S01 |
- 01. Draw a circle passing three given points.
S02 |
- 02. Solve x^4 + x^3 + x^2 + x + 1 = 0
S03 |
- 03. What is ex-central triangle ?
S04 |
- 04. How to find the 3rd perfect number ?
S05 |
- 05. Sequence in Pascal triangle : Sum[n*(n+1)/2] = n*(n+1)*(n+2)/3!
S06 |
- 06. Intersections of y = a*x^2 + b*x + c with its inverse
S07 |
- 07. Intersections of y = a*x^2 + b*x + c with y = 1/x
S08 |
- 08. Intersections of y = x^2 - 6*x + 8 with its inverse.
S09 |
- 09. Properties of numbers by its factors
S10 |
- 10. Equation of e^x + e^(2*x) + e^y + e^(2*y) = 12
S11 |
- 11. Pedal triangle of triangle ABC
S12 |
- 12. Amicable numbers
S13 |
- 13. Absolute Abs(x^2 - 6*Abs(x) + 8) = n
S14 |
- 14. Series and sequences
S15 |
- 15. Solve x^7 +2*x^6 -5*x^5 -13*x^4 -13*x^3 -5*x^2 +2*x +1 = 0
S16 |
- 16. Solve x^5 - 1 = 0
S17 |
- 17. Compare ((x-h)/4)^2-((y-k)/4)^2 =1 & ((x-h)/4)^2-((y-k)/4)^2 =-1
S18 |
- 18. Ortho-center theory
S19 |
- 19. Transformation matrix of a circle
S20 |
- 20. Parametric equation : x = sec(t) and y = tan(t)
Home Works
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S01. Home Work 1 : Draw a circle passing three given points
Geometrical method
- 1. Construction
- Draw a triangle using the points A, B, C.
- Bisect the sides of the triangle.
- Bisectors are concurrent to a point E which is the ex-center.
- Use EA as radius to draw a circle.
- The circle will pass points A, B, C.
- 2.
Study subjects :
Five centers of a triangle.
- a. What is ex-center ?
- b. How to prove that the biscetors of sides are concurrent.
Analytic geometric method
-
Study subjects :
Find equation of the circle passing three points.
- Method
- a. Substitute 3 points into x^2 + y^2 +a*x + b*y + c = 0.
- b. Solve 3 linear equations of a,b,c.
- c. Use the completing square to find center and radius.
Go to Begin
S02. Home Work 2 : Solve x^4 + x^3 + x^2 + x + 1 = 0 using DeMoivre's theorem
Questions
- 1. What is the DeMoivre theorem ?
- 2. How to solve x^5 - 1 = 0 using DeMoivre's theorem ?
- 3. Solve x^4 + x^3 + x^2 + x + 1 = 0
- Prove that cis(72), cis(144), cis(216) and cis(288) are solutions.
- Prove that cos(72) + cos(144) + cos(216) + cos(288) = -1.
- Prove that sin(72) + sin(144) + sin(216) + sin(288) = 0.
- How to use construction method to find solutions ?
Study subjects :
DeMoivre's theorem and application.
Go to Begin
S03. Home Work 3 : What is ex-central triangle ?
Construction
- 1. Draw a triangle ABC ?
- 2. Draw three es-centers of triangle ABC.
- 3. Join the centers J, K, L with vertices of the triangle.
Questions
- 1. Prove that J, C, K are colinear.
- 2. Prove that I is the in-center of triangle ABC.
- 3. Prove that I is also the orthocenter of triangle JKL.
- 4. Prove that ABC is pedal triangle of triangle JKL.
Study subjects :
The ex-central triangle.
Study subjects :
The five centers of triangle.
Go to Begin
S04. Home Work 4 : How to find 3rd perfect number ?
- Properties of perfect number.
- Perfect number = 1^3 + 3^3 + 5^3 + ....
- Perfect number = (2^(n-1))*(2^n-1).
- 2^n - 1 is a prime number for a perfect number
- Third perfect number is between 400 and 500.
Study subjects :
Perfect numbers.
Go to Begin
S05. Home Work 5 : Sequence in Pascal triangle
- 1. What is Pascal triangle ?
- 2. Prove that Sum[n*(n+1)/2] = n*(n+1)*(n+2)/3!.
- 3. Prove that Sum[C(n+1,2)] = C(n+2,3). Where C(n,r) is coeff of binomial expansion.
Study subjects :
Sequence and Pascal triangle.
Go to Begin
S06. Intersection of y = a*x^2 + b*x + c with its inverse
- Find intersection of following quadratic functions with its inverse
- 1. y = x^2 + 2.
- 2. y = x^2 + 0.25.
- 3. y = x^2 .
- 4. y = x^2 - 2.
-
Study subjects :
Intersections of quadratic function with its inverse.
- Click Start after entering application program.
- Click subject 6 in upper box.
- Click program 1, 2, 3 or 4 to find graphic answer.
Study subjects :
Inverse of quadratic functions.
Go to Begin
S07. Intersection of y = a*x^2 + b*x + c with y = 1/x
- Find intersection of following quadratic functions with y = 1/x
- 1. y = 4*x^2 - 3.
- 2. y = 4*x^2 + 2*x + 3.
- 3. y = 4*x^2 + 2*x - 5.
-
Study subjects :
Intersections of quadratic function with y = 1/x.
- Click Start after entering application program.
- Click subject 6 in upper box.
- Click program 7, 8 or 9 to find graphic answer.
Go to Begin
S08. Estimate intersections of y = x^2 - 6*x + 8 with its inverse.
