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ROAD TO A*
FURTHER MATHS
THE FE2+ CORP. PRESENTS
WRITTEN BY:
MR FE2+
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1
(A) Polar Curve
1.
2.
3.
4. ∫ d
5. ∫
6. ,
,
(B) Application of Calculus
1. ∫
dx or
∫
dt
2. ∫ dx or
∫
dt
3. ∫ dx or
∫
dt
4. ̅ ∫ dx
∫ dx , where ∫ dx5. ̅ ∫ dx , where ∫ dx
6. ∫ dx
7.
8.
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2
(C) Calculus
1. ,
2. ⁄ ,
3.
4.
5.
(D) Roots of Equations
1. ∑ ∑ ∑
2.
3.
4.
(E) Summation
1. [ ]
2. AP
3.
4. GP
5. ∑ ∑ ∑
6. ∑
(F) Partial Fraction
1.
2.
3.
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3
4.
5.
6.
(G) Curve Sketching
1. Asymptote
2. Improper fraction : do long division
3. Turning point :
4. Intersection between curve and axes : when y=0 and x=0 5. Intersection between curve and asymptotes (equate equation of asymptote and
equation of curve)
6. Label graph and asymptotes
7.
=
(H) Second Order Differential Equation
1. Auxiliary equation : aM²+bM+c=0 2. ( )
3. ( )
4.
Particular Integral:
5.
6.
7.
8. 9.
10. Special case particular integral***
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4
(I) Reduction Formulae
1. ∫ ∫
2. ∫ ∫
3. ∫ ∫
4.
∫ ∫ 5. ∫ ∫ ∫ ∫
6. ∫ ∫
7. ∫ ∫
8. ∫ ∫[ ] ∫
(J) Differentiation1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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5
(K) Integration
1. ∫
2. ∫
3. ∫
∫
4. ∫
5. ∫ ∫
6. ∫
7. ∫
8. ∫
9. ∫ 10.
∫
11. ∫ ∫ ∫
12. ∫ ∫
13. ∫ ∫
∫
14. ∫
15. ∫ ∫
16.
(a) ∫ () ()
(b) ∫
(c) ∫ () ()
(d) ∫
17. ∫
18.
(a) √
19. ∫ ∫
∫
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6
(L) Trigonometry
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24. √
25. √
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7
(M) Complex Number
1.
2.
3.
4.
5.
(N) System of Equation
1. 3 equations with 3 unknowns (x, y, and z)
2. Eliminate 1 unknown, e.g. x
3. If we get same equations , infinite no. of solutions, consistent, linearly
dependant, determinant = 0.
4. If we get different equations (which do not contradict), unique solution,
determinant 0.
5. If we get contradict equations, for example : y+2z=2 and y+2z=3, no solution,
inconsistent, determinant = 0.
(O) Vector Space
1. Basis for spaces must independent.
2.
3.
4.
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8
(P) Proof by Induction(step 3 )
1. Summation:
2. Divisibility:
3. Differentiation:
Note:
4. Matrix:
5. de Moivre’s theorem:
6. Reduction formula(example):
7. Iterative formula(example):
8. Inequality(example):
9. Eigenvalue and eigenvector:
10. ***Try to think how to use your assumption (step 2, where n = k) in step 3 !!!
(Q) Extra Informations
1.
2. ∑ 3.
4. √
√ √
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9
5. Cubic equation with ∑ 6. Cubic equation with ∑
7. ∫ *
+ ∫
[]
8.
For both odd and even: n=1,2,3,…
9. 10.
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