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CBSE Class 9 Mathematics Chapter 2 Introduction to Linear Polynomials Worksheet with Answers

Linear polynomials are the simplest algebraic expressions where the variable appears to the power of 1. Chapter 2 of NCERT Class 9 Mathematics introduces students to variables, constants, coefficients, and how they combine to form linear polynomials. This worksheet tests your understanding of finding zeros, graphing straight lines, identifying slope and y-intercept, and applying linear models to real-world scenarios like costs, heights, and growth patterns. Print this sheet, attempt all sections, and check your answers with the detailed key at the end.

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Key takeaways

  • ✓Linear polynomials have degree 1 and general form ax + b, where a ≠ 0; the zero is found by solving ax + b = 0, giving x = −b/a.
  • ✓The coefficient of the variable in a linear polynomial represents the slope or constant rate of change in real-world applications.
  • ✓Linear growth means adding a fixed quantity at regular intervals; linear decay means subtracting a fixed quantity at regular intervals.
  • ✓In the linear relationship y = ax + b, 'a' is the slope (rate of change) and 'b' is the y-intercept (value when x = 0).
  • ✓Two linear graphs with the same slope are parallel; if they have different slopes, they intersect at exactly one point.
  • ✓Real-world problems—telecom bills, water draining, plant growth—are modelled using linear polynomials and solved by finding zeros or evaluating expressions.
  • ✓Graphing a linear polynomial requires plotting just two points and drawing a straight line through them; positive slope tilts upward, negative slope tilts downward.

Quick Chapter Recap: Introduction to Linear Polynomials

Chapter 2 builds foundational algebraic skills by introducing variables, constants, and coefficients. A linear polynomial has the form ax + b where a is non-zero, making the highest power of the variable equal to 1. The chapter emphasizes understanding the zero of a polynomial—the value of the variable that makes the polynomial equal to zero. For a linear polynomial ax + b, the zero is x = −b/a. Students also learn to distinguish between algebraic expressions and polynomials, classify polynomials by degree, and recognise linear, quadratic, and cubic forms. Practical applications include linear growth (constant addition) and linear decay (constant subtraction), such as water draining from a tank or a phone's value depreciating each year. The chapter concludes with graphing linear relationships y = ax + b, where 'a' represents the slope and 'b' the y-intercept. Understanding these concepts enables students to model and solve everyday problems using algebraic thinking. Key NCERT examples include Raju's pens and pencils expression, Bela's pocket money running out, and telecom billing structures. Mastery of linear polynomials sets the stage for solving linear equations and systems in later chapters.
  • Linear polynomial general form: ax + b, where a ≠ 0
  • Degree of a linear polynomial is always 1
  • Zero of ax + b is found by solving ax + b = 0, giving x = −b/a
  • Linear growth: constant positive change; linear decay: constant negative change
  • In y = ax + b, 'a' is slope (rate of change), 'b' is y-intercept
  • Graphs of linear polynomials are always straight lines

Section A: Multiple Choice Questions (MCQs)

This section contains 6 multiple-choice questions designed to test your conceptual clarity and quick recall of definitions, formulas, and properties of linear polynomials. Each question has four options; choose the correct one. Topics include identifying linear polynomials, computing zeros, recognising slope and y-intercept, and understanding linear growth and decay patterns. These MCQs mirror the style of CBSE board objective questions and help build confidence for internal assessments and competitive exams. Attempt all six questions, mark your answers clearly, and cross-check with the answer key at the end of this worksheet. Pay attention to wording—terms like 'coefficient', 'constant term', and 'zero' are used precisely as per NCERT terminology. Time allocation: approximately 10 minutes for this section. Remember that every linear polynomial ax + b has exactly one zero when a is non-zero, and constant polynomials like 5 or −7 have degree 0 and no zero. Understanding these fine distinctions will help you avoid common mistakes in the board exam and internal tests throughout the year.
  • Q1. Which of the following is a linear polynomial? (a) 5x² + 2, (b) 3x + 7, (c) 8, (d) x³ − 1
  • Q2. The zero of the polynomial 2x − 10 is: (a) 2, (b) 5, (c) −5, (d) 10
  • Q3. In the linear relationship y = 4x + 9, the y-intercept is: (a) 4, (b) 9, (c) 0, (d) 13
  • Q4. If the height of water in a tank is given by h(t) = 5 − 0.25t, after how many months will the tank be empty? (a) 10, (b) 20, (c) 5, (d) 25
  • Q5. The slope of the line represented by y = −3x + 2 is: (a) 2, (b) −3, (c) 3, (d) −2
  • Q6. Which statement is true about linear polynomials? (a) They always have two zeros, (b) Their graphs are parabolas, (c) They have degree 1, (d) They cannot have negative coefficients

