Math RCB HYE
Surds
Surds are irrational numbers that cannot be expressed as fractions or recurring decimals, while recurring decimals are decimals that repeat:
Examples: √3, √6, √19
The important laws of exponents are given below:
a^m×a^n = a^ m+n
a^m/a^n = a^ m-n
(a^m)^n = a^mxn
a^n/b^n = (a/b)^n
a^0 = 1.
a^-m = 1/a^m
Scientific Notation
Definition: A method to express very large or very small numbers in the form a×10n where 1≤∣a∣<10 and n is an integer.
Example: 5000 = 5×10^3; 0.0004 = 4×10^−4
Operations:
Multiplication: (a×10^m)×(b×10^n)=ab×10^m+n
Division: b×10^n/a×10^m=ba×10^m−n
Absolute Values
Definition: The distance of a number from zero on the number line, denoted as ∣x∣∣x∣.
Properties:
∣x∣≥0
∣x∣=x if x≥0
∣x∣=−x if x<0
Example: ∣3∣=3, ∣−3∣=3
Changing the Subject of the Formula
Definition: Rearranging an equation to solve for a different variable.
Steps:
Identify the variable to isolate.
Use order of operations and inverse operations to move other terms to the opposite side of the equal sign.
Simplify the equation.
Ratio and Proportion
Ratio: A comparison of two quantities, expressed as a:b.
Proportion: An equation stating that two ratios are equal, e.g., ab=cd; ba=dc.
Cross-Multiplication: For proportions, ad=bc.
Percentages
Definition: A way to express a number as a fraction of 100.
Conversion:
Percent to Decimal: ‘x’/100
Decimal to Percent: ‘x’ x 100
Example: 25% of 200 = 25/100 x 200 = 50
Direct and Inverse Proportion
Direct Proportion: Two quantities increase or decrease together, y=kx.
Inverse Proportion: One quantity increases while the other decreases, y=k/x.
Graphical Representation:
Direct: Straight line through the origin.
Inverse: Hyperbola.
Simple and Compound Inequality
Simple Inequality: Involves one inequality sign (e.g., x<5).
Compound Inequality: Involves two inequalities, connected by "and" or "or" (e.g., 2<x<5).
Solving: Treat each part of the inequality separately; ensure to reverse the inequality sign when multiplying or dividing by a negative number.
Quadratic Equations:
Trigonometry
Vertical line test
Determines if a graph represents a function:How to perform: Draw a vertical line through the graph
What it means: If the line intersects the graph at only one point, the graph is a function
What it doesn't mean: If the line intersects the graph at more than one point, the graph is not a function
Horizontal line test
Determines if a function is one-to-one:How to perform: Draw a horizontal line through the graph
What it means: If the line intersects the graph at only one point, the function is one-to-one
What it doesn't mean: If the line intersects the graph at more than one point, the function is not one-to-one
Exponent Laws:
1. Product of Powers Rule
When multiplying like bases, add the exponents.
am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n
2. Quotient of Powers Rule
When dividing like bases, subtract the exponents.
aman=am−n(as long as a≠0)\frac{a^m}{a^n} = a^{m-n} \quad \text{(as long as } a \ne 0)anam=am−n(as long as a=0)
3. Power of a Power Rule
When raising a power to another power, multiply the exponents.
(am)n=am⋅n(a^m)^n = a^{m \cdot n}(am)n=am⋅n
4. Power of a Product Rule
When raising a product to a power, apply the exponent to each factor.
(ab)m=am⋅bm(ab)^m = a^m \cdot b^m(ab)m=am⋅bm
5. Power of a Quotient Rule
When raising a fraction to a power, apply the exponent to numerator and denominator.
