Below are the PDF links for NCERT Solutions for Class 9 Maths for the 2023 session. You can also download separate PDFs for all the exercises. Our WEBSITE for Class 9 Maths are available in both Hindi and English for free. Remember that both NCERT Books and Exemplar problem books are essential for CBSE exams.

Additionally, we provide chapter-wise assignment test papers, previous year question papers from CBSE and other schools, chapter-wise tests, the syllabus for the academic year 2023, and various other online study materials.

## NCERT Solutions For Class 9 Maths

Class: | 9 |

Subject: | Maths – गणित |

Contents: | NCERT Solutions |

**NCERT Solutions for class 9 Maths Free PDF Download**

### CHAPTER 1: NUMBER SYSTEMS

Representation of terminating/non-terminating recurring decimals on the number line (and successive magnification method). Presentation of square roots of 2, 3 and other non-rational numbers. Rationalization of real numbers, laws of integral powers and rational exponents with positive real bases in *Number Systems*.

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**NCERT Solutions For Class 9 Maths **FAQ

## What Are The Theorems That May Be Asked For Proving In Exams For Class 9 Chapter 6: Lines And Angles?

In exams, you may be required to prove the following theorems, while the others are provided for application and understanding:

- When two lines intersect, the vertically opposite angles are equal.
- The total of the angles within a triangle always adds up to 180 degrees.

## Is There Any Theorem In Chapter 10: Quadrilateral To Prove?

Indeed, there’s just a single significant theorem (The diagonal divides a parallelogram into two congruent triangles) that could be requested for proof.

## How Many Theorems Are There In Chapter 9: Areas Of Parallelograms And Triangles To Prove?

There is only one theorem requested for proof, which is quite important for all examinations: “Parallelograms on the same base and between the same parallels have equal area.”

## Is There Only One Theorem In Chapter 10: Circle, Which Is Asked To Prove In Examinations?

- Equal chords of a circle subtend equal angles at the center.
- The angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle.

### Class 9 Maths Important Points Related To All Chapters.

**NCERT Solutions for Class 9**

NCERT Sols for Class 9 Maths, Science, Social Science (S. St.) and Hindi are given to study online. These are based on latest CBSE Syllabus for 2023.

- Class 9 Maths
- Class 9 Science (Notes)
- Class 9 Social Science
- Class 9 Hindi
- Class 9 English
- Class 9 Book
- Ask Your Questions

### Chapter 1: **Number Systems**

In this chapter, we dive into the world of numbers. It covers natural numbers, whole numbers, integers, and how to represent them on number lines. You’ll also learn how to find rational or irrational numbers between two values, which is a common question in CBSE exams. Additionally, you’ll explore converting irrational numbers into the form “p/q,” a frequent exam question. Understanding how to plot square root numbers on a number line is also crucial for exams.

### Chapter 2: **Polynomials**

Chapter 2 focuses on polynomials. You’ll encounter important concepts like the remainder theorem and factor theorem. This chapter covers polynomial terms, factors, and zeroes. You’ll also learn to identify monomials, binomials, and trinomials in polynomial expressions. Understanding the degree of a constant polynomial and other related concepts is crucial for tests. You’ll also classify linear, quadratic, and cubic polynomials, which are commonly asked in exams. Factorization techniques like middle-term splitting and using identities are also explained.

### Chapter 3: **Coordinate Geometry**

Chapter 3 introduces coordinate geometry, which helps you locate objects using the x-axis and y-axis. You’ll learn to plot points on a Cartesian plane and find the coordinates of a point using a two-dimensional coordinate system. Concepts like abscissa, ordinate, origin, quadrants, and more are explained.

### Chapter 4: **Linear Equations In Two Variables**

This chapter delves into linear equations with two variables. You’ll explore techniques to find multiple solutions to equations, allowing you to draw lines on graphs. Understanding the standard form of linear equations and terms like coefficients and constants is essential. The chapter also shows real-life applications of linear equations.

### Chapter 5: **Introduction To Euclid’s Geometry**

Chapter 5 introduces geometry, its origin, and Euclid’s fundamental contributions. Postulates and axioms, especially Euclid’s Fifth postulate, are important topics. Definitions of geometric terms like point, line, surface, curves, and more are also covered.

### Chapter 6: **Lines And Angles**

This chapter explores concepts like line segments, rays, angles, vertex, collinear, non-collinear points, and various types of angles. You’ll encounter acute, right, obtuse, straight angles, as well as adjacent, reflex, complementary, supplementary, and vertically opposite angles. Theorem on Vertically Opposite angles is emphasized, and Exercise 6.1 deals with this topic.

