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Paradigm
Specialising in O
Level Mathem
atics
Ultimate
Exam Guide &
Cheat Sheet
Paradigm Math De
partment
EQUATIO
N^
& I
NEQUALITIES
Completing The Square
𝟐^ −
+^
=^
𝟐^ −
𝟓 𝒙 𝟐^
+^
=^ 𝟐
[(𝒙
𝟐^ −
𝟓 𝒙 𝟐^
+^
𝟐^ −
𝟐^ +
𝟗 ] 𝟐
=^ 𝟐
[(𝒙
−^
]
=^ 𝟐
−^
Quadratic Inequalities [Divide negative, flip sign if it’s
𝟐]
Question Type 1 : Quadratic Inequalities Find the range of values of
𝑥^
for which
𝑥^ −
)^ ≤
(^1
+^ 𝑥
Step 1: Flush everything to the left and rearrange according to^ 𝑎
+^ 𝑏𝑥
+^
Step 2: Simplify and rearrange according to
+^
Step 3: Solve your quadratic inequalities Question Type 2 : Reverse Quadratic Inequalities Find the values of 𝑝^ and 𝑞^ for which
𝑥^
<^ −
3 or
𝑥^ >
is the solution set of
+^
𝑞^ >
NATURE OF ROOTS
Question Type 1: Nature of Roots Find the range of 𝑎, for which the graph is Step 1: Equate Curve and Line if applicable*
,^ >,
Step 2: Flush everything to the left and rearrange according to^ 𝑎
+^ 𝑏𝑥
+^
Step 3: Determine Nature of Roots base on the condition Step 4: Simplify and rearrange according to
+^
Step 5: Solve your quadratic inequalities Step 6: Reject if Necessary Question Type 2 : Proving Questions (Prove/Show/Explain) Show that the expression is negative for all real values of
Prove that the expression is always positive for all values of
Step 1: Flush everything to the left and rearrange according to
+^
Step 2: Apply
−^
but don’t put the sign Step 3: Simplify and rearrange according to
+^
Step 4: Use Completing The Square/otherwise (See Condition) Step 5: For all real values of 𝑥, ____ is positive/negative (Proven) Example^ 𝑏
[ −
(𝑘^
+^2
2 )]
−^
4 (^3
)(^2
𝑘^ −
=^ 𝑘
+^
𝑘^ +
=^ 𝑘
𝑘^ +
=^ (
𝑘^ −
Since
−^
>^
0 ,^
−^4
𝑎𝑐^
>^0
and line intersects the curve for all real values of k.
SURDS
Mensuration with Surds Cone Surface Area:
Volume:
Pyramid Surface Area: Add all sides Volume: 1 Base area x Height 3 Hemi
[open] /
Volume:
Sphere Surface Area:
Volume:
Prism Surface Area: Add all sides Volume: Cross Section x Length Cylinder Surface Area:
+^2
Volume:
𝑏^ +
𝑑^ =
𝑏^ +
implies that
𝑎^ =
𝑥^ and
𝑐^ =
E.g.
