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Class 8 Mathematics NCERT Exemplar Solutions
Chapter 1 Rational Numbers
This chapter explores the properties of rational numbers, which are numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Topics include closure, commutativity, associativity, and the distributive law. Students also learn about additive and multiplicative inverses and how to represent rational numbers on a number line.
For example, $\frac{2}{3} + \frac{4}{5} = \frac{10 + 12}{15} = \frac{22}{15}$, demonstrating addition under a common denominator.
Chapter 2 Data Handling
Data handling involves organizing, representing, and interpreting information. Students learn to construct frequency tables, bar graphs, and histograms. The concept of probability is introduced informally as the ratio of favourable outcomes to total outcomes: $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$.
Example: The probability of getting a head when tossing a coin once is $P(\text{Head}) = \frac{1}{2}$.
Chapter 3 Square-Square Root & Cube-Cube Root
Students explore perfect squares and cubes, and methods to find their roots. For example, $12^2 = 144$ and $\sqrt{144} = 12$. They also study the prime factorization method to find square roots and cube roots. The cube of $5$ is $5^3 = 125$, while $\sqrt[3]{125} = 5$.
Chapter 4 Linear Equation in One Variable
This chapter introduces equations of the form $ax + b = c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Students learn to isolate the variable by performing inverse operations on both sides.
Example: Solve $3x + 5 = 14$. Subtracting $5$ from both sides gives $3x = 9$, hence $x = 3$.
Chapter 5 Understanding Quadrilaterals & Practical Geometry
This section covers properties of polygons, especially quadrilaterals. Students learn about internal angles, diagonals, and special types such as parallelograms, rhombuses, and trapeziums. The sum of interior angles of an $n$-sided polygon is $(n - 2) \times 180^\circ$. Practical Geometry teaches constructions using a compass and ruler.
Chapter 6 Visualising the Solid Shapes
Students learn to represent 3D objects on 2D surfaces using nets, isometric views, and oblique sketches. Solids like cubes, cuboids, cones, and cylinders are studied with their faces, edges, and vertices.
For example, a cube has 6 faces, 8 vertices, and 12 edges, satisfying Euler’s formula $F + V - E = 2$.
Chapter 7 Algebraic Expressions, Identities & Factorisation
This chapter discusses expressions containing variables, constants, and operations. Students learn to add, subtract, and multiply algebraic expressions and use identities like $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$. Factorisation techniques include taking common factors and using identities.
Chapter 8 Exponents & Powers
Students explore powers with integer exponents and laws of exponents such as $a^m \times a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{mn}$. Scientific notation using powers of 10 is also introduced for expressing very large or very small numbers.
Chapter 9 Comparing Quantities
This chapter extends percentage, ratio, and proportion concepts to profit and loss, discount, simple and compound interest. For compound interest, the formula used is $A = P\left(1 + \frac{r}{100}\right)^n$ where $A$ is the amount, $P$ is principal, $r$ is rate, and $n$ is time in years.
Chapter 10 Direct & Inverse Proportions
Students study situations where one quantity varies directly or inversely with another. In direct proportion, $\frac{x_1}{y_1} = \frac{x_2}{y_2}$, and in inverse proportion, $x_1 y_1 = x_2 y_2$. Practical examples include speed-time relationships and cost-quantity problems.
Chapter 11 Mensuration
This chapter deals with the measurement of plane and solid figures. Students learn formulas for the surface area and volume of cubes, cuboids, cylinders, and cones. For example, the volume of a cylinder is $V = \pi r^2 h$, and the surface area is $A = 2\pi r(h + r)$.
Chapter 12 Introduction to Graphs
Students learn to plot points on the Cartesian plane and interpret graphs representing linear relationships. Coordinates are written as ordered pairs $(x, y)$, where $x$ and $y$ denote the distances from the axes.
Chapter 13 Playing with Numbers
This chapter revises divisibility rules, number patterns, and general forms of numbers. For instance, any two-digit number can be represented as $10x + y$, where $x$ is the tens digit and $y$ is the units digit. Students learn to analyze divisibility by 2, 3, 5, 9, and 11 using these representations.