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Mathematics (Junior Form Syllabus)
1
Algebra
A.
Monomials & Polynomials
1. Monomials
- Expression of no./product of a no. & >= 1 variables
E.g. 2, xy, 5ab
- Degree of a monomial = sum of exponents of all variables
E.g. Degree of 2xy2 is 3
2. Polynomials
- Expression contains a group of monomials
E.g. 4y + 5x2, 3y3 + 2y2 + 1
- Degree of polynomial = highest degree of all monomials
E.g. Degree of 6x5 – 4x2 + 3 is 5
3. Long Division For Polynomial Divisors
E.g. (2x3 + 4x2 – 7x + 5) /( x + 3)
2x2 – 2x – 1______
x + 3)2x3 + 4x2 – 7x + 5
-]2x3 + 6x2__________
- 2x2 – 7x
-]
- 2x2 - 6x____
- x + 5
-]_ _______- x – 3
8
B.
Simultaneous Linear Equations
1. Solving Methods (Text)
- Elimination
- Substitution
- Equivalent values of 1 unknown
2. Statements for answers in text of graph
- LHS = RHS & overlapped: Infinite many solutions (same equation)
- RHS = 0 & //: No solution (same slope)
C.
Identities
1. Algebraic Identities
- a2 – b2 = (a+b)(a-b)
- (a±b)2 = a2 ± 2ab + b2
2. Extended Problems: Add a virtual object (e.g. add terms or constants but change nothing, multiply and divide a
term by same object).
D.
Indices
1. Law and meanings
- ±an = positive if even n, remains same sign if odd n
- Law of indices
a0 = 1
a-n = 1/an
(am)n = amn
am/n = n√am
(a/b)n = an/bn
am/an = am-n
2. Scientific Notations: x = p(10n) [1 =< p < 10; n is integer]
aman = am+n
(ab)n = anbn
E.
Quadratic Polynomials & Equations
1. Cross method factorization: Satisfy both the product of constants & the sum of coefficients
2. Solving: Change to general form, factorize, then move objects to divide 0. Solve remaining linear equation.
F.
Inequalities
1. Graphical representation: solid dot = included; empty = excluded
2. Reversing signs: exchange whole content with another side, moves negative coefficient of variable and
reciprocal of variable.
3. And: Take the overlapped part (common)
If 2 solutions points at opposite edges, no solution
4. Or: Take all parts (both solution)
If 2 solutions overlapped, x can be any real numbers
Mathematics (Junior Form Syllabus)
2
Arithmetic
A. Percentages
1. Percentage Change =
2.
3.
4.
5.
|Original–New| x 100%
Original
New Value = Original x (1 ± %change)
Errors
- Max Abs Err = Accuracy / 2
- Relative Err = (Max Abs Err) / Given Value
- %error = (Relative Err) x 100%
Taxation
- Property Tax = (Annual rent - allowance) x %Given
- Profits Tax = (Gross profits – Operating Expenses) x %Given
- Salaries Tax: (Annual Income – Allowance) -> Net taxable income table
Interests
- Simple Interest = (Principal x %rate x time)
- Amount = Principal + Interest
- Growth or Depreciation = Original x (1 ± constant %rate) Periods
- Compound Interest = Growth of principal – Principal
B. Mensuration
1. Units reminder: 10cm2 = 10000mm2
| 10cm or 100mm x 10cm or 100mm
2. Length
- Circumference = 2r
- Arc = (Circumference)(Proportion)
3. Area
- Circle = r2
- Sector = (Circle Area)(Proportion)
- Curved surface of cylinder = (Circumference)(h)
- Curved surface of cone = rl
| l stands for lateral height
- Curved surface of sphere = 4r2
- Triangle = (base)(h)/2
- Rectangle or //gram = Product of Dimensions or (side)(h)
- Trapezium = (sum of // sides)(h)/2
4. Volume
- Cuboids or Cylinder or Prism = (Cross section area)(height)
- Cone or Pyramid = (base area)(h)/3
- Sphere = 4r3/3
5. Similar figures: (L1 / L2)3 = V1 / V2
or
(L1 / L2)2 = A1 / A2
C. Proportion
1. Direct proportion: both value increase/decrease simultaneously
E.g. when a α b, 3a α 3b or (3+a) α (3+b)
2. Inverse proportion: one value increase while the other decrease
E.g. when (value) = ab, (value) = a/3 x 3b
Mathematics (Junior Form Syllabus)
3
Trigonometry
A. Trigonometric Ratios:
cosθ = (adj. side)/hypotenuse
sinθ = (opp. side)/hypotenuse
tanθ = (opp. side)/(adj. side)
B. Trigonometric Relations
1. Trigonometric Identities
Shortcut Formula 1
Shortcut Formula 2
Original
sinθ = tanθ x cosθ
cosθ = sinθ / tanθ
tanθ = sinθ / cosθ
sin2θ + cos2θ = 1
sin2θ = 1 – cos2θ
cosθ = 1 – sin2θ
o
o
tanθ = 1 / tan(90 -θ)
cosθ = sin(90 -θ)
sinθ = cos(90o-θ)
o
o
1/tanθ = tan(90 -θ)
cosθ / sin θ = tan(90 -θ)
1/cosθ = tanθ / sinθ
-tanθ = tan(θ-90o)*
-cosθ = sin(θ-90o)*
-sinθ = cos(θ-90o)*
* For all (90o-θ) and (θ-90o) formula, only applicable for 0o < θ < 90o (Junior forms syllabus).