+^
𝑏^ +
√^5
=^ −
−^12
√^5
=^
−^12
𝒂^ =
3 𝑎^
+^ 𝑏
=^
−^29
−^6
)^ +
𝑏^ =
−^29
𝒃^ =
(𝑥^
+^3
−^
4 )^
>^0
(𝑥^
+^3
−^
4 )^
<^
𝑥^ <
−^3
or^
𝑥^ >
−^3
<^
𝑥^ <
Multiplication : √𝑎^ ×^ √ 𝑏^ = √𝑎𝑏 𝑎^ × √𝑏 =^ 𝑎√𝑏 𝑐√𝑎 ×^ 𝑑√𝑏 =^ 𝑐𝑑√ 𝑎𝑏 𝑎 √ ×^ √ 𝑎^ = 𝑎 𝑏√𝑎 ×^ 𝑏√𝑎 =^ (^2) 𝑏𝑎 Number ×^ N umber, Surd ×^ Surd Division : 𝑎 ට 𝑏 √= (^) 𝑎 √𝑏 Addition & Subtraction : 2 √^3 +^5 √^3 =^7 √^3 4 √ 2 − √^2 =^ 3 √^2 Similar Terms can be Added or subtracted Key To Solving Surds: Simplify all Surds to their simplest forms √^50 =^ 5 √^2 √^27 =^ 3 √^3 Rationalisation of Surds : (We don’t like SqRoots in Denominator) Note: Change Signs while Rationalizing 3 ×^2 √
√^2 2 √
=^
15 √^3
×^
√^3 √^3
=^
√^315
√^3
2 ×
−^
√^3
√^3
=^
4 (^2
−^
√^3
=^
√^3
Step 1:
𝑥^ <
3 or
𝑥^ >
Step 2:
+^
𝑥^ −
>^
0 (Reverse and Form Back Original) Step 3:
+^
𝑥^ −
(Expand) Step 4:
−^
(Rearrange according to question) Step 5:
𝑞^ =
−^6
,^ 𝑝^ −
1 (Compare coefficient) Determinants (Curve & Axis)^2 𝑏 −^4 𝑎𝑐^ < 0 No Roots No Real Roots or Imaginary Roots or Graph is always positive (Completely above x - axis) or Graph is always negative (Completely below x - axis) (^2) 𝑏− 4 𝑎𝑐 =^0 1 Roots Real & Equal or Real & Repeated Roots (^2) 𝑏− 4 𝑎𝑐
^0 2 Roots
Real & Distinct Roots or Different Roots (^2) 𝑏− 4 𝑎𝑐 ≥^0 1/2 Roots Graph has real roots or Graph Intersects the x - axis
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Paradigm
Specialising in O
Level Mathem
atics
Ultimate
Exam Guide &
Cheat Sheet
Paradigm Math De
partment
POLY
NOMIALS
Question Type 1: Solving Unknowns Question Type 2: Remainder & Factor Theorem Question Type 3: Reverse Engineer to get Polynomial Equation Question Type 4: Cubic Factorisation^3 (𝑥
+^
=^
(𝑥^
+^ 𝑦
+^
−^
=^
(𝑥^
−^ 𝑦
+^
Question Type 5 : Solve Cubic Equations
Partial
Fractions
Question Type 5 (Hence) : Solve Cubic Equations Type 1: Nature of Roots After long division, you can use
−^
to determine the number of roots of the Quadratic Equation. Recap: 𝑏^^2
−^
=^
root 𝑏^^2
−^
>^
roots 𝑏^^2
−^
<^
root If asked for TOTAL roots, remember to add Initial Solutions. Type 2: Replacement Level 1:^3 𝑥
−^
=^
Hence, solve
−^
=^
𝑦^ =
0 ,^ 𝑦
=^
1 ,^ 𝑦
=^
Level
Hence ,^ solve
+^2
𝑦^ +
+^
)^2
=^0
𝑦^ +
0 ,^
𝑦^ +
1 ,^
𝑦^ +
𝑦^ =
−^2
,^ 𝑦^
=^ −
1 ,^ 𝑦
=^
Level 3: Hence, Solve
𝑦^ =
1 = 𝑦^
=^
=^
𝑦^ =
0 ,^ 𝑦
=^
1 ,^ 𝑦
=^
12^ Greetings from Paradigm! Hope this Cheatsheet can help you in your O Levels Preparation. We wish you all the best If you’re looking for more Resources & Study Advice, Simply Scan this QR Code! Method 1: Fully expand and compare coefficient Method 2: Substitution Method Given that the identity^3 𝑥
𝑥^ −
𝑥^ −
(𝑥^
+^2
)^ +
𝑥^ −
1 ) + C, for all real values of x, find the value of A, of B and of C by substitution. Step 1: Sub
𝑥^
=^1
(to get C) Step 2: Sub
𝑥^
=^ −
2 (to get B) Step 3: Sub in convenient value of
𝑥^
e.g.
