( 7) ( 7) 4(1)(9) 4 Substitute 1, 7 and 9 into .

2(1) 2

7 13 7 13 7 13 i.e. or

2 2 2

b b acx a b c x

a

− − ± − − − ± −= = = − = =

± + −=

<

<

Nature of Roots of a Quadratic Equation

Consider the quadratic equation ,02 =++ cbxax where .0≠a

Discriminant

)4( 2 acb −====∆∆∆∆ 0>∆ 0=∆ 0<∆

Nature of roots 2 distinct real roots 1 double real root no real roots

Sum and Product of Roots of a Quadratic Equation

If α and β are the roots of the quadratic equation ,02 =++ cbxax where ,0≠a then

sum of roots

a

b−=+= βα , product of roots

a

c== αβ

Solving Equations Reducible to Quadratic Equations

Equations of Higher Degree

e.g. Solve 076 24 =−− xx .

By substituting ux =2 into the equation 076 24 =−− xx , we have

1or7

0)1)(7(

0762

−==

=+−

=−−

uu

uu

uu

Since ux =2 , we have

7

(rejected)1or7 22

±=

−==

x

xx

x +3 x −1

� +3x −x = +2x

◄ For any real number x, x2 ≥ 0.

◄ x4 = (x2)2 = u2

Number and Algebra

Page 3

Fractional Equations Equations with Square Root Signs

e.g. Solve 01

2=−

−x

x. e.g. Solve 02 =+ xx .

2or1

0)2)(1(

02

02

0)1(2

01

2

2

2

=−=

=−+

=−−

=+−

=−−

=−−

xx

xx

xx

xx

xx

xx

(rejected)4

1or0

0)14(

04

4

)2()(

2

02

2

2

22

==

=−

=−

=

−=

−=

=+

xx

xx

xx

xx

xx

xx

xx

Exponential Equations Logarithmic Equations

e.g. Solve 0222 =− xx . ……(1) e.g. Solve 0log)(log 3

2

3 =− xx . ……(2)

Substitute ux =2 into (1). Substitute ux =3log into (2)

1or0

0)1(

02

==

=−

=−

uu

uu

uu

1or0

0)1(

02

==

=−

=−

uu

uu

uu

Since ux =2 , we have Since ux =3log , we have

0

12or(rejected)02

=

==

x

xx

3or1

1logor0log 33

==

==

xx

xx

Complex Numbers

Complex number a bi+ , where a and b are real numbers and 2 1, . . 1i i e i= − = − .

a bi+ is purely real if and only if 0b = .

a bi+ is purely imaginary if and only if 0 and 0a b= ≠ . Addition Subtraction idbcadicbia )()()()( +++=+++ idbcadicbia )()()()( −+−=+−+

Multiplication Division

ibcadbdac

dibiacbiadicbia

)()(

))(())(())((

++−=

+++=++

2 2 2 2

a bi a bi c di ac bd bc adi

c di c di c di c d c d•

+ + − + −= = +

+ + − + +

Equality

a bi c di+ = + if and only if and a c b d= = .

◄ Multiply both sides by x – 1. ◄ Square both sides.

Check the answers since

squared both sides.

◄222 )2(2 uxx ==

Number and Algebra

Page 4

Graphs of Quadratic Functions

Features of the graphs of quadratic functions

Consider a quadratic function y = ax

2 + bx + c.

a > 0 a < 0

The graph opens upwards.

The graph opens downwards.

Finding the Optimum Values of Quadratic Functions

(a) To complete the square for x2 ± kx, add

2

2

k. Then

22

2

2

2

2222

2

±=

+

±=

+±k

xk

xk

xk

kxx

(b) For a quadratic function y = a(x – h)2 + k, (i) if a > 0, then the minimum value of y is k when x = h, (ii) if a < 0, then the maximum value of y is k when x = h. Laws of Indices

For integral indices p and q, let a, b be real numbers. For rational indices p and q, let a, b be non–negative numbers.