(ab)m=ambm(as long as b≠0)\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \quad \text{(as long as } b \ne 0)(ba)m=bmam(as long as b=0)
6. Zero Exponent Rule
Any non-zero base raised to the power of 0 is 1.
a0=1(as long as a≠0)a^0 = 1 \quad \text{(as long as } a \ne 0)a0=1(as long as a=0)
7. Negative Exponent Rule
A negative exponent means the reciprocal of the base raised to the positive exponent.
a−m=1am(as long as a≠0)a^{-m} = \frac{1}{a^m} \quad \text{(as long as } a \ne 0)a−m=am1(as long as a=0)
RATIONAL NUMBERS:
Numbers can be expressed as a/b, where a and b are integers (natural numbers, whole numbers, and integers), terminating decimals, and non-terminating decimals that repeat.
IRRATIONAL NUMBERS:
Surds (5√3), non-terminating decimals (continuing indefinitely and are non repeating, ex: π)
Length of arc and chord, perimeter, and area of sector and segment
Length of arc and chord - theta/360 pir^2
Perimeter of sector - theta/ 360*pi*r*2
Unit 1 – Numerical and Abstract reasoning
Number Systems
Rational and Irrational Numbers
Surds and Recurring Decimal
Laws of Exponents
a^m×a^n = a^{m+n}
a^m/a^n = a^{m-n} (as long as a ≠ 0)
(a^m)^n = a^{m \times n}
a^n/b^n=(a/b)^n
a^0=1 (as long as a ≠ 0)
a^{-m}=1/a^m (as long as a ≠ 0)
Estimation, Approximation and Significant Figures
Scientific Notation
Definition: a × 10^n where 1 ≤ |a| < 10 and n is an integer.
Example: 5000 = 5 × 10^3; 0.0004 = 4 × 10^{-4}
Operations:
Multiplication: (a × 10^m) × (b × 10^n) = ab × 10^{m+n}
Division: b × 10^n / a × 10^m = (b/a) × 10^{n−m}
Absolute Values
Definition: Distance from zero, denoted as |x|.
Properties:
|x| ≥ 0
|x| = x if x ≥ 0
|x| = −x if x < 0
Example: |3| = 3, |−3| = 3
Changing Subject of the Formula
Steps:
Identify the variable to isolate.
Use inverse operations to rearrange.
Simplify.
Ratio and Proportion
Ratio: a:b
Proportion: ab=cd; (a/b)=(c/d)
Cross-Multiplication: ad=bc
Percentages and Applications
Definition: A fraction of 100
Conversion: Percent to Decimal: x/100, Decimal to Percent: x × 100
Example: 25% of 200 = 25/100 × 200 = 50
Direct and Inverse Proportion
Direct: y = kx
Inverse: y = k/x
Graphical Representation: Direct - straight line through the origin; Inverse - hyperbola
Simple and Compound Inequality
Simple: e.g., x < 5
Compound: e.g., 2 < x < 5
Solving: Treat separately, reverse sign when multiplying/dividing by a negative.
Sequences
Arithmetic, Geometric, and Quadratic sequences
Unit 2 – Thinking with Models
Introduction to Functions
Vertical Line Test
Domain and Range
Linear Functions
Coordinate Geometry
Distance: √((x2 - x1)² + (y2 - y1)²)
Midpoint: ((x1 + x2)/2, (y1 + y2)/2)
Parallel Lines: Same slope, different y-intercepts.
Perpendicular Lines: Slopes that are negative reciprocals.