𝑥^ =
0 (to get A) You will find A, B, and C easily. Step 1: Listing down all the different variations and conditions Step 2: Always check the coefficient of the highest power, unsure put 𝑘 Step 3: Substitution Method and Apply Remainder and Factor Theorem The highest order term of a polynomial 𝑓(𝑥 )^ is 3 𝑥 4.^ Three of the roots of the equation 𝑓(𝑥 )^ = 0 are (^1 0) , 3 and −^1
.^ When 𝑓(𝑥 )^ Is divided by (𝑥^ −^2 ), the remainder is 30. Find the expression for 𝑓( 𝑥)^ in descending powers of 𝑥. ( 𝑓𝑥 )^ = (𝑥 )(^3 𝑥 −^1 )(𝑥 +^ 1 )( 𝑥^ − 𝑎) Sub 𝑥^ = 2 𝑓(^2 )^ = (^2 ) (^5 ) (^3 ) (^2 − 𝑎)^ =^30 60 −^30 𝑎^ = 30 𝑎^ = 1 𝑓(𝑥 )^ = (𝑥 )(^3 𝑥 −^1 )(𝑥 +^ 1 )( 𝑥^ − 1 ) 𝑓(𝑥 )^ = 3 𝑥 4 − (^3) 𝑥 −^3 (^2) 𝑥 +^ 𝑥 Solve the cubic expression
−^20
𝑥^ +
9 completely. Use Mode
3 ,^4
(Don’t Write This) Substitute
𝑥^ =
−^1
)^1
=^6
(−^1
−^
−^1
)^ +
By^ Factor Theorem
,^ (𝑥
+^
1 )^ is a factor. Using Long Division ,^ (Show Working) You will get
𝑥^ +
9 as the Quotient. Use Mode
3 ,^3
)^ =
+^1
)(^2
𝑥^ −
(^3 𝑥
−^1
Sub
𝑥)^ =
𝑥^ =
−^1
,^ 𝑥^
,^ 𝑥^
Note: If Mode 3,3 shows i maginary number, use (^2) 𝑏 −^4 𝑎𝑐^ and show it is <^0 . *Look at the question. They may ask you to express answers in surd forms.^ Apply Quadratic Formula and simplify. 𝑥^ = −𝑏 ±^ √ (^2) 𝑏− 4 𝑎𝑐 2 𝑎 Important terms
Coefficient of a Variable: Number in front of the variable (algebra) 2 is the coefficient of
,^4
is the coefficient od
2 Degree of a Polynomial: The degree of a polynomial is the highest power in the equation Adding, Subtraction & Multiplication follows algebra rules Division of Polynomials Note: When you encounter polynomials such as Divide
−^
9 𝑥^
+^10
by
𝑥^ +
Please write it as
+^
+^
Fil up the missing terms with
Remainder Theorem
:^
Factor Theorem
)^ =
𝑥)^
+^
𝑅^
)^ 𝑥
=^ (
𝑥^ −
)^ =
)^
𝑏)^
=^0
Step 1: Mode 3 ,^4 to find the factor Step 2: Choose the smallest solution and prove that it is a Factor Step 3: Write by Factor Theorem (𝑥^ −^ 𝑎 )^ is a factor Step 4: Conduct Long division to obtain your quotient Step 5: Using your quotient, apply quadratic equation to get your roots Step 6: If roots are irrational, use (^2) 𝑏 −^4 𝑎𝑐^ and show it is <^