1. p q p qa a a

+× =

2. , where 0p

p q

q

aa a

a

−= ≠

3. ( )p q pqa a=

4. ( ) p p pab a b=

5. , where 0

p p

p

a ab

b b

= ≠

6. 1

, where 0p

pa a

a

−= ≠

7. 0 1, where 0a a= ≠

8. ( )p

q qq p pa a a= = for integers p and q where a, q > 0.

Number and Algebra

Page 5

Laws of logarithms

If , , 0 and 1, then logy

ax a a x a y x= > ≠ = .

For , , , 0 and , 1M N a b a b> ≠ ,

1. log 1 0a =

2. log 1a a =

3. log ( ) log loga a aMN M N= +

4. log log loga a a

MM N

N= −

5. log ( ) log ( is any real number)k

a aM k M k=

6. loga Ma M=

7. ,log

loglog

a

NN

b

b

a = where b > 0 and 1≠b

Note: loga x is undefined for 0x ≤ .

Graphs of exponential functions and logarithmic functions

Percentages

new value initial value (1 + percenatge change)= ×

new value initial value

percentage change 100%initial value

−= ×

Selling problems

A book is marked at $78 for sale. A customer is offered a 10% discount. Find the selling price.

selling price = marked price × (1 – discount percent) $78 (1 10%) $70.2= × − =

Simple and compound interest

A P I= + simple interest %I P r n= × ×

compound interest (1 %)nI P r P= + −

where A: amount, P: principal, I: interest, r: interest rate per period, n: number of periods. Ratio

Given : 3 : 4a b = and : 6 : 7b c = . Find : :a b c .

Since . . . of 4 and 6 is 12L C M : : 9 :12 :14a b c =

: 3: 4

: 6 : 7

: : 9 :12 :14

a b

b c

a b c

=

=

=

Number and Algebra

Page 6

Variation

Direct variation

, where 0y x y kx k∝ = ≠ .

Inverse Variation

1

, or where 0k

y y xy k kx x

∝ = = ≠

Joint variation 1. z varies jointly as x and y

, where 0z xy z kxy k∝ = ≠ .

2. z varies directly as x and inversely as y

, where 0x kx

z z ky y

∝ = ≠ .

Partial variation 1. z partly varies as x and partly varies as y

1 2 1 2where , 0z k x k y k k= + ≠ .

2. z partly constant and partly varies as y

1 2 2where 0z k k y k= + ≠ .

3. z partly varies as x and partly varies inversely as y

21 1 2where , 0

kz k x k k

y= + ≠ .

Errors

absolute error = measured value true value−

Note: maximum absolute error = largest possible error, in this case the true value is not known.

relative error absolute error

true value= or

maximum absolute error

measured value

percentage error relative error 100%= × Arithmetic sequence

, , 2 , 3 ,a a d a d a d+ + + L . Common difference d= . First term (1)T a=

nth term ( ) ( 1)T n a n d= + −

Let the sum of the first n terms be ( ) (1) (2) (3) ( )S n T T T T n= + + + +L , then

(1) ( )( )

2

T T nS n n

+ =

or [ ]2 ( 1)2

na n d+ − .

If , ,x y z form an arithmetic sequence, then 2

x zy

+= .

Geometric sequence

2 3, , , ,a ar ar ar L . Common ratio r= . First term (1)T a=

nth term 1( ) nT n ar −=

Let the sum of the first n terms be ( ) (1) (2) (3) ( )S n T T T T n= + + + +L , then

( 1)( ) , 1

1

na r

S n rr

−= ≠

−

Sum to infinity , 1 11

ar

r= − < <

−

If , ,x y z form a geometric sequence, then 2 , . .y xz i e y xz= = ± .

Number and Algebra

Page 7

The nth term of an arithmetic sequence is 2

n−. Find the first term and the common ratio.

1 2 1 1

(1) , (2) 1, 12 2 2 2

T T d− − − −

= = = − = − − = .