Section B: Fill in the Blanks

Fill in the blanks with the correct word, number, or expression. This section assesses your ability to recall key terminology and apply formulas in straightforward contexts. Each blank tests one precise concept: the definition of a variable, the formula for the zero of a linear polynomial, the meaning of slope, the classification of polynomials by degree, or the interpretation of the y-intercept in a real-world scenario. Read each sentence carefully and ensure your answer fits grammatically and mathematically. These questions are designed to reinforce NCERT definitions and the exact language used in the textbook. For example, the zero of a polynomial p(x) is the value of x for which p(x) equals zero, not just 'the solution'. Use standard mathematical notation where needed. Time allocation: approximately 8 minutes for this section. After completing, verify each answer with the answer key provided at the end. Consistent practice with fill-in-the-blank questions sharpens your precision in exams, where partial marks depend on exact terminology and correct symbolic representation of concepts learned in Class 9 Mathematics Chapter 2.
  • Q7. A polynomial of degree 1 is called a __________ polynomial.
  • Q8. The zero of the polynomial ax + b is given by the formula x = __________.
  • Q9. In the expression 7y + 12, the coefficient of y is __________ and the constant term is __________.
  • Q10. If a linear relationship has the form y = mx + c, then m represents the __________ and c represents the __________.
  • Q11. The graph of every linear polynomial is a __________ line.

Section C: True or False

Read each statement carefully and write True or False. These five statements test common misconceptions and deeper understanding of linear polynomials, their properties, graphing behaviour, and real-world modelling. Some statements are deliberately worded to trap careless readers, so think critically before answering. For example, a constant polynomial like 7 has degree 0, not degree 1, and it has no zero—students often confuse this. Similarly, parallel lines share the same slope but have different y-intercepts; if both slope and intercept are identical, the lines coincide completely. Understanding these subtleties is crucial for CBSE board exams, where true/false or assertion-reason questions appear frequently. Time allocation: approximately 7 minutes for this section. After marking True or False, check the answer key for explanations. If you marked an answer incorrectly, revisit the relevant section in your NCERT textbook or notes. Consistent practice with such statements builds exam confidence and prevents silly errors. These questions align with the NCERT Class 9 Mathematics curriculum and reflect the kind of reasoning expected in internal assessments and board papers across all CBSE schools in India.
  • Q12. The polynomial 9 is a linear polynomial.
  • Q13. Every linear polynomial has exactly one zero.
  • Q14. If two linear graphs have the same slope but different y-intercepts, they are parallel and will never intersect.
  • Q15. The zero of the polynomial −4x + 20 is x = −5.
  • Q16. In a linear decay model, the coefficient of the variable is always negative.

Section D: Short Answer Questions (2-3 marks each)

Answer the following five questions in 2-3 sentences or show clear working. Each question is designed to test application of formulas, interpretation of word problems, and step-by-step algebraic manipulation. You may be asked to find the zero of a given polynomial, determine the value of a linear expression at a specific point, set up a linear model from a word problem, or identify slope and y-intercept from an equation. Write neatly, show all working, and box or underline your final answer. Marks are awarded for method as well as the correct answer, so even if you make a small arithmetic error, clear working can earn you partial credit. Time allocation: approximately 25 minutes for this section. These questions mirror CBSE board exam short-answer style and are directly based on NCERT examples and exercise problems from Chapter 2. Practice writing concise, logical steps: state the formula, substitute values, simplify, and conclude. For instance, if asked to find the zero of 6x − 18, write: '6x − 18 = 0 ⇒ 6x = 18 ⇒ x = 3'. Verify by substituting back. This habit reduces errors and improves exam performance. Refer to Class 9 Mathematics solutions and notes if you need to revise any concept before attempting these questions.
  • Q17. Find the zero of the polynomial 5x + 25.
  • Q18. A telecom plan charges ₹150 as a fixed fee and ₹8 per GB of data used. Write a linear polynomial representing the total bill for using g GB of data.
  • Q19. If the polynomial p(x) = 3x − 12, find p(5).
  • Q20. The height of a candle is given by h(t) = 20 − 2t cm, where t is time in hours. After how many hours will the candle burn out completely?
  • Q21. Identify the slope and y-intercept of the linear relationship y = −7x + 14.