2.
Special Angles
0o
0
sinθ
1
cosθ
0
tanθ
N.B. Alternate method:
30o
45o
60o
90o
*
1/2
√2/2 or 1/√2
√3/2
1
√3/2
√2/2* or 1/√2
1/2
0
1/√3
1
√3
∞
Draw a half equilateral triangle
(30o-90o-60o)
facing sides
1, 2, √3
Draw a right triangle
(45o-45o-90o)
facing sides
1, 1, √2
* To get this for particular cases, both numerator and denominator times √2(Rationalization of denominator)
C. Gradient, Bearing, Evaluation and Depression Angles
1. Gradient = rise/run or rise in run
2. Bearing
- Compass bearing: count from N and S
- True bearing: 0xxo or xxxo must be 3 digit
3. Angle of evaluation/depression
- a is the angle of elevation of C (from A)
- b is the angle of depression of B (from A)
- c is the angle of elevation of A (from B)
C
A
B
Coordinate Geometry
A. Slope Conditions
1. L1 // L2 <-> ML1 = ML2
2. L1  L2 <-> ML1 x ML2 = -1
3. // x-axis:
M = 0, Eqn: y = (y-intercept)
4. // y-axis:
M = ∞, Eqn: x = (x-intercept)
5. Pass origin:
M = y-coordinate/x-coordinate
B. Formula (Location & Distance)
1. Distance = √[(y1 - y2)2 + (x1 – x2)2]
2. Intersection point: Solve the equation of 2 lines by simultaneous equation.
3. Section formula: pt = [ sx1 + rx2 sy1 + ry2 ]
| s, r are ratios
r+s
,
r+s
4. Mid-pt. formula: pt = (x1 + x2)/2 , (y1 + y2)/2
5. Centroid formula: pt = (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3
| x, y = median
C. Formula (Slope & Intercepts)
1. 2 pt. formula:
y - y1
y2 – y1*
=
x - x1
x2 – x1
2. Slope intercept formula: y = mx + c
3. Intercept formula:
x/a + x/b = 1
4. General formula:
Ax + Bx + C = 0
-A/B = Slope
-C/B = y-intercept
-C/A = x-intercept
| * = the slope
| a, b = x, y-intercept
(Alternative method: turn to slope intercept formula)
(Alternative method: substitute (0,y))
(Alternative method: substitute (x,0))
D. If given points and formula, try substituting it into equation to get answer
Mathematics (Junior Form Syllabus)
4
Geometry
A. Triangles, lines and Polygons
Reference
Proof
Base  isos.
sides opp. equal s
 sum of  = 180o
ext.  of 
o
s at a pt. = 360
adj. s on st. line = 180o
vert. opp. s
sum of ext. s of polygon = 360o
 sum of polygon = (side – 2) x 180o / side [for int. ]
B. //grams and // lines
Reference
corr. s, // lines
alt. s, // lines
int. s // lines
opp. sides of //gram
opp. s of //gram
diags. of //gram
(Definition)
Square
Rectangle
Proof
corr. s equal
alt. s equal
int. s supp.
opp. sides equal
opp. s equal
diags. bisect each other
2 sides equal and //
Rhombus
C. Congruence:
SSS, SAS, ASA, RHS, (AAS)
D. Similarity
1. References:
- Equiangular
- 3 sides proportional
- 2 sides proportional include an 
2. Common Similar Triangles:
E. Geometrical Theorems
1. Mid-pt. thm.
- In: Line cuts 2 mid-pt.
- Out: Base lines //, Upper line = Base line/2
2. Intercept thm. (Alt.: cut into 2 triangles or 1 //gram + 1 triangle)
- In: All involved lines //
- Out: The ratio of opp. side is the same
3. Equal ratios, Converse of equal ratios
- In/Out:
Base and upper line //
- Out/In:
Equal ratio of sides
4.  bisector thm.
- In: Angle bisector, a point on bisector line  both side lines
- Out: The  construction lines have same length
5.  bisector thm.
- In:  bisector of a line, a pt. on the bisector line
- Out: The construction lines to 2 edges of base line have same length