Express 0.16& as a fraction.

0.16 0.1 0.06 0.006 0.0006

0.06 1 6 10.1

1 0.1 10 90 6

= + + + +

= + = + =−

& L

Simultaneous equations in two unknowns

2

0Ax By C

y ax bx c

+ + =

= + +

By substitution

2( ) 0Ax B ax bx c C+ + + + =

2 ( ) ( ) 0Bax A Bb x Bc C+ + + + =

Since , , , , ,A B C a b c are known, x can be find by quadratic formula.

Having found x, ( )Ax C

yB

− += .

By graphical method

The coordinates of intersecting points 1 1( , )x y and 2 2( , )x y represents the solutions.

There may be 0, 1, or 2 intersecting points, depending on the number of solutions. Inequalities

Rule 1 If a b< and b c< , then a c< . Rule 2 Addition to both sides does not change the inequalities sign. then a b a c b c< + < +

Rule 3 For c > 0, a < b then ac bc< .

For c < 0, a < b then ac bc> . Graphical representation

Number and Algebra

Page 8

Solving ( ) , ( ) , ( ) and ( )f x k f x k f x k f x k> < ≥ ≤ graphically:

Step 1 Draw the graph ( )y f x= and y k= .

Intersecting points should be labeled.

Solve 2 3 2 6x x+ + >

Step 2 Identify of x which ( )f x k> .

4 or 1x x< − > .

Note: ‘>’ is different from ‘≥’. If the equation is changed to

‘ ( )f x k≥ ’, then the solution is 4 or 1x x≤ − ≥ .

Solving a Linear Inequality in Two Unknowns Graphically

Linear programming (a) Solve graphically the system of inequalities:

2 2

3

0

x y

x y

x

+ ≥

+ ≤ ≥

(b) Hence find the maximum and minimum value of ( , ) 2f x y x y= − subject to the above

constraints. Consider the intersecting points

(x,y) ( , ) 2f x y x y= −

(0,1) –2

(0,3) –6

(4, 1)− 6

Maximum value = 6 and minimum value = –6 Alternative method:

Draw a dotted line of 2 0x y− = .

To determine the extremes of 2x y− , move the dotted line vertically and find the two critical

positions such that the line is about to leave the shaded region. The value of x – 2y at the two extreme positions give the maximum and minimum.

Number and Algebra

Page 9

Different Numeral Systems

Numeral in the systems

Numeral system Numerals

Denary system 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Binary system 0 and 1

Hexadecimal system 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F

In a number, the position of each didgit has fixed place value. e.g. In 430510, the place value of 3 is 103. In C8A16, th epace value of 8 is 16. The value of a number can be expressed in the expanded form.

e.g. 3 2 1

104305 4 10 3 10 0 10 5 1= × + × + × + ×

4 3 2 1

211101 1 2 1 2 1 2 0 2 1 1= × + × + × + × + ×

2 1

168 12 16 8 16 10 1C A = × + × + ×

Converting a binary number or a hexadecimal number into a denary number

4 3 2 1

211101 1 2 1 2 1 2 0 2 1 1 29= × + × + × + × + × =

2 1

168 12 16 8 16 10 1 3210C A = × + × + × =

Converting a denary number into a binary number or a hexadecimal number.

10 229 11101= 10 163210 8C A=

2 29 remainder

2 14 1

2 7 0

2 3 1

2 1 1

0 1

↑

↑

↑

↑

↑

LL

LL

LL

LL

LL

16 3210 remainder

16 200 10 ( )

16 12 8

16 0 12 ( )

A

C

↑

↑

↑

LL

LL

LL

Transformations of functions

The following table summarizes the transformations of the graph of a function ( )y f x= and the

corresponding transformations of the function.