Section E: Long Answer and HOTS Questions (4-5 marks each)

This section contains three higher-order thinking skills (HOTS) questions that require multi-step reasoning, setting up equations from word problems, graphing, or comparing linear relationships. Each question is worth 4-5 marks and tests your ability to synthesize concepts from different parts of the chapter. You may need to form a linear polynomial from given conditions, solve for unknowns, interpret graphs, or apply the slope-intercept form y = ax + b in unfamiliar contexts. Show all steps clearly, including forming equations, solving them, and interpreting the result in the context of the problem. Time allocation: approximately 30 minutes for this section. These questions resemble CBSE board exam long-answer problems and require careful reading and logical structuring of your solution. For example, if asked to find the relationship between two quantities given two data points, you will set up two simultaneous equations and solve for the slope and intercept. Always state what your variables represent, write equations clearly, and conclude with a sentence explaining the result. NCERT Class 9 Mathematics emphasizes such problem-solving skills, and CBSETUTOR.ai offers 24×7 AI-powered doubt solving where students can upload photos of such questions and get instant step-by-step explanations for just ₹999/month across all subjects for classes 6 to 12, with a 3-day free trial available.
  • Q22. A plant is 1.5 feet tall today and grows at a constant rate of 0.4 feet per month. (i) Write a linear polynomial h(t) representing its height after t months. (ii) Find its height after 6 months. (iii) After how many months will it reach 5.5 feet?
  • Q23. A mobile phone costs ₹12,000 and depreciates by ₹1,000 every year. (i) Write a linear polynomial v(t) for its value after t years. (ii) Find its value after 4 years. (iii) When will its value be ₹6,000?
  • Q24. Two friends start walking from the same point. Friend A's distance is given by d₁ = 5t + 10 km, and Friend B's distance is d₂ = 3t + 20 km, where t is time in hours. (i) Find when they will be at the same distance from the start. (ii) What is that distance?

Section F: Case Study Question (4 marks)

Read the following real-world scenario carefully and answer the sub-questions that follow. Case studies integrate mathematics with everyday contexts and test your ability to extract data, form equations, and interpret results. This format is increasingly popular in CBSE board exams and internal assessments. The scenario below involves a water tank, linear decay, and zero calculation. You will apply the concepts of linear polynomials, zeros, and rate of change to answer four sub-parts, each worth 1 mark. Show working for computational questions and write short explanations where needed. Time allocation: approximately 10 minutes for this section. Case studies develop critical thinking and demonstrate how Class 9 Mathematics is relevant beyond the classroom. They also help you practice reading comprehension in a mathematical context—an essential skill for competitive exams and higher studies. After attempting, cross-check each sub-question with the detailed answer key provided at the end of this worksheet to ensure you have interpreted the data correctly and applied the right formulas. CBSETUTOR.ai helps students with such integrated questions through personalized AI tutoring at a flat ₹999/month for all classes, making premium support affordable for every student.

Answer Key: Section A (MCQs)

