NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials (PDF) – 2026-27

NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials provide clear, step-by-step answers to every exercise and in-text question from the chapter Introduction to Linear Polynomials of the NCERT textbook Ganita Manjari. Prepared by subject experts as per the latest NCERT (CBSE) syllabus for 2026-27, these NCERT Solutions for Class 9 Maths help you understand each concept, write exam-ready answers, and check your own solutions. You can read them online below or download the free Class 9 Maths Chapter 2 question-answer PDF.

NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials

• Class: Class 9
• Subject: Maths
• Chapter: Chapter 2 – Introduction to Linear Polynomials
• Textbook: Ganita Manjari (NCERT)
• Study material: NCERT Solutions – questions with answers, free PDF

These solutions answer all the exercise questions of Chapter 2 Introduction to Linear Polynomials — including the in-text questions, short-answer and long-answer questions, and activities — with complete explanations so you can follow the method, not just the final answer. Read the full solutions below.

NCERT Solutions Class 9 Maths Chapter 2 Introduction to Linear Polynomials View Download

NCERT Solutions for Class 9 Maths Chapter 2 PDF Download

You can read the NCERT Solutions for Class 9 Maths Chapter 2 online above, or download the complete question-answer PDF to study Introduction to Linear Polynomials offline at any time.

---

NCERT Solutions for Class 9 Maths Chapter 2 PDF Download Link – Click Here to Download Solutions PDF

---

Questions Covered in This Chapter

These NCERT Solutions answer all 65 questions of this chapter. The questions solved are:

1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. How is it different from the algebraic expression in Example 1?
3. A wire of length 20 cm is bent in different ways to form rectangles. For example, we can have a rectangle with length 7 cm and width 3 cm. We can also have one of length 5.5 cm and width 4.5 cm. (Think of a few more ways of forming such rectangles.)
4. Can you identify the terms, variables and coefficients of this algebraic expression?
5. Can you point out any similarity or difference between the algebraic expressions obtained in Examples 1 and 3?
6. Find the degrees of the following polynomials: (i) 2x² – 5x + 3 (ii) y³ + 2y – 1 (iii) – 9 (iv) 4z – 3
7. Write polynomials of degrees 1, 2 and 3.
8. What are the coefficients of x² and x³ in the polynomial x⁴ – 3x³ + 6x² – 2x + 7?
9. What is the coefficient of z in the polynomial 4z³ + 5z² – 11?
10. What is the constant term of the polynomial 9x³ + 5x² – 8x –10?
11. Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm?
12. If a player paid ₹750, how many matches did he play?
13. We have learnt that to evaluate the value of an algebraic expression, we substitute a value of the variable in the given expression. Consider Example 3, where the wire is bent to form a rectangle. Here, the area of the rectangle, 10x – x², is a function of x. Can you interpret this as an input-output process? What value does the expression take when x = 6 cm?
14. Find the value of the linear polynomial 5x – 3 if: (i) x = 0 (ii) x = –1 (iii) x = 2
15. Find the value of the quadratic polynomial 7s² – 4s + 6 if: (i) s = 0 (ii) s = –3 (iii) s = 4
16. The present age of Salil’s mother is three times Salil’s present age. After 5 years, their ages will add up to 70 years. Find their present ages.
17. The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
18. Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total ₹88, how many coins does she have of each type?
19. A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
20. If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?
21. Predict the number of squares in the next three stages of the pattern and write the sequence of numbers up to Stage 7 of the pattern.
22. Using the expression 2n – 1, can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles?
23. What amount will be left on the 15th day? How many days will it take for the entire amount to be spent?
24. For how many km will the fare be ₹130?
25. A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the nth month.
26. A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, … hours? Find a linear expression to represent the number of members at the end of the nth hour.
27. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.
28. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.
29. Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.
30. What is the cost for travelling 15 km? For how many kilometres will the cost of the journey be ₹700?
31. What will be the height of the water at the end of 5 months?
32. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (i) Find the height after 7 months. (ii) Make a table of values for t varying from 0 to 10 months and show how the height, h, increases every month. (iii) Find an expression that relates h and t, and explain why it represents linear growth.
33. A mobile phone is bought for ₹10,000. Its value decreases by ₹800 every year. (i) Find the value of the phone after 3 years. (ii) Make a table of values for t varying from 0 to 8 years and show how the value of the phone, v, depreciates with time. (iii) Find an expression that relates v and t, and explain why it represents linear decay.
34. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (i) Find the population of the village after 6 years. (ii) Make a table of values for t varying from 0 to 10 years and show how the population, P, increases every year. (iii) Find an expression that relates P and t, and explain why it represents linear growth.
35. A telecom company charges ₹600 for a certain recharge scheme. This prepaid balance is reduced by ₹15 each day after the recharge. (i) Write an equation that models the remaining balance b(x) after using the scheme for x days. Explain why it represents linear decay. (ii) After how many days will the balance run out? (iii) Make a table of values for x varying from 1 to 10 days and show how the balance b(x), reduces with time.
36. Can you guess what the numbers 20 and 150 in the equation y = 20x + 150 represent?
37. A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was ₹400. When she accessed 14 modules, her bill was ₹500. If the monthly bill y depends on the number of modules accessed, x, according to the relation y = ax + b, find the values of a and b.
38. A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was ₹800. When she used it for 15 hours, her bill was ₹1100. If the monthly bill y depends on the hours of the use of the badminton court, x, according to the relation y = ax + b, find the values of a and b.
39. Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit. (Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this information to find a and b, and thus, the linear relationship between °C and °F.)
40. Identify other points on the line by completing the following table. (x: 1, 2, 5, 7, 9, 12, 20; y: 3, …, …, 15, …, …, …)
41. Let us plot the points (– 3, 6), (– 2, 4), (0, 0), (1, – 2), (2, – 4), (3, – 6) in the coordinate plane on a graph paper as shown in Fig. 2.7. Join the points (– 3, 6) and (3, – 6) using a ruler. Doing so, observe that all five points lie on a straight line. Can you guess the equation of this line by looking at the relationship between the x and y coordinates of each point?
42. Draw the graphs of y = ½x, y = x, y = 2x by selecting suitable points on these lines. (Hint: In order to graph y = ½x, we could take the points (0, 0) and (4, 2). Can you verify that these lie on the line?)
43. Fig. 2.9 shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = ax, a > 0 as a varies? What happens when a > 1 and when a < 1? (Hint: You may also plot the equations y = 3x and y = ⅓x on the same axes.)
44. Fig. 2.11 shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = – ax, a > 0, as a varies? What will happen when a > 1 and when a < 1?
45. Differentiate between the graphs of the equations y = 3x + 1, and y = –3x + 1.
46. Does this help you to conclude anything about the linear equation y = ax + b when a is fixed but b varies? (Hint: In these equations a = 2, and b takes the values –1, 1 and 5, respectively.)
47. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. (i) y = 4x, y = 2x, y = x
48. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. (ii) y = – 6x, y = – 3x, y = – x
49. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. (iii) y = 5x, y = –5x
50. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. (iv) y = 3x – 1, y = 3x, y = 3x + 1
51. Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’. (v) y = –2x – 3, y = –2x, y = 2x + 3
52. Write a polynomial of degree 3 in the variable x, in which the coefficient of the x² term is –7.
53. Find the values of the following polynomials at the indicated values of the variables. (i) 5x² – 3x + 7 if x = 1 (ii) 4t³ – t² + 6 if t = a
54. If we multiply a number by 5/2 and add 2/3 to the product, we get –7/12. Find the number.
55. A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?
56. If you have ₹800 and you save ₹250 every month, find the amount you have after (i) 6 months (ii) 2 years. Express this as a linear pattern.
57. The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.
58. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (i) y = –3x + 4 (ii) 2y = 4x + 7 (iii) 5y = 6x – 10 (iv) 3y = 6x – 11. Are any of the lines parallel?
59. If the temperature of a liquid can be measured in Kelvin units as x K and in Fahrenheit units as y °F, the relation between the two systems of measurement of temperature is given by the linear equation y = (9/5)(x – 273) + 32. (i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K. (ii) If the temperature is 158 °F, then find the temperature in Kelvin.
60. The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work w and distance d), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.
61. The graph of a linear polynomial p(x) passes through the points (1, 5) and (3, 11). (i) Find the polynomial p(x). (ii) Find the coordinates where the graph of p(x) cuts the axes. (iii) Draw the graph of p(x) and verify your answers.
62. Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that: (i) p(0) = 5. (ii) The polynomial p(x) – q(x) cuts the x-axis at (3, 0). (iii) The sum p(x) + q(x) is equal to 6x + 4 for all real x. Find the polynomials p(x) and q(x).
63. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages? (ii) Complete the following table. (iii) Find a rule to determine the number of matchsticks required for the nth stage. (iv) How many matchsticks will be required for the 15th stage of the pattern? (v) Can 200 matchsticks form a stage in this pattern? Justify your answer.
64. Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that: (i) The graph of p(x) passes through the points (2, 3) and (6, 11). (ii) The graph of q(x) passes through the point (4, –1). (iii) The graph of q(x) is parallel to the graph of p(x). Find the polynomials p(x) and q(x). Also, find the coordinates of the point where these lines meet the x-axis.
65. What do all linear functions of the form f(x) = ax + a, a > 0, have in common?