Transformation of the graph Transformation of the function

Translate upwards by k units ( )y f x k= +

Translate downwards by k units ( )y f x k= =

Translate leftwards by k units ( )y f x k= +

Translate rughtwards by k units ( )y f x k= −

Reflect about the x-axis ( )y f x= −

Reflect about the y-axis ( )y f x= −

Enlarge along the y-axis k times the original, where 1k > ( )y kf x=

Reduce along the y-axis k times the original, where 0 1k< <

Enlarge along the x-axis 1

k times the original, where 1k >

( )y f kx=

Reduce along the x-axis 1

k times the original, where 0 1k< <

� A, B, C, D, E and F represents 10, 11, 12, 13, 14 and 15

respectively.

Measures, Shape and Space

Page 10

Mensuration of common plane figures and solids

Plane figures Perimeter Area

Trapezium

a b c d+ + + ( )

2

a b h+

Rhombus

4a 2

xy

Circle

2 rπ 2rπ

Sector

o

22

360

rr

π θ+ 2

o360r

θπ ×

Solids Volume Surface Area

Cube

3a 26a

Cuboid

blh 2( )bh bl hl+ +

b

a

c d h

a

a a

a

x

y

a

r

a

l b

h


Page 11

Prism

Ah Base perimeter 2h A× +

Cylinder

2r hπ 22 2rh rπ π+

Pyramid

1

3Ah total area of lateral facesA +

Cone

21

3r hπ 2

rl rπ π+

Sphere

34

3rπ 24 rπ

Eulaer’s formula (for a convex polyhedron only)

2V E F− + = where V: number of vertices, E: number of edges, F: number of faces.

A h


Page 12

Similar plane figures and soilds

1 12 3

2 3

1 1 1 1 1 1 1

2 2 2 2 2 2 2

, ,l A l V l A V

l A l V l A V

= = ∴ = =

1 2: length of figure 1, : corresponding length of figure 2l l

1 2: face area of figure 1, : corresponding face area of figure 2A l

1 2: volume of figure 1, : volume length of figure 2V l

Congruent Triangles

Properties of congruent triangles:

If △ABC ≅ △XYZ, then

(i) ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z,

(ii) AB = XY, BC = YZ, CA = ZX.

Conditions for congruent triangles:

(i) SSS (ii) SAS

(iii) AAS (iv) ASA

(v) RHS

Similar Triangles

Properties of similar triangles:

If △ABC ~ △XYZ, then

(i) ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z,

(ii) ZX

CA

YZ

BC

XY

AB== .

Conditions for similar triangles:

(i) AAA (ii) 3 sides prop.

(iii) ratio of 2 sides, inc. ∠

ZX

CA

YZ

BC

XY

AB==


Page 13

Transformation and Symmetry in 2-D figures Reflectional symmetry and rotational symmetry

order of rotational symmetry = 3 3-fold rotational symmetry Transformation of 2-D figures

Translation Reflection

Rotation Enlargement

Geometry

Abbreviation Meaning Abbreviation Meaning

adj. ∠s on st. line

a + 47o = 180o

∠s at a pt.

b + 75o + 135o = 360o

vert. opp. ∠s

c = 50o

corr. ∠s, AB // CD

a = 40o

alt. ∠s, AB // CD

b = 50o

int. ∠s, AB // CD

c + 150o = 180o

Centre of rotation

axis of relection


Page 14

∠ sum of △

°=++ 180cba

ext. ∠ of △

bac +=

base ∠s isos. ∆

If AB = AC, then CB ∠=∠

sides opp. eq. ∠s

If ∠A = ∠C. then

AB BC= .

prop. of isos.△

If AB = AC and one of the following conditions (i) BD = DC

(ii) CADBAD ∠=∠

(iii) BCAD ⊥

prop. of equil.△

If AB = BC = AC, then °=∠=∠=∠ 60CBA

Pyth. theorem

In △ABC, if

°=∠ 90C , then 222

cba =+

Converse of Pyth. thm.

In △ABC, if 222

cba =+ , then °=∠ 90C

Mid-pt. Thm.

