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Class 9 Mathematics NCERT Exemplar Solutions
Chapter 1 Number Systems
This chapter explores different types of numbers — natural numbers, integers, rational numbers, and irrational numbers — which together form the real number system. Students learn to represent irrational numbers like $\sqrt{2}$ and $\pi$ on the number line using successive magnification. The laws of exponents for real numbers are introduced to simplify expressions such as $a^{m} \times a^{n} = a^{m+n}$.
Example: $\sqrt{3}$ is irrational, and $2^{3/2} = 2 \times \sqrt{2}$ shows how powers and roots relate.
Chapter 2 Polynomials
Students study algebraic expressions called polynomials, their degrees, coefficients, and terms. The chapter covers zeros of polynomials and the relationship between coefficients and zeros for quadratic polynomials. Identities like $(x + y)^2 = x^2 + 2xy + y^2$ are applied to simplify expressions.
Example: For $p(x) = x^2 - 5x + 6$, the zeros are $x = 2$ and $x = 3$ since $p(x) = 0$ gives $(x-2)(x-3)=0$.
Chapter 3 Coordinate Geometry
This chapter introduces the Cartesian system, where every point on a plane is represented by an ordered pair $(x, y)$. Students learn the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ and the section formula for dividing a line segment in a given ratio.
Example: The midpoint of the line joining $(2, 3)$ and $(4, 7)$ is $\left(\frac{2+4}{2}, \frac{3+7}{2}\right) = (3, 5)$.
Chapter 4 Linear Equation in Two Variables
Students learn to represent linear equations of the form $ax + by + c = 0$ on a coordinate plane and interpret their graphical solutions. Each equation represents a straight line. The concept of solutions as intersection points of lines is emphasized.
Example: For $x + y = 6$, points $(0,6)$, $(2,4)$, and $(3,3)$ satisfy the equation and lie on the same line.
Chapter 5 Introduction to Euclid's Geometry
This chapter discusses the foundation of geometry laid by Euclid through definitions, axioms, and postulates. It explains concepts like points, lines, and planes, and shows how Euclid’s five postulates lead to logical reasoning in geometry.
Example: “A straight line may be drawn joining any two points” is Euclid’s first postulate.
Chapter 6 Lines and Angles
Students explore relationships between intersecting and parallel lines, including alternate interior, corresponding, and vertically opposite angles. The sum of the angles on a straight line is $180^\circ$, and vertically opposite angles are equal.
Example: If two lines intersect, then $\angle 1 = \angle 3$ (vertically opposite angles).
Chapter 7 Triangles
This chapter introduces congruence criteria such as SAS, ASA, and RHS. The Pythagoras Theorem $(AC)^2 = (AB)^2 + (BC)^2$ is proved and applied to problems involving right triangles. Concepts of inequalities and medians are also introduced.
Example: In a right triangle with sides $3$ cm and $4$ cm, the hypotenuse is $\sqrt{3^2 + 4^2} = 5$ cm.
Chapter 8 Quadrilaterals
Students learn about different quadrilaterals and their properties, including parallelograms, rectangles, rhombuses, and squares. The diagonal properties and mid-point theorems are discussed. The chapter emphasizes geometric reasoning to prove relations between sides and angles.
Example: In a parallelogram $ABCD$, $AB \parallel CD$ and $AD \parallel BC$.
Chapter 9 Areas of Parallelograms and Triangles
This chapter derives and applies area formulas for parallelograms and triangles using geometrical proofs. It explains how two triangles on the same base and between the same parallels have equal areas. The area of a triangle is given by $A = \frac{1}{2} \times b \times h$.
Chapter 10 Circles
Students study chords, tangents, and theorems related to circles. Key results include that the perpendicular from the center to a chord bisects the chord and tangents drawn from an external point to a circle are equal in length.
Example: If two tangents $PA$ and $PB$ are drawn to a circle, then $PA = PB$.
Chapter 11 Construction
This chapter teaches constructions of triangles, bisectors, and perpendiculars using compass and ruler. Students learn to construct triangles when base, base angle, and sum/difference of remaining angles are given.
Example: Construct $\triangle ABC$ where base $BC = 5$ cm, $\angle B = 60^\circ$, and $\angle C = 45^\circ$.
Chapter 12 Heron’s Formula
This chapter provides Heron’s formula to calculate the area of a triangle when all three sides are known: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter. It is used to find areas of various shapes by dividing them into triangles.
Example: For sides $a=3$, $b=4$, $c=5$, $s=6$, area $A = \sqrt{6(6-3)(6-4)(6-5)} = 6$ sq. units.
Chapter 13 Surface Area and Volumes
Students learn formulas for surface areas and volumes of 3D solids such as cubes, cuboids, cylinders, cones, and spheres. For example, volume of a cylinder is $V = \pi r^2 h$ and that of a cone is $V = \frac{1}{3}\pi r^2 h$. Real-life problems link geometry to practical applications.
Chapter 14 Statistics and Probability
The chapter introduces data collection, organization, and presentation through frequency tables and graphs. Students calculate measures of central tendency — mean, median, and mode. Probability is defined as $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$.
Example: Probability of getting a 6 when rolling a die is $\frac{1}{6}$.
Design of the Question Paper - Set I
This section outlines the structure, types of questions, and marks distribution in the Class 9 Mathematics question paper, helping students prepare strategically.
Design of the Question Paper - Set II
A continuation of Set I, this section provides alternative question paper designs with variations in question formats and difficulty levels for better exam preparation.