Step7: Souble check all answers with Mode 3 ,^4
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Paradigm
Specialising in O
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atics
Ultimate
Exam Guide &
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Paradigm Math De
partment
PARTIAL FRACTIO
NS
Step 1: If improper fraction and Conduct Long Division Improper = Degree of numerator
≥^
Degree of denominator Dividend^ Divisor
Quotient
+^
Remainder^ Divisor Step 2: Ensure Denominator is Fully Factorised Step 3: Apply Partial Fraction Rules Step 4: Eliminate all fractions by multiplying the Denominator Step 5: Solve using Substitution Method Express 4 − 4 𝑥− (^28) 𝑥 (^2) 𝑥(𝑥 +^2 ) as a sum of its partial fractions. 4 −
−^
(𝑥^ +
𝐴 + 𝑥^
+^
(𝑥^
+^2
−^
(𝑥^
+^2
)^ +
𝑥^ +
+^
2 Let
𝑥^
=^0
,^ 𝐵^
=^2
Let
𝑥^
=^ −
2 ,^ 𝐶
−^5
Let
𝑥^
=^1
,^ 𝐵^
=^2
,^ 𝐶^
=^ −
5 ,^ 𝐴
=^
−^3
4 −^4 𝑥− (^28) 𝑥 (^2) 𝑥( 𝑥+ 2 )^
=^
−^
3 + 𝑥^
(^2 2) 𝑥
(^5) 𝑥+
BI
NOMIAL THEOREM
Question Type 1: Binomial Expansion Hence Expansion Type 1: Selective Expansion You selectively rainbow the terms to obtain the algebra you need. Hence Expansion Type 2: Full Expansion You have no choice but to expand everything fully Hence Replacement Type 1: Number (𝟐𝒙
+^ 𝟑
)^ (
𝟏^ −
𝟏𝟏^ =
𝟑^ −
𝟐𝟗 𝟐^
𝒙^ +
𝟏𝟐𝟏 𝟒
−^
𝟐𝟕𝟓^ 𝟖^
𝟑^ 𝒙
+^ ⋯
Observe that x is replaced with 0.1. 𝟑.^ 𝟐
×^ 𝟎
.^ 𝟗𝟓
𝟏𝟏^
=^
𝟑^ −
𝟐𝟗 𝟐^
+^
𝟏𝟐𝟏^ 𝟒
𝟎.^ 𝟏
)^𝟐
−^
𝟐𝟕𝟓^ 𝟖
𝟎.^ 𝟏
)^𝟑
+^
=^
𝟏.^ 𝟖𝟏𝟖𝟏𝟐𝟓
Hence Replacement Type 2: Algebra (𝟏^
+^ 𝒌
𝟓^ )𝒘𝒊𝒕𝒉
+^
𝒙^ −
(𝟏^
+^ 𝒌
𝟓^ )=
𝟏^ +
+^
(𝟏^
+^ 𝒙
−^
𝟓^ =
𝟏^ +
+^
Question Type 2: Finding Specific Term^ Question Type 3: When
𝒏^
is unknown Timeline to O Levels Jun