If AM = MB and AN = NC, then MN // BC

and 1

2MN BC=

Intercept Thm.

If AB// CD // EF, then

: :BD DF AC CE= .

∠ sum of polygon

Sum of all interior angles of n-sided

polygon = o( 2) 180n − ×

sum of ext. ∠s of polygon

Sum of all exterior angles of n-sided

convex polygon = 360o

prop. of trapezium

o180P R∠ + ∠ = o180Q S∠ + ∠ =

opp. sides equal

AD = BC and AB = DC


Page 15

opp. ∠s of //gram

∠A = ∠C and

∠B = ∠D

diag. of //gram

AO = CO and BO = DO

prop. of rhombus

prop. of //gram diagonals bisect interior angles

diagonals are ⊥

prop. of rectangle

prop. of // gram equal diagonals

prop. of square

prop. of rhombus prop of rectangle

line from centre ⊥ chord bisects chord

If ABON ⊥ , then AN = BN.

line joining centre to mid-pt. of chord ⊥ chord

If AN = BN, then

ABON ⊥

equal chords, equidistant from centre

If ABOM ⊥ ,

CDON ⊥ and AB = CD, then OM = ON.

chords equidistant from centre are equal

If ABOM ⊥ ,

CDON ⊥ and OM = ON, then AB = CD.

∠ at centre twice ∠ at ☼

ce

q = 2p

∠ in semi-circle

If AB is a diameter,

then ∠APB = 90º.

converse of ∠ in semi-circle

If ∠APB = 90º, then AB is a diameter.

∠s in the same segment

The angles in the same segment of a

circle are equal. i.e. x = y

equal ∠s, equal arcs

If ∠AOB = ∠COD,

then )

AB =)

CD


Page 16

equal ∠s, equal chords

If ∠AOB = ∠COD, then AB = CD

equal arcs, equal ∠s

If )

AB =)

CD , then

∠AOB = ∠COD

equal arcs, equal chords

If )

AB =)

CD , then AB = CD

equal chords, equal ∠s

If AB = CD, then

∠AOB = ∠COD

equal chords, equal arcs

If AB = CD, then )

AB =)

CD

arcs prop. to ∠s at centre

)AB :

)CD = x : y

arcs prop. to ∠s at ☼ce

)AB :

)CD = m : n

opp. ∠s, cyclic quad.

∠A + ∠C = 180º and

∠B + ∠D = 180º

ext.. ∠, cyclic quad

∠DCE = ∠A

Converse of ∠s in the same segment

If p = q, then A, B, C and D are concyclic.

opp. ∠s supp.

If a + c = 180o (or o180b d+ = ), then A,

B, C and D are concyclic.

ext. ∠ = int. opp. ∠

If p = q, then A, B, C and D are concyclic.

tangent⊥radius

If PQ is the tangent to the circle at T, then

PQ⊥OT.

converse of tangent⊥

radius

If PQ⊥OT, then PQ is

the tangent to the circle at T.


Page 17

tangent properties

If two tangents, TP and TQ, are drawn to a circle from an external point T and touch the circle at P and Q respectively, then (i) TP = TQ,

(ii) ∠POT = ∠QOT,

(iii) ∠PTO = ∠QTO.

∠ in alt. segment

A tangent-chord angle of a circle is equal to

an angle in the alternate segment.

∠ATQ = ∠ABT

converse of ∠ in alt. segment

If ∠ATQ = ∠ABT, then

PQ is the tangent to the circle at T.

Special lines in a triangle

AD is the median of BC in △ABC. CD is the angle bisector of ∠ACB in △ABC

.

CD is the altitude of AB in △ABC. MN is the perpendicular bisector of AC in △ABC.

