Number Patterns
Definition: A sequence of numbers following a specific rule.
Examples:
- 2, 4, 6, 8, 10 (even numbers)
- 1, 4, 9, 16, 25 (perfect squares)
- 1, 2, 4, 8, 16 (powers of 2)
- 5, 10, 15, 20, 25 (multiples of 5)
- 1, 1, 2, 3, 5, 8, 13 (Fibonacci)
- 10, 20, 30, 40, 50 (add 10)
- 100, 90, 80, 70, 60 (subtract 10)
- 3, 6, 12, 24, 48 (multiply by 2)
Exercises:
- Find the next three numbers: 3, 6, 12, 24, ...
- Find the pattern and next number: 2, 5, 10, 17, ...
- Identify the pattern: 1, 4, 9, 16, 25, ...
- Find the 10th term of the sequence: 2, 4, 6, 8, ...
- Find the 20th term of the sequence: 5, 8, 11, 14, ...
- Find the 12th term of the sequence: 3, 7, 11, 15, ...
- Find the 50th term of the sequence: 1, 4, 7, 10, ...
- Determine the common difference of the sequence: 7, 12, 17, 22, ...
- Find the first term of a sequence if the 8th term is 29 and the common difference is 3.
- Find the number of terms in the sequence: 5, 8, 11, ..., 50
Mensuration
Definition: The study of measurement of geometric figures including length, area, and volume.
Examples:
- Perimeter of a rectangle: P = 2(l + w), e.g., l=5cm, w=3cm → P=16cm
- Area of a triangle: A = ½ × b × h, e.g., b=6cm, h=4cm → A=12cm²
- Volume of a cube: V = a³, e.g., a=3cm → V=27cm³
- Surface area of a sphere: SA = 4πr², e.g., r=7cm → SA≈615.75cm²
- Volume of a cylinder: V = πr²h, e.g., r=3cm, h=5cm → V≈141.37cm³
- Perimeter of a square: P = 4a, e.g., a=6cm → P=24cm
- Area of a circle: A = πr², e.g., r=5cm → A≈78.54cm²
- Surface area of a cuboid: SA = 2(lb + bh + hl), l=2, b=3, h=4 → SA=52cm²
Exercises:
- Find the area of a triangle with base 10cm and height 6cm.
- Calculate the volume of a cylinder with radius 4cm and height 10cm.
- Determine the perimeter of a square with side 8cm.
Algebraic Processes
Definition: Operations on algebraic expressions like expanding, factorizing, and simplifying.
Expand Examples:
- (x + 2)(x + 3) = x² + 5x + 6
- (x - 1)(x + 4) = x² + 3x - 4
- (2x + 3)(x + 5) = 2x² + 13x + 15
Factorize Examples:
- x² + 5x + 6 = (x + 2)(x + 3)
- x² - x - 6 = (x - 3)(x + 2)
- 2x² + 7x + 3 = (2x + 1)(x + 3)
Quadratic Examples:
- Equation: x² + 5x + 6 = 0 → x = -2, -3
- Equation: x² - 3x - 10 = 0 → x = 5, -2
- Equation: 2x² + 7x + 3 = 0 → x = -1, -3/2
Exercises:
- Expand: (x + 4)(x + 5)
- Factorize: x² + 8x + 15
- Solve: x² - 7x + 10 = 0
- Expand: (2x - 3)(x + 6)
- Factorize: 2x² + 5x + 2
- Solve: x² + 3x - 18 = 0
- Expand: (x + 2)(x² + 3x + 4)
- Factorize: x² - 9
- Solve: 3x² + 11x + 6 = 0
- Expand: (x - 5)(x² + x + 1)
Simultaneous Equations
Definition: A set of two or more equations with the same variables that are solved together to find common solutions.
Substitution Method
Solve one equation for one variable and substitute into the other equation to find the values of the variables.
Example:
- Equation 1: x + y = 7
- Equation 2: x - y = 3
- Solution: From Eq1: x = 7 - y → substitute into Eq2: (7 - y) - y = 3 → 7 - 2y = 3 → y = 2 → x = 5
Elimination Method
Add or subtract equations to eliminate one variable and solve for the other variable.
Example:
- Equation 1: 2x + 3y = 12
- Equation 2: 3x - 2y = 1
- Solution: Multiply Eq1 by 2 and Eq2 by 3 to eliminate y: 4x + 6y = 24, 9x - 6y = 3 → Add: 13x = 27 → x = 27/13 → Substitute x into Eq1 to find y
Exercises:
- Solve by substitution: x + 2y = 8, 3x - y = 5
- Solve by substitution: 2x + y = 7, x - y = 1
- Solve by elimination: 2x + 3y = 13, 3x - 2y = 4
- Solve by elimination: x + y = 9, x - y = 3
- Solve by substitution: 4x - y = 11, 2x + 3y = 7
- Solve by elimination: 5x + 2y = 20, 3x - y = 7
- Solve by substitution: 2x + 5y = 19, 3x - 2y = 4
- Solve by elimination: x + 4y = 13, 2x - y = 1
- Solve by substitution: x - 3y = -2, 2x + y = 7
- Solve by elimination: 3x + 2y = 16, 5x - 3y = 11