1. **Number**: Numbers are the basic building blocks of mathematics. They include natural numbers, whole numbers, integers, rational and irrational numbers. Example: Simplify $\frac{6}{8}$. Step: Find the greatest common divisor (GCD) of 6 and 8, which is 2. Simplify: $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$. 2. **Algebra and graphs**: Algebra involves variables and constants to form expressions and equations. Graphs visually represent these relationships. Example: Solve $2x + 3 = 7$. Step: Subtract 3 from both sides: $2x = 4$. Divide both sides by 2: $x = 2$. 3. **Coordinate geometry**: Studies geometric figures using coordinate systems. Example: Find the distance between points $(1,2)$ and $(4,6)$. Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Calculation: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. 4. **Geometry**: Deals with shapes, sizes, and properties of space. Example: Find the area of a triangle with base 5 and height 3. Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. Calculation: $\frac{1}{2} \times 5 \times 3 = 7.5$. 5. **Mensuration**: Measurement of geometric figures. Example: Find the volume of a cube with side length 4. Formula: $V = s^3$. Calculation: $4^3 = 64$. 6. **Trigonometry**: Study of relationships between angles and sides of triangles. Example: Find $\sin 30^\circ$. Value: $\sin 30^\circ = \frac{1}{2}$. 7. **Transformations and vectors**: Transformations change figures; vectors have magnitude and direction. Example: Vector addition: $\vec{A} = (2,3)$, $\vec{B} = (1,4)$. Sum: $\vec{A} + \vec{B} = (2+1, 3+4) = (3,7)$. 8. **Probability**: Measure of likelihood of an event. Example: Probability of rolling a 3 on a fair six-sided die. Calculation: $\frac{1}{6}$. 9. **Statistics**: Collection, analysis, interpretation of data. Example: Find mean of 2, 4, 6, 8. Calculation: $\frac{2+4+6+8}{4} = \frac{20}{4} = 5$.