Below are the correct answers with brief explanations for each multiple-choice question. Cross-check your responses and understand the reasoning behind each correct option. If you chose a wrong answer, revisit the relevant concept in your NCERT textbook or Class 9 Mathematics notes. Understanding why other options are incorrect is as important as knowing the right answer, because it prevents pattern-based guessing and builds genuine conceptual clarity. For example, in Q1, option (a) is quadratic (degree 2), option (c) is a constant polynomial (degree 0), and option (d) is cubic (degree 3); only option (b) has degree 1, making it linear. Similarly, in Q4, setting h(t) = 0 and solving for t tests your ability to find the zero of a linear polynomial in a real-world context. Each explanation references NCERT terminology to align with the Class 9 Mathematics curriculum used across all CBSE schools in India. These explanations also prepare you for assertion-reason and objective-type questions in board exams and competitive tests.
  • A1. (b) 3x + 7. Explanation: A linear polynomial has degree 1. Here, the highest power of x is 1. Option (a) is quadratic, (c) is constant, (d) is cubic.
  • A2. (b) 5. Explanation: Set 2x − 10 = 0 ⇒ 2x = 10 ⇒ x = 5. Verify: 2(5) − 10 = 0.
  • A3. (b) 9. Explanation: In y = 4x + 9, the y-intercept is the constant term, which is 9. It is the value of y when x = 0.
  • A4. (b) 20. Explanation: Set h(t) = 0 ⇒ 5 − 0.25t = 0 ⇒ 0.25t = 5 ⇒ t = 5 ÷ 0.25 = 20 months.
  • A5. (b) −3. Explanation: In y = −3x + 2, the coefficient of x is −3, which is the slope.
  • A6. (c) They have degree 1. Explanation: By definition, a linear polynomial has degree 1. Option (a) is false (only one zero), (b) is false (graphs are straight lines), (d) is false (coefficients can be negative).

Answer Key: Sections B, C, D, E, and F with Detailed Solutions

This section provides complete answers and step-by-step solutions for fill-in-the-blanks, true/false, short-answer, long-answer HOTS, and the case-study question. Each answer includes the correct response, working where applicable, and a brief explanation linking back to NCERT concepts. For fill-in-the-blanks, the exact term or expression is given. For true/false, the correct choice is stated with reasoning. For short-answer questions, full working is shown with verification where possible. For long-answer questions, multi-step solutions are laid out clearly, demonstrating how to set up equations, solve them, and interpret results in context. The case-study answers show how to extract data from the scenario and apply linear polynomial concepts systematically. Use this key not just to mark your work, but to learn from mistakes and understand the logic behind each solution. If you find a concept difficult, revisit the relevant NCERT example or exercise problem, and consider using CBSETUTOR.ai for personalized doubt-solving and practice at ₹999/month for all classes with a 3-day free trial. Consistent review of answer keys improves retention and builds the problem-solving stamina needed for CBSE board exams and internal assessments throughout the academic year.
  • A7. linear
  • A8. −b/a
  • A9. 7; 12
  • A10. slope; y-intercept
  • A11. straight
  • A12. False. Explanation: The polynomial 9 is a constant polynomial with degree 0, not degree 1.
  • A13. True. Explanation: A non-zero linear polynomial ax + b has exactly one zero, x = −b/a.
  • A14. True. Explanation: Same slope means parallel; different y-intercepts mean they never meet.
  • A15. False. Explanation: Set −4x + 20 = 0 ⇒ −4x = −20 ⇒ x = 5, not −5.
  • A16. True. Explanation: In decay, the quantity decreases, so the rate (coefficient) is negative.
  • A17. Set 5x + 25 = 0 ⇒ 5x = −25 ⇒ x = −5. Verification: 5(−5) + 25 = 0. Answer: x = −5.
  • A18. Total bill = fixed fee + (rate per GB × number of GB) = 150 + 8g. Answer: p(g) = 150 + 8g.
  • A19. p(x) = 3x − 12. Substitute x = 5: p(5) = 3(5) − 12 = 15 − 12 = 3. Answer: p(5) = 3.
  • A20. Set h(t) = 0 ⇒ 20 − 2t = 0 ⇒ 2t = 20 ⇒ t = 10 hours. Answer: The candle burns out after 10 hours.
  • A21. Comparing y = −7x + 14 with y = ax + b: slope a = −7, y-intercept b = 14. Answer: Slope = −7, y-intercept = 14.
  • A22. (i) Initial height = 1.5 ft, growth = 0.4 ft/month. h(t) = 1.5 + 0.4t. (ii) h(6) = 1.5 + 0.4(6) = 1.5 + 2.4 = 3.9 ft. (iii) Set h(t) = 5.5 ⇒ 1.5 + 0.4t = 5.5 ⇒ 0.4t = 4 ⇒ t = 10 months.
  • A23. (i) Initial value = 12000, depreciation = 1000/year. v(t) = 12000 − 1000t. (ii) v(4) = 12000 − 1000(4) = 12000 − 4000 = ₹8000. (iii) Set v(t) = 6000 ⇒ 12000 − 1000t = 6000 ⇒ 1000t = 6000 ⇒ t = 6 years.
  • A24. (i) Set d₁ = d₂ ⇒ 5t + 10 = 3t + 20 ⇒ 2t = 10 ⇒ t = 5 hours. (ii) Distance = 5(5) + 10 = 35 km (or 3(5) + 20 = 35 km). Answer: They meet after 5 hours at 35 km.
  • Case Study A: (i) V(t) = 240 − 15t. (ii) V(5) = 240 − 15(5) = 240 − 75 = 165 litres. (iii) Half-full = 120 litres. Set 240 − 15t = 120 ⇒ 15t = 120 ⇒ t = 8 hours. (iv) Set V(t) = 0 ⇒ 240 − 15t = 0 ⇒ t = 16 hours.