-^ Complete Topical Revision (TYS) & Notes Jul^ -^ Kickstart Prelim Papers Aug -^ Prepare for Prelims Sep -^ Complete Yearly TYS + Prelim Papers Oct -^ Final Revision. Ready to ACE O’s! Towards Os, I will be sharing in - depth on - MUST Know Questions for Os - AM Paper 2 Analysis See you inside! Case Fraction N(𝑥) D(𝑥) Form of denominator, D(𝑥 ) Partial Fraction Form (where A, B and C are^ unknown constants) 1 N(x ) (ax^ +^ b) (cx^ +^ d) Linear Factors A ax +^ b +^ B cx +^ d 2 N(x ) (ax^ +^ b (^2) ) Repeated Linear Factors A ax +^ b +^ B (ax^ +^ b) 2 N(x ) (ax^ +^ b) (cx^ +^ d) 2 Linear and Repeated^ Linear Factors A ax +^ b +^ B cx +^ d +^ C (cx^ +^ d (^2) ) 3 N(x ) (ax^ +^ b (^2) )(x +^ c 2 ) Linear and Quadratic^ (which cannot be^ factorised) Factors A ax +^ b Bx+ (^) +^ C (^2 2) x+^ c
(𝑥^
7 )^ =
+^ (
7 )^1
)^ 𝑥
+^ ⋯
=^ 𝑥
+^189
General Term Formula:
- Sub in values, 2.^ SPLIT to individual indices 3.^ Rearrange and merge powers of
- Equate to required power (Independent of
𝑥^
means
- Find coefficient When power is unknown, we need to expand
𝑟 When n^ is^ unknown
𝑛𝐶^0
=^
𝑛𝐶^1
=^
𝑛𝐶^2
=^
𝑛(𝑛 −^1 ) 2!^ , where
2!^
=^
2 ×
𝑛𝐶^3
=^
𝑛(𝑛 −^1 ) (𝑛− 2 ) 3!^ , where
=^
3 ×
2 ×
𝑛𝐶^4
=^
𝑛(𝑛 −^1 ) (𝑛− 2 )(𝑛 −^3 ) 4!^ , where
4!^
=^4
×^
3 ×
2 ×
& so on… Last but not least,
You can use either 1. Binomial Expansion Method or 2. Finding Specific Term Method (𝑎^
+^ 𝑏
=^ ( 𝑛)^ ( 0 𝑛𝑎) −^0 ( (^0) 𝑏) +^ ( 𝑛)^ ( 1 𝑛𝑎) −^1 ( (^1) 𝑏) +^ ( 𝑛)^2 (𝑎) 𝑛−^2 (𝑏) 2 + ⋯^ +^ ( 𝑛)^ ( 𝑛 (^0) 𝑎) 𝑛(𝑏) Note:
)^ can be written as
𝑟 Evaluate
)^ using calculator: Press 3 , Shift
÷^
,^1 ,^ =
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Paradigm
Specialising in O
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atics
Ultimate
Exam Guide &
Cheat Sheet
Paradigm Math De
partment
LOGARITHM
log
is read as logarithm of
y^ to base
a
log
log
10
𝑎^ =
lg
log
log
ln
𝑒^ =
Laws of Logarithm Product Law:
log
+^
log
=^
log
Quotient Law:
log
−^
log
log
𝑥𝑎 𝑦
Power Law:
𝑟^ ×
log
log
𝑟
Changing Base: Changing Log to Index Form:
log
𝑦^
→^
𝑥^ =
Simplifying Question^ Step 1: Check Base^ Step 2: Decide MERGE or SPLIT^ Step 3: Simplify (Bring Down
Powers, Special Logs etc) Remember that we can convert integers back into Logarithm as well. For us to do that, we must determine what base we want to change If we want base 5,
log
log
log
