View
232
Download
7
Category
Preview:
Citation preview
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
Measures, Shape and Space
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
Measures, Shape and Space
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==
Measures, Shape and Space
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
Measures, Shape and Space
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
Measures, Shape and Space
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
Measures, Shape and Space
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.
Measures, Shape and Space
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
Measures, Shape and Space
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)
Measures, Shape and Space
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−
+ =
Measures, Shape and Space
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
Measures, Shape and Space
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= −
Recommended