Special points of a triangle Intersecting point of 3 medians Intersecting point of 3 angle bisectors – Centroid – Incentre


Page 18

Intersecting point of 3 altitudes Intersecting point of 3 ⊥ bisectors – Orthocentre – Circumcentre

1. These 4 points exist for any triangle. But they may or may not coincide. 2. There exists exactly one circle with in–centre as centre such that it touches all the 3 sides of

the triangle. (i.e. the 3 sides are tangents.) 3. There exists exactly one circle with circumference as centre such that it passes through all the

3 vertices of the triangle. Coordinates Geometry

Let the points A, B be 1 1( , )x y and 2 2( , )x y respectively.

Distance formul, 2 2

1 2 1 2( ) ( )AB x x y y= − + −

mid–point of AB 1 2 1 2,2 2

x x y y+ + =

If : :AP PB m n= , then 1 2 1 2,nx mx ny my

Pm n m n

+ + = + +

Straight lines

Slope of AB 2 1

2 1

y y

x x

−=

−

Let 1 2,m m be slopes of lines L1, L2 respectively and L1, L2 are not vertical lines.

Then 1 2 1 2// ,L L m m= , 1 2 1 2, 1L L m m⊥ = − .

Equation of straight lines

Point slope form

1 1( )y y m x x− = −

General form

0Ax By C+ + =

Slope A

B

−= if 0B ≠ (if B = 0, is undefined)

x–intercept C

A

−= if 0A ≠ (if A = 0, the line does not cut the x–axis)

y–intercept C

B

−= if 0B ≠ (if B = 0, the linen does not cut the y–axis)


Page 19

Equation of a circle

Let (h,k) be the centre and r be the length of the radius. Centre–radius form

2 2 2( ) ( )x h y k r− + − =

General form

2 2 0x y Dx Ey F+ + + + =

Centre ,2 2

D E− − =

Radius

2 2

2 2

D EF

+ −

Polar coordinates

Polar coordinates of P = ( , )r θ , r OP= , tan slope of OPθ =

Trigonometric ratios of an acute angle θθθθ

hypotenuse

side oppositesin =θ

hypotenuse

sideadjacent cos =θ

sideadjacent

side oppositetan =θ

Trigonometric ratios any angle θθθθ Sign of trigomometric ratios

Trigonometric identities

1. sin

tancos

θθ

θ= 2. 2 2sin cos 1θ θ+ =

3. osin(90 ) cosθ θ− = 4. ocos(90 ) sinθ θ− =

5. o 1tan(90 )

tanθ

θ− = 6. osin(90 ) cosθ θ+ =

7. ocos(90 ) sinθ θ+ = − 8. o 1tan(90 )

tanθ

θ−

+ =


Page 20

9. sin(180 ) sinθ θ° − = 10. cos(180 ) cosθ θ° − = −

11. tan(180 ) tanθ θ° − = − 12. sin(180 ) sinθ θ° + = −

13. cos(180 ) cosθ θ° + = −

14. tan(180 ) tanθ θ° + =

15. sin( ) sin(360 ) sinθ θ θ− = ° − = −

16. cos( ) cos(360 ) cosθ θ θ− = ° − =

17. tan( ) tan(360 ) tanθ θ θ− = ° − = − 18. sin(360 ) sinθ θ° + =

19. cos(360 ) cosθ θ° + = 20. tan(360 ) tanθ θ° + =

Graphs of trigonometric functions Graphs of y = sinx

Graphs of y = cosx

Graphs of y = tanx


Page 21

Application of Trigonometry

Sine Formula

sin sin sin

a b c

A B C= =

Cosine Formula

2 2 2 2 cosc a b ab C= + − Area of a triangle

Area 1

sin2

ab C=

Heron’s Formula

Area ( )( )( )s s a s b s c= − − − where 2

a b cs

+ += .

Angle between a line and a plane A. Projection on a plane (a) Projection of a point on a plane (b) Projection of a line on a plane

If PQ is perpendicular to any straight line If AB intersects π at A, and C is the

(L1 and L2) on π passing through Q, then projection of B on π, then

(i) PQ is perpendicular to π, (i) BC is perpendicular to π,

(ii) Q is the projection of P on π, (ii) AC is projection of AB on π,

(iii) PQ is the shortest distance between (iii) BC is the distance between B and π.