Solving Question Type 1: Log to Ind ices Form log
=^
Usually this is in the form of 𝑥^ =
log
𝑥^ =
Solving Question Type 2: CCC (Change, Combine, Cancel) log
+^
)^2
+^ log
−^
)^2
=^ log
(^23
𝑥^ −
log
[( 3
𝑥^ +
(𝑥^
−^2
)]^ =
log
(^23
𝑥^ −
- Change Base (^2) 𝑥
−^4
=^
2 𝑥^
−^1
- Combine Log (^2) 𝑥
−^2
𝑥^ −
- Cancel Log (𝑥^
−^3
+^
1 )^
=^0
𝑥^ =
3 or
𝑥^ =
−^1
(Reject)
- Sub in to make sure Log is Positive Solving Question Type 3: Substitution*^ log
=^
4 log
+^3
log
=^
4 ×
log
log
𝑥^5
+^3
log
=^
(^4) log𝑥^5
+^
Substitute
𝑦^ =
log
𝑦^ =
4 + 𝑦^
−^3
𝑦^ +
𝑦^ =
4 or
𝑦^
=^ −
log
=^
4 or log
𝑥^5
=^ −
𝑥^ =
=^
or^
𝑥^ =
1 5
Logarithmic Graphs (Asymptotes,
𝒚^
&^ 𝒙
𝑦^ =
log
,^ where 𝑎^ >
𝑦^ =
log
,^ where 𝑎^ >
𝑦^ =
log
,^ where 0 <
𝑎^
<^1
EXPO
NE
NTIAL
Solving Type 1: Compare Power^5
𝑥^ =
Solving Type 2: Use Log/Ln to solve
=^
𝑥^3
=^5
𝑙𝑛^
𝑜𝑛^
ℎ^ 𝑠𝑖𝑑𝑒𝑠
𝑥^ =
Advance Type 2: Split & Rearrange Solve
𝑥+^3
−𝑥 2 𝑥 4
×^
=^7
3 ×
𝑥^
𝑥^ =
343 64 𝑥 16
×^
=^
×^
𝑥lg
=^
(^343) lg 64 𝑥 16
÷^
𝑥^ =
343 64
𝑥^ =
lg 343 64 ÷^ lg 𝑥 16
×^
𝑥^7
64
𝑥^ =
0.^3558
(^16
×^
=^
34364
𝑥^ =
0.^36
(^2
d.^ p.^ ) 112
𝑥^ =
343 64 Solving Type 3: Substitution^ (
+^
−^3
𝑦^ +
=^0
(𝑦^
−^
𝑦^ −
=^
𝒚^ =
𝟐^
𝒚^
=^
𝒙^ 𝟑
=^
𝟐^ or
Sub ln on both sides to solve Exponential Graphs (Sub
𝒙^
=^ 𝟎
to get y
- intercept)
𝑦^ =
ln^
𝑥^
𝑦^ =
ln(
)^
𝑦^ =
log
Restrictions on logarithms: For log
=^
𝑦,^
𝑥^ >
0 ,^
𝑎^ >
0 ,^
𝑎^ ≠
E.g. Show that the equation log
(^43
𝑥^ −
)^ −
log
−^
3 )^
=^1
has no real solutions: log
4 𝑥−
=^1
4 𝑥−
−^
=^3
𝑥^ −
𝑥^ =
When
𝑥^
=^2
, both log
(^43
𝑥^ −
)^ &
log
−^
)^3
are undefined. ∴^ the equation has no real solns.
1.^ Change Base 2.^ Substitute 3.^ Solve
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Paradigm
Specialising in O
Level Mathem
atics
Ultimate
Exam Guide &
Cheat Sheet
Paradigm Math De
partment
DIFFERE
NTIATIO
N
Algebra
𝑦^ =
𝑛^ 𝑑𝑦 𝑑𝑥^
=^ 𝑛
[𝑎𝑥
𝑛−^1
]
Chain Rule
:^
𝑦^ =
𝑑𝑦 𝑑𝑥^
=^ 𝑛
[(𝑓
−^1 ]
′[𝑓
]
Pro duct Rule:
𝑦^ =
𝑑𝑦 𝑑𝑥^
=^ 𝑢
𝑑𝑦 𝑑𝑥^
+^ 𝑣
𝑑𝑢 𝑑𝑥 Quotient Rule:
=^
𝑢 𝑣
𝑑𝑦= 𝑑𝑥^
𝑑𝑢𝑣− 𝑑𝑣 𝑑𝑣𝑢 𝑑𝑥 (^2) 𝑣
Differentiation: Trigonometry
(PTB)