### Chapter 7: **Triangles**

Chapter 7 focuses on triangle congruency criteria. You’ll work with SAS and ASA criteria of congruence, as well as RHS and SSS criteria. The chapter also involves inequalities of sides and angles.

### Chapter 8: **Quadrilaterals**

Chapter 8 explores the angle sum property of quadrilaterals, types of quadrilaterals like rectangles, trapeziums, squares, rhombuses, parallelograms, and their properties. Mid-Point Theorem and its converse are important parts of this chapter.

### Chapter 9: **Areas Of Parallelograms And Triangles**

In this chapter, you’ll determine the area of parallelograms and triangles using theorems and corollaries. It also explains the relationship between triangles and parallelograms when they share the same base and are between the same parallel lines.

### Chapter 10: **Circles**

Chapter 10 covers circles and their components like segments, arcs, semicircles, chords, diameter, minor and major sectors. You’ll explore theorems related to circles, including perpendicular bisectors and equidistant chords.

### Chapter 11: **Constructions**

The Constructions chapter teaches you how to create angles and triangles using a ruler and compass. You’ll construct various angles and triangles using these tools.

### Chapter 12: **Heron’s Formula**

Chapter 12 introduces Heron’s Formula, which helps you find the area of triangles and quadrilaterals by splitting them into triangles. It provides practical applications of Heron’s Formula related to everyday life.

### Chapter 13: **Surface Areas And Volumes**

This chapter covers mensuration concepts. You’ll calculate surface areas and volumes of solid figures like cubes, cuboids, cones, cylinders, spheres, and hemispheres. Various exercises illustrate these concepts.

### Chapter 14: **Statistics**

Chapter 14 deals with statistics, including primary and secondary data, data representation using histograms, bar charts, and polygons. It also covers central tendency measures like mean, mode, and median.

### Chapter 15: **Probability**

Chapter 15 focuses on probability, where you’ll calculate probabilities using favorable and total outcomes. It includes examples related to coins, cards, dice, and real-life experiences. Probability is a fundamental concept for future classes as well.

### Are the statements true or false? Give your answer with reasons. Every irrational number is a real number.

True, because the collection of all rational and irrational numbers is a real number.

### Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Diagonals of parallelogram bisect each other.

Therefore, PO = OR and SO = OQ

In ΔPQS, PO is median. [∵ SO = OQ]

Hence, ar(PSO) = ar(PQO) … (1)

[∵ A median of a triangle divides it into two triangles of equal areas.]

Similarly, in ΔPQR, QO is median. [∵ PO = OR]

Hence, ar(PQO) = ar(QRO) … (2)

And in ΔQRS, RO is median. [∵ SO = OQ]

Hence, ar(QRO) = ar(RSO) … (3)

From the equations (1), (2) and (3), we get

ar(PSO) = ar(PQO) = ar(QRO) = ar(RSO)

Hence, in parallelogram PQRS, diagonals PR and QS divide it into four triangles in equal area.

### A circle has only finite number of equal chords. True or False?

False.

Because, there are infinite number of equal chords in a circle.

### Find the following values without direct multiplication: 103×107

103 × 107

= (100 + 3)(100 + 7)

= (100)^2 + (3 + 7)100 + 3 × 7

[∵〖(x + a)(x + b) = x〗^2 + (a + b)x + ab]

= 10000 + 1000 + 21

= 11021

### Name each part of the plane made up of x-axis and y-axis lines.

चतुर्थांश

### If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a

Given equation of line:

3y = ax + 7.

Putting x = 3 and y = 4,

we have, 3 × 4 = a × 3 + 7

⇒ 12 = 3a + 7

⇒ 12 – 7 = 3a

⇒ a = 5/3

### Does Euclid’s fifth postulate justify the existence of parallel lines? Explain.

Yes, Euclid’s fifth postulate provides justification for the existence of parallel lines. When a third line intersects two lines in a manner where the sum of the interior angles on the same side is less than 180°, it implies that both lines intersect each other. However, if the sum of these angles equals 180°, then the lines do not intersect each other, confirming that they are parallel.

### What do you mean by collinear points or non-collinear points?

If three or more points lie on the same line, they are called collinear points; otherwise they are called non-collinear points.

### In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.

Draw the bisectors of ∠A, ∠B and ∠C, which intersect each other at O. Point O is equidistant from the three sides of ΔABC i.e.

OP = OQ = OR.

### ABCD is a quadrilateral in which P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. AC is one of its diagonals. Show that SR || AC and SR=1/2 AC.

In triangle ACD,

S is positioned at the midpoint of side DA (as given).

R is located at the midpoint of side DC (as given).

Therefore, we can conclude that SR is parallel to AC, and the length of SR is half that of AC, as per the Mid-point theorem.

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