P and π. B. Angles between lines and planes

(a) Angle θ between two intersecting straight (b) Angle θ between a straight line and a lines. plane.

θ is the acute angle formed by the two θ is the acute angle between the straight lines. It is called the angle of intersection. line and its projection on the plane.

(c) Angle θ between two intersecting planes

(i) AB is the line of intersection of π1 and π2.

(ii) θ is the acute angle between two interescting straight lines on the planes, and which are perpendicular to AB.

C. Line of greatest slope Lines of greatest slope

(a) AB is the line of intersection of the planes

ABCD and ABEF. (b) PX, L1, L2 and L3 are perpendicular to AB. They are lines of greatest slope of the inclined plane ABEF. Angle of elevation and angle of depression

Data Handling

Page 22

Simple Statistical Diagrams and Graphs

Pie chart Stem-and-leaf diagram

e.g.

e.g.

The data presented above are: Angle of the sector representing the 15 kg, 16 kg, 21 kg, 21 kg, 27 kg, colour ‘Blue’ = °=×°=° 144%40360x . 33 kg, 34 kg, 36 kg, 42 kg, 51 kg

Histograms

e.g. The histogram on the right is constructed

from the frequency distribution table below.

Hourly wage

($)

Class boundaries

($)

Class mark

($)

Frequency

25 – 29 24.5 – 29.5 27 15

30 – 34 29.5 – 34.5 32 22

35 – 39 34.5 – 39.5 37 11

40 – 44 39.5 – 44.5 42 4

45 – 49 44.5 – 49.5 47 3

Frequency polygon

e.g. The following table shows the amount of time spent by 60 customers in a supermarket.

Time spent (min)

Class mark (min)

No. of customers

11 – 20 15.5 9

21 – 30 25.5 21

31 – 40 35.5 16

41 – 50 45.5 8

51 – 60 55.5 4

61 – 70 65.5 2

Weights of 10 dogs Stem (10 kg) leaf (1 kg) 1 5 6 2 1 1 7 3 3 4 6 4 2 5 1

Data Handling

Page 23

Cumulative frequency polygon The following table shows the age distribution of 50 teachers in a school

Age 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

Frequency 2 9 18 11 6 4

The cumulative frequency table of the above data is constructed below

Age less than 19.5 24.5 29.5 34.5 39.5 44.5 49.5

Cumulative Frequency

0 2 11 29 40 46 50

The cumulative frequency polygon of the above data is constructed below

Measures of central tendency

Mean

Ungrouped mean 1 2 3 nx x x xx

n

+ + + +=

L

Grouped mean 1 1 2 2 3 3

1 2 3

n n

n

f x f x f x f xx

f f f f

+ + + +=

+ + + +L

L

Weighted mean 1 1 2 2 3 3

1 2 3

n n

n

w x w x w x w xx

w w w w

+ + + +=

+ + + +L

L

Median For ungrouped data, if number of data, n, is: (i) odd, median = middle datum (ii) even, median = mean of the 2 middle data

For data grouped into classes, the median can be determined from its cumulative frequency polygon/curve. e.g. The cumulative frequency polygon on the right

shows the donations by a group of students to a charity fund. From the graph, the number of students is 100. From the graph, the median donation by the group of students is $52.