Rule: Differentiate Trigo, X^ Angle 𝑑൛sin 𝑑𝑥^ ൣ 𝑓(𝑥 )൧ൟ^ =^ cos [൫𝑓 (𝑥) ൧.^ 𝑓 ′(𝑥) 𝑑൛cos 𝑑𝑥^ ൣ 𝑓( 𝑥)൧ ൟ^ = −^ sin ൣ 𝑓( 𝑥)൧
.^ 𝑓′( 𝑥) 𝑑൛tan 𝑑𝑥^ ൣ 𝑓( 𝑥)൧ ൟ^ = sec 2 ൣ𝑓 (𝑥) ൧.^ 𝑓′ (𝑥) Remember Power, Trigonometry, Bracket 𝑦^ = 𝑠𝑖𝑛 𝑛(𝑓 (𝑥) )^ 𝑑𝑦= 𝑑𝑥^ 𝑛.^ sin 𝑛−^1 (ൣ𝑓 )𝑥൧ ∗^ cos (ൣ 𝑓 )𝑥൧^ ∗^ 𝑓 ′(𝑥) 𝑦^ = 𝑐𝑜𝑠 𝑛(𝑓 (𝑥) )^ 𝑑𝑦= 𝑑𝑥^ 𝑛.^ cos 𝑛−^1 (ൣ𝑓 )𝑥൧ ∗^ − sin (ൣ 𝑓𝑥 )൧^ ∗ 𝑓′( 𝑥) 𝑦^ = 𝑡𝑎𝑛 𝑛(𝑓 (𝑥) )^ 𝑑𝑦= 𝑑𝑥^ 𝑛.^ tan 𝑛−^1 ൣ𝑓( 𝑥)൧ ∗^ sec 2 ൣ𝑓 (𝑥) ൧^ ∗^ 𝑓 ′(𝑥)
If you are given other Trigo, Apply identities to convert to sin, cos, tan Differentiation: Exponential & Logarithm Exponential
[Differentiate Power, Multiply Original Eq] 𝑑ൣ 𝑑𝑥^
=^
𝑓(𝑥
).^ 𝑓
Advance: Law of Indices Logarithm: [Differentiate Bracket, Over Bracket] 𝑑൛𝑎𝑙𝑛 𝑑𝑥^ (ൣ 𝑓 )𝑥൧ ൟ^ = 𝑎^ ( 𝑓′(𝑥 ) 𝑓(𝑥) ) Advance: Law of Logarithm
Equation of Tangent &
N
ormal
Key Concepts: 1. Always Sketch out Graph for Visualization 2.^ 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡
=^
−^
1 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑇𝑎𝑛𝑔𝑒𝑛𝑡
3.^ 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
:^ 𝑦^
−^
𝑦^1
=^ 𝑚
(𝑥^
−^ 𝑥
Increasing &
Decreasing Function
Increasing Function
𝑑𝑦 𝑑𝑥
Decreasing Function
𝑑𝑦 𝑑𝑥
Trick Question: Find the range of values for the gradient of curve,
is
increasing. This indicates that
(^2) 𝑑
𝑦>^2 𝑑𝑥
Question Type 1: Finding Range Apply
concept in Quadratic Inequality to find Range
Question Type 2: Prove/Show Method 1: Observation Method
Dissect Fraction/Term and explain individually. e.g. Since (^12) (𝑥+^3 )
(^22) (𝑥+^3 )
>^
Method 2: Completing the Square Method Apply this when you see a
Quadratic Equation.
Rate of Change^ Write OUT what is given & what you need!^ Rate of Change of
𝑥^
Rate of Change of y =
𝑑𝑦 𝑑𝑡
×
When dealing with Mensuration, remember to recall the Formulas of Volume & Surface Area (E. Maths) Advance Rate of Change Question: 1. Split & Cancel (Rate of Change of X is
3 x the Rate of Change of Y)
- Double Split 3. Mensuration (eg Cone Questions)
Maximum and Minimum^ Always remember that stationary point has a gradient of 0.