Study subjects :
Intersections of quadratic function with y = 1/x.
- Click Start after entering application program.
- Click subject 6 in upper box.
- Click program 10 find sketch program.
- Type in coefficients 1, -6, 8.
- Estimate the answers from the diagram.
Go to Begin
S09. Properties of numbers by its factors
Find the properties of the number 27, 28, 29 30 by their factros.
- Find the factors of a given number.
- Sum the factors of the number exluding number itself.
- Give the properties of the number (abudunt, deficient, perfect, prime).
- See Q02 of QA web-page.
Go to Begin
S10. Equation : e^x + e^(2*x) + e^y + e^(2*y) = 12
Home work questions
- 1. Find equation of tangent when x = ln(3).
- 2. Find y when x = ln(2).
- 3. How to sketch the curve of y verse x ?
Reference
Go to Begin
S11. Pedal triangle of triangle ABC
Home work questions
- 1. What is pedal triangle of triangle ABC ?
- 2. Prove that in-center of pedal triangle is also orthocenter of triangle ABC.
- 3. Prove that vertices A, B, C are es-center of the pedal triangle.
- 4. Prove that triangle ABC is ex-central triangle of its pedal triangle.
Reference
Go to Begin
S12. Amicable numbers
Home work questions
- 1. What are amicable number pairs ?
- 2. Prove that 220 and 284 are amicable pairs.
- 3. If 1184 and n are amicable pairs, find n.
Reference
1.
Study subjects :
Amicable number pairs
2.
Study subjects :
Who discover amicable number pairs ?
Go to Begin
S13. Absolute operation : Abs(x^2 - 6*Abs(x) + 8) = m
Home work questions
- 1. How many real roots Abs(x^2 - 6*Abs(x) + 8) = 0.5 ?
- 2. How many real roots Abs(x^2 - 6*Abs(x) + 8) = 1 ?
- 3. How many real roots Abs(x^2 - 6*Abs(x) + 8) = 3 ?
Reference
Go to Begin
S14 : Series
Questions
- 1. Prove that Sum[n^2] = n*(n+1)*(2*n)/6.
- 2. Prove that Sum[n^3] = (n*(n+1)/2)^2.
- 3. Prove that Sum[n*(n+1)/2] = n*(n+1)*(n+2)/6
Reference
1.
Study subjects :
Sequence of 1, 4, 9, 16, ...
2.
Study subjects :
Sequence of 1, 8, 27, 64, ...
3.
Study subjects :
Sequence of 1, 3, 6, 10, ....
Go to Begin
S15 : Solve equations
Questions
- 1. Solve x + 1 = 0
- 2. Solve x^2 + x + 1 = 0
- 3. Solve x^4 - 7*x^2 + 1 = 0
- 4. Change (x+1)*(x^2+x+1)*(x^4 - 7*x^2 + 1) to polynomial form
- 5. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0
Reference
Go to Begin
S16 : Solve x^n - 1 = 0
Questions
- 1. Solve x5 - 1 = 0
- 1. Use DeMoivre's thereom : x = cis(A) for A = 72, 144, 216, 288, 360
- 2. Use constructions : Draw unit circle and angle A = 72, 144, 216, ...
- 3. 1st root r1 = cos(72) + i*sin(72)
- 4. 2nd root r2 = cos(144) + i*sin(144)
- 5. 3rd root r3 = cos(216) + i*sin(216)
- 6. 4th root r4 = cos(288) + i*sin(288)
- 7. 5th root r5 = cos(360) + i*sin(360) = 1
- 2. Express (x - 1)*(x^4 + x^3 + x^2 + x + 1) as polynomial form
- 3. Solve x^4 + x^3 + x^2 + x + 1 = 0
Reference
Go to Begin
S17 : Compare ((x-h)/4)^2-((y-k)/4)^2 =1 & ((x-h)/4)^2-((y-k)/4)^2 =-1
Compare the following
- 1. Assymptotes
- 2. Coordinates of center
- 3. Coordinates of vertices
- 4. Coordinates of focus
- 5. Focal length
- 6. Equation of Principal axis
- 7. Equation of directrix
- 8. Equation in polar form
Reference
Go to Begin
S18 : Ortho-center Theorem
Ortho-center
- 1. What is ortho-center of triangle ABC
- 2. Relation between ortho-center of circum center : Prove that AO : UR = 2 : 1
- O is ortho-center and U is circum-center
- AOD perpendicular to AB and UR perpendicular to AB
- 3. Trinangle ABC has pedal triangle PQR : Prove O is in-center of triangle PQR
Reference
Go to Begin
S19 : Transformation matrix of a circle
Study transformation of circle (x-h)^2 + (y-k)^2 = r^2
- Find eqnation and transformation matrix to transform circle as image of y = x
- Find eqnation and transformation matrix to transform circle as image of y = -x
- Find eqnation and transformation matrix to transform circle as image of x-axis
- Find eqnation and transformation matrix to transform circle as image of y-axis
- Find matrix to transform circle as line section x-axis (Projection on x-axis)
- Find matrix to transform circle as line section y-axis (Projection on x-axis)
- Find matrix to transform circle as ellipse (Principal axis paralle to x-axis)
- Find matrix to transform circle as ellipse (Principal axis paralle to y-axis)
Reference
Go to Begin
S20 : Parametric equation
Equations
- 1. Graph of x = sec(t) and y = tan(t)
- 2. Graph of x = tan(t) and y = sec(t)
Questions
- Compare the two graphs in rectangular form
- compare the two graphs in polar form
Reference
Go to Begin
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