Mode The data item with the highest frequency. Model class The class with the highest frequency

Data Handling

Page 24

Measures of Dispersion

Range

Ungrouped data: range = largest datum – smallest datum Grouped data: range = highest class boundary – lowest class boundary. Inter-quartile range

Ungrouped data:

inter-quartile range = upper quartile (Q3) − lower quartile (Q1)

Grouped data:

The quartiles can be read from the corresponding cumulative frequency polygon (or curve).

inter-quartile range = upper quartile (Q3) − lower quartile (Q1). Box-and-whisker diagram The box-and-whisker diagram is an effective way to present the lower quartile, the upper quartile, the median, the maximum and the minimum values about a data set

Standard deviation

2 2 2 2

1 2 3( ) ( ) ( ) ( )nx x x x x x x x

nσ

− + − + − + + −=

L

For grouped data,

2 2 2 2

1 1 2 2 3 3

1 2 3

( ) ( ) ( ) ( )n n

n

f x x f x x f x x f x x

f f f fσ

− + − + − + + −=

+ + + +

L

L

Standard Score

For a set of data with mean x and standard deviation σ, the standard score z of a given datum x is

given by σ

xxz

−= .

Normal distribution

Characteristics of a normal curve:

1. It is bell-shaped.

2. It has reflectional symmetry about xx = .

3. The mean, median and mode are all equal to x .

Data Handling

Page 25

Probability

Theoretical probability

In the case of an activity where all the possible outcomes are equally likely, the probability of an event E, denoted by P(E), is given by

Theoretical probability outcomes possible ofnumber total

event the tofavourable outcomes ofnumber )(

EEP =

where 1)(0 ≤≤ EP .

Note that P(impossible event) = 0 and P(certain event) = 1 Experimental probability The experimental probability P(E) of an event E is defined as follows:

trialsofnumber total

experimentan in occurs event timesofnumber )(

EEP = or

EEP event offrequency relative)( =

For a large number of trials, the experimental probability is close to the theoretical probability which is deduced from theories. Permutation An arrangement of a set of objects selected from a group in a definite order is called a permutation. (i) The number of permutations of n distinct objects is n!.

e.g. Arrange 5 students in a row. Number of arrangements = 5! = 120

(ii) The number of permutations of n distinct objects taken r at a time, denoted by n

rP , is

)!(

!

rn

n

−.

e.g. Choose 5 out of 7 students, and arrange them in a row.

Number of arrangements 25207

5 == P

Combination A selection of r objects from n distinct objects regardless of their order is called a combination.

The number of combinations of n distinct objects taken r at a time, denoted by n

rC , is

!)!(

!

rrn

n

−.

e.g. Choose 2 from a list of 12 friends to invite to a party.

Number of ways of choosing 6612

2 == C

Additional law of probability

Mutually exclusive events – events that cannot occur at the same time.. If events A and B are mutually exclusive, then

( ) ( ) ( )P A B P A P B∪ = +

If events 1 2 3, , , , nE E E EL are mutually exclusive events, then

1 2 3 1 2 3( ) ( ) ( ) ( ) ( )n n

P E E E E P E P E P E P E∪ ∪ ∪ ∪ = + + + +L L

If events A and B are not mutually exclusive, then ( ) ( ) ( ) ( )P A B P A P B P A B∪ = + − ∩ where ( ) 0P A B∩ ≠

Multiplication Law of Probability If the probability that one event occurs is not affected by the occurrence of another event,

these two events are called independent events. Multiplication law of probability for independent events: If A and B are two independent events, then )()()( BPAPBAP ×=∩

If the probability that one event occurs is affected by the occurrence of another event, these two events are called dependent events.

Data Handling

Page 26

Multiplication law of probability for dependent events: If A and B are two dependent events, then ( ) ( ) ( | )P A B P A P B A∩ = × , where

( | )P B A is the probability that event B occurs given that event A has occurred.

Expected value (i.e expectation)

1 1 2 2 3 3( )n n

E X Px P x P x P x= + + + +L , where 1 2, ,P P L are probabilities of events 1 2, ,E E L and

1 2, ,x x L are the corresponding value of 1 2, ,E E L .

Complementary events If A and A’ are complementary events, then (1) A’ happens if A does not happen and B does not happen if A happens; (2) ( ') 0P A A∩ = (i.e. mutually exclusive);

(3) ( ') 1 ( )P A P A= −

revision summary of dse mathematics (compulsory part)

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