Essential Math Formulas and Concepts: A Quick Reference
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Essential Math Formulas and Concepts
Volume of Shapes
(a) Cube:
- Formula: V = s3 (where s is the length of a side)
- Example: A cube with side length 4 cm
V = 43 = 64 cm3
(b) Cylinder:
- Formula: V = πr2h (where r is the radius and h is the height)
- Example: A cylinder with radius 2 cm and height 6 cm
V = π(22)(6) = 24π cm3 (approximate value)
Surface Area of Shapes
(a) Cube:
- Formula: SA = 6s2 (where s is the length of a side)
- Example: A cube with side length 3 cm
SA = 6(32) = 54 cm2
(b) Cone:
- Formula: SA = πr(r + √(r2 + h2)) (where r is the radius and h is the height)
- Example: A cone with radius 5 cm and height 8 cm
SA = π(5)(5 + √(52 + 82)) = 129.74 cm2 (approximate value)
Stem and Leaf Plot
(a) Mean:
- Example: Data: 5, 7, 8, 9, 10, 12, 15
Mean = (5 + 7 + 8 + 9 + 10 + 12 + 15) / 7 = 9
(b) Median:
- Example: Data: 5, 7, 8, 9, 10, 12, 15
Median = 9
(c) Mode:
- Example: Data: 5, 7, 8, 9, 9, 9, 10, 12, 15
Mode = 9
(d) Range:
- Example: Data: 5, 7, 8, 9, 10, 12, 15
Range = 15 - 5 = 10
(e) Lower Quartile:
- Example: Data: 5, 7, 8, 9, 10, 12, 15
Lower Quartile = 7
Box & Whisker Graphs
(a) Inter-Quartile Range (IQR):
- Example: Class A: Q3 = 60, Q1 = 40
IQR = Q3 - Q1 = 60 - 40 = 20
- Class B: Q3 = 80, Q1 = 30
IQR = Q3 - Q1 = 80 - 30 = 50
(b) Comparison:
- Class average (center): Compare the mean of each class.
- Example: Class A mean = 50, Class B mean = 60
Class B has a higher class average.
- Spread of the middle 50%: Compare the IQR values for each class.
- Example: Class A IQR = 20, Class B IQR = 50
Class B has a larger spread of the middle 50%.
Probability
(a) Tree Diagram:
- Example: Tossing a fair coin twice
(b) Probabilities:
- Example: P(H, H) = ¼ (from the tree diagram)
- Example: P(At least one Tail) = ¾ (from the tree diagram)
Probability and Venn Diagram
(a) P(B):
- Example: P(B) = Number of outcomes in B / Total number of outcomes
(b) P(A ∪ B):
- Example: P(A ∪ B) = Number of outcomes in A or B / Total number of outcomes
(c) P(A | B) (conditional probability):
- Example: P(A | B) = Number of outcomes in both A and B / Number of outcomes in B
(d) Independence of events A and B:
- Example: P(A | B) = P(A) (events A and B are independent) or P(A | B) ≠ P(A) (events A and B are not independent)
Solving Quadratic Equations
(a) Solve the quadratic equation y = x2 + 7x + 10
- Example: y = x2 + 7x + 10
Set y = 0 and solve for x.
x2 + 7x + 10 = 0
(x + 5)(x + 2) = 0
x = -5 or x = -2
(b) Solve the quadratic equation y = x(x - 2) - 2
- Example: y = x(x - 2) - 2
Set y = 0 and solve for x.
x(x - 2) - 2 = 0
x2 - 2x - 2 = 0
Use the quadratic formula or factorize.
(x + 1 - √3)(x + 1 + √3) = 0
x = -1 - √3 or x = -1 + √3
(c) Solve the quadratic equation y = x2 - 6x + 12
- Example: y = x2 - 6x + 12
Set y = 0 and solve for x.
x2 - 6x + 12 = 0
Use the quadratic formula or factorize.
(x - 3 + √3i)(x - 3 - √3i) = 0
x = 3 + √3i or x = 3 - √3i