Chapter at a Glance

• An algebraic expression is built from numbers, variables and operation signs. In 4 x + 5 y + 3 the parts 4 x , 5 y and 3 are the terms , x and y are the variables , 4 and 5 are the coefficients and 3 is the constant .
• A univariate polynomial (one-variable polynomial) uses a single variable and its powers. The highest power present is its degree : 5 y ³ + y ² + 2 y − 1 is cubic (degree 3), x ² + 5 x + 1 is quadratic (degree 2), 3 z + 7 is linear (degree 1), and a non-zero constant such as 8 = 8 x ⁰ has degree 0.
• A polynomial is an input–output machine: put a value of the variable in, get a number out. For 2 x + 3, an input of 4 gives 11 and an input of −6 gives −9. This is what is meant by calling 2 x + 3 a function of x .
• The defining behaviour of a linear polynomial: equal steps in the variable give equal steps in the value . That constant step is what makes 1, 3, 5, 7, … (from 2 n − 1) a linear pattern . Linear growth adds a fixed amount each interval; linear decay subtracts one.
• A linear relationship between two quantities is written y = ax + b . Two known pairs ( x , y ) are enough to pin down a and b , because they give two equations in those two unknowns.
• Graphically, a is the slope and b the y-intercept : the line meets the y-axis at (0, b ). Growth means a positive slope, decay a negative slope. Changing b with a fixed slides the line up or down without tilting it — so lines with equal slope and different intercepts are parallel .

How to Download NCERT Solutions for Class 9 Maths Chapter 2 PDF

Follow these simple steps to get the Introduction to Linear Polynomials questions-and-answers PDF from Ganita Manjari.

1. Search NCERT Solutions for Class 9 Maths Chapter 2 aglasem and open this page.
2. Read the exercise questions with answers for Introduction to Linear Polynomials shown above.
3. Click the Download PDF link to save the Introduction to Linear Polynomials solutions to your device.

NCERT Solutions for Class 9 Maths – All Chapters

There are more chapters to study besides Introduction to Linear Polynomials in Maths. Here are the NCERT Solutions for all chapters of Class 9 Maths.

• Chapter 1 Orienting Yourself the Use of Coordinates
• Chapter 2 Introduction to Linear Polynomials
• Chapter 3 The World of Numbers
• Chapter 4 Exploring Algebraic Identities
• Chapter 5 I M Up and Down and Round and Round
• Chapter 6 Measuring Space Perimeter and Area
• Chapter 7 The Mathematics of Maybe Introduction to Probability
• Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions

NCERT Solutions for Class 9 – All Subjects

Just like Chapter 2 of Maths, you can get the exercise questions with answers for every other subject of Class 9. Here are the NCERT Solutions for all subjects of Class 9.

• English
• Hindi
• Maths
• Sanskrit
• Science
• Social Science

NCERT Solutions for Class 9 Maths Chapter 2 – An Overview

The key highlights of this study material are as follows.

Aspects	Details
Class	Class 9
Subject	Maths
Chapter Number	Chapter 2
Chapter Name	Introduction to Linear Polynomials
Book Name	Ganita Manjari
Book By	NCERT (National Council of Educational Research and Training)
Educational Resource Here	NCERT Solutions of Class 9 Maths Chapter 2 for all exercises
More Questions Answers of This Subject	NCERT Solutions for Class 9 Maths
Download Book Chapter	NCERT Book Class 9 Maths
All Questions Answers For This Class	NCERT Solutions for Class 9
Complete Solutions	NCERT Solutions

NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials – FAQs

What are the NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials?

They are the complete, step-by-step answers to all the exercise and in-text questions of Chapter 2 Introduction to Linear Polynomials from the NCERT Class 9 Maths textbook Ganita Manjari, written by experts as per the latest NCERT syllabus.

How can I download the Class 9 Maths Chapter 2 solutions PDF for free?

Open this page on aglasem, read the Introduction to Linear Polynomials questions with answers, and click the “Download Solutions PDF” link. The Class 9 Maths Chapter 2 NCERT Solutions PDF is completely free to download.

Are these NCERT Solutions as per the latest 2026-27 syllabus?

Yes. The NCERT Solutions for Class 9 Maths Chapter 2 are based on the latest NCERT textbook Ganita Manjari and the current 2026-27 CBSE syllabus, so the questions and answers match what you study in class.

Where can I get NCERT Solutions for the other chapters of Class 9 Maths?

You can find the answers to every chapter on the NCERT Solutions for Class 9 Maths page, and solutions for every subject on the NCERT Solutions for Class 9 page.

How do NCERT Solutions help in exam preparation?

They show the correct method to solve each question, help you write answers the way they are expected in exams, let you check and correct your own work, and save revision time — which together improve your marks in Class 9 Maths.

If you have any queries on NCERT Solutions for Class 9 Maths Chapter 2 Introduction to Linear Polynomials, then please ask in the comments below.