Equation of Straight Lines
Simultaneous Equations
Quadratic Functions
Vertex, Roots, Intercepts, Axis of Symmetry
Factorization
Quadratic Equations
Unit 3 – Spatial Reasoning
Pythagoras Theorem
Trigonometric Ratios in Right-Angle Triangles
Sine, Cosine, Tangent
Angle of Elevation and Depression
Sine and Cosine Rule
Area of Right and Non-Right Angle Triangles
Bearings
Perimeter and Area of 2-D Shapes
Length of Arc and Chord, Perimeter, and Area of Sector and Segment
Length of arc = θ/360 × 2πr
Area of sector = θ/360 × πr²
Volume and Surface Area of 3-D Shapes
Prisms, Pyramids, Sphere, and Combined Shapes
Unit 4 – Reasoning with Data
Data Processing
Mean, Median, Mode, Quartiles, Percentiles for Discrete and Grouped Data
Graphical Representations
Bo
Unit 1 – Numerical and Abstract reasoning
Number Systems
Rational and Irrational Numbers
Surds and Recurring Decimal
Laws of Exponents
a^m×a^n = a^{m+n}
a^m/a^n = a^{m-n} (as long as a ≠ 0)
(a^m)^n = a^{m \times n}
a^n/b^n=(a/b)^n
a^0=1 (as long as a ≠ 0)
a^{-m}=1/a^m (as long as a ≠ 0)
Estimation, Approximation and Significant Figures
Scientific Notation
Definition: a × 10^n where 1 ≤ |a| < 10 and n is an integer.
Example: 5000 = 5 × 10^3; 0.0004 = 4 × 10^{-4}
Operations:
Multiplication: (a × 10^m) × (b × 10^n) = ab × 10^{m+n}
Division: b × 10^n / a × 10^m = (b/a) × 10^{n−m}
Absolute Values
Definition: Distance from zero, denoted as |x|.
Properties:
|x| ≥ 0
|x| = x if x ≥ 0
|x| = −x if x < 0
Example: |3| = 3, |−3| = 3
Changing Subject of the Formula
Steps:
Identify the variable to isolate.
Use inverse operations to rearrange.
Simplify.
Ratio and Proportion
Ratio: a:b
Proportion: ab=cd; (a/b)=(c/d)
Cross-Multiplication: ad=bc
Percentages and Applications
Definition: A fraction of 100
Conversion: Percent to Decimal: x/100, Decimal to Percent: x × 100
Example: 25% of 200 = 25/100 × 200 = 50
Direct and Inverse Proportion
Direct: y = kx
Inverse: y = k/x
Graphical Representation: Direct - straight line through the origin; Inverse - hyperbola
Simple and Compound Inequality
Simple: e.g., x < 5
Compound: e.g., 2 < x < 5
Solving: Treat separately, reverse sign when multiplying/dividing by a negative.
Sequences
Arithmetic, Geometric, and Quadratic sequences
Unit 2 – Thinking with Models
Introduction to Functions
Vertical Line Test
Domain and Range
Linear Functions
Coordinate Geometry
Distance: √((x2 - x1)² + (y2 - y1)²)
Midpoint: ((x1 + x2)/2, (y1 + y2)/2)
Parallel Lines: Same slope, different y-intercepts.
Perpendicular Lines: Slopes that are negative reciprocals.
Equation of Straight Lines
Simultaneous Equations
Quadratic Functions
Vertex, Roots, Intercepts, Axis of Symmetry
Factorization
Quadratic Equations
Unit 3 – Spatial Reasoning
Pythagoras Theorem
Trigonometric Ratios in Right-Angle Triangles
Sine, Cosine, Tangent
Angle of Elevation and Depression
Sine and Cosine Rule
Area of Right and Non-Right Angle Triangles
Bearings
Perimeter and Area of 2-D Shapes
Length of Arc and Chord, Perimeter, and Area of Sector and Segment
Length of arc = θ/360 × 2πr
Area of sector = θ/360 × πr²
Volume and Surface Area of 3-D Shapes
Prisms, Pyramids, Sphere, and Combined Shapes
Unit 4 – Reasoning with Data
Data Processing
Mean, Median, Mode, Quartiles, Percentiles for Discrete and Grouped Data
Graphical Representations
Box and Whisker, Cumulative Frequency Curves, Histogram, Bar Charts, Stem and Leaf, Pie Charts
Measures of Dispersion
Range, Interquartile Range
Sets and Venn Diagram Operations (up to 3 sets)