However, we are unsure the nature of the point. First Derivative Test
𝑥^ =
𝑎^ −
0.^1
𝑥^ =
𝑎^
𝑥^ =
𝑎^ +
0.^1
𝑑𝑦 𝑑𝑥 Slope
Minimum Point? Maximum Point? Point of Inflexion? Second
Derivative Test
Minimum Point:
(^2) dy dx
Maximum Point:
(^2) d
y<0^2 dx
d If
2 y^2 dx
Use First Derivative Test. Maximum and Minimum
Cone Surface Area:
Volume:
Pyramid Surface Area: Add all sides Volume: 1 Base area x Height 3 Hemi
[open] /
Volume:
Sphere Surface Area:
Volume:
Prism Surface Area: Add all sides Volume: Cross Section x Length Cylinder Surface Area: 2
+^
Volume:
Page
Paradigm
Specialising in O
Level Mathem
atics
Ultimate
Exam Guide &
Cheat Sheet
Paradigm Math De
partment
IN
TEGRATIO
N
Algebra^ Fundamental:
[Power +1. Divide New Power] ∫^ 𝑝𝑥
𝑛^ 𝑑𝑥
𝑛+^1 𝑛^ +
+^ 𝐶
Definite Integrals (Upper Limit
-^ Lower Limit) ∫^
𝑏 𝑎^
=^ [
𝑛𝑝𝑥
+^1 ] 𝑛+ 1
𝑏^ =𝑎
[(
𝑛𝑝𝑏
+^1 ) 𝑛+ 1
−^
𝑛+^1 ) 𝑛+^1
]
Algebra :^ [Power +1, Divide New Power, Divide Coeff
𝒙 ]
+^
=^
+^
+^1 (𝑛^
+^1
)^
+^ 𝐶
Exponential^ ∫
𝑞𝑥+
𝑟^ =
𝑞𝑥+ 𝑟 𝑞^
+^ 𝐶
Common Exam Questions:
1)^
2 𝑥+ 𝑥𝑒𝑥 (^) 𝑒
∫^
𝑥^ 𝑒
+^1
=^
+^
𝑥^ +
2)^
∫^
+^
2 𝑥^
+^1
Trigonometry^ ∫
+^
𝑟)^
=^
+^
𝑞^
+^ 𝐶
∫^ 𝑝𝑐𝑜𝑠
+^
𝑟)^
𝑑𝑥^
+^ 𝑟
𝑞^
+^ 𝐶
∫^ 𝑝
+^
𝑟)^
𝑑𝑥^
+^
𝑞^
+^ 𝐶
Note: You can only integrate
,^ 𝑐𝑜𝑠
,^ sec
.^ Any other Trigo needs to be converted.
=^
2 𝑥^
2 𝑥^
=^ 𝑠𝑒𝑐
2 𝑥^
−^1
2 𝑥^
−^
[Double Angle]
2 𝑥^
+^
[Double Angle]
Logarithm^ ∫
1 𝑝𝑥 +^
=^
1 ∫ 𝑝^
+^
=^
1 ln 𝑝^
+^
𝑞)^
+^ 𝐶
Note: Ln integration happens only when denominator is Linear. Ensure that numerator is the coefficient of 𝑥^ in the denominator.
Integration and Partial Fraction and Ln^ ∫
𝟑^ +
𝟐^ +
𝒙^
−^ 𝟏
𝒙^ +
=^ ∫
+^
𝟐− 𝒙^
𝟏 𝟐^ 𝒙
−^
=^ 𝟒
𝒙^ +
𝟐^
𝐥𝐧^
𝒙^ +
𝟏 − 𝒙^
(𝒙^
+^ 𝟏
)^ +
Notice the mix of integration techniques here Equation of Curve Remember to always +C when integrating. Know clearly what to substitute to find unknown constants. Be mindful of Question: Gradient of Tangent/Normal Area Under
Graphs
Kinematics^ Drawing Pathways is the KEY to excel in this segment. Question Type 1: Instantaneous s, v, or a^ 1. Sub in
𝑡^ to find Answer
Question Type 2: Maximum/Minimum Velocity^ 1.
𝑑𝑣 𝑑𝑡^
=^0
- Find 𝑡^ when it occurs
- Substitute 𝑡^ into 𝑣^ to find maximum/minimum value
Question Type 3: Total Distance or Average Speed^ Find Total Distance:^ 1.
Start ing Position (START) Substitute
𝑡^ =
0 and
𝑠^ =
0 (*Origin)
2.^ Turn (TURN) Substitute
𝑣^
=^0
, find 𝑡, find
3.^ Final Position (STOP) Use 𝑡, find
- Draw Pathway to find the Total Distance Note: If question is asking for distance travelled in 5 th^ Second, that is distance travelled between
ℎ^ and
ℎ^ second. Note: If there are 2 particles colliding, displacement by both particles should be same as point of collision.