Difficulty Level, Time Allocation, and Preparation Tips

This worksheet is designed at a medium to challenging difficulty level, suitable for students who have completed the NCERT Class 9 Mathematics Chapter 2 textbook and exercise problems. It combines straightforward recall (MCQs, fill-in-the-blanks) with application-based short answers and higher-order thinking questions (HOTS) that require multi-step reasoning and real-world interpretation. Suggested total time for completion is 90 minutes under exam conditions, though you may take longer during initial practice. Allocate approximately 10 minutes for MCQs, 8 minutes for fill-in-the-blanks, 7 minutes for true/false, 25 minutes for short answers, 30 minutes for long HOTS questions, and 10 minutes for the case study. To prepare effectively, first revise all NCERT examples, solved problems, and exercise questions from Chapter 2. Pay special attention to Examples 6, 7, 9, 10, and 11 in the textbook, which illustrate finding zeros, linear growth and decay, and forming linear relationships from data. Practice writing clear, step-by-step solutions as shown in the answer key. If you struggle with word problems, break them into parts: identify what is given, what is asked, form the polynomial or equation, solve, and verify. Use CBSETUTOR.ai for instant doubt-solving, personalized practice, and step-by-step explanations by uploading photos of problems—all for ₹999/month across classes 6-12, with a 3-day free trial. Consistent practice with worksheets like this builds exam confidence, improves speed, and ensures you can tackle any question format in CBSE board exams and internal assessments.
  • Difficulty Level: Medium to Challenging
  • Suggested Time: 90 minutes (under exam conditions)
  • Section-wise time: MCQs—10 min, Fill-ups—8 min, True/False—7 min, Short answers—25 min, Long HOTS—30 min, Case study—10 min
  • Revise NCERT Examples 6, 7, 9, 10, 11 before attempting this worksheet
  • Practice forming linear polynomials from word problems and finding zeros by setting the expression equal to zero
  • Always verify your final answer by substituting back into the original expression or equation
  • Use graph paper to sketch linear graphs for visual understanding of slope and intercept

Frequently asked questions

What is a linear polynomial and how is it different from other polynomials?+
A linear polynomial has degree 1, meaning the highest power of the variable is 1. Its general form is ax + b where a ≠ 0. Quadratic polynomials have degree 2 (e.g., x² + 5x + 6) and cubic polynomials have degree 3 (e.g., 2y³ + y − 1). Linear polynomials graph as straight lines; quadratics graph as parabolas.
How do I find the zero of a linear polynomial?+
Set the polynomial equal to zero and solve for the variable. For ax + b, solve ax + b = 0 to get x = −b/a. Example: For 3x − 9, set 3x − 9 = 0, so 3x = 9, x = 3. Verify by substituting: 3(3) − 9 = 0. The zero is x = 3.
What is the difference between a linear polynomial and a linear equation?+
A linear polynomial is an expression like 2x + 10. A linear equation is formed when you equate that expression to a number, such as 2x + 10 = 64. Solving the equation gives the value of x that makes the statement true. The polynomial is the building block; the equation is the statement to solve.
What do slope and y-intercept mean in a linear relationship?+
In the linear relationship y = ax + b, 'a' is the slope—the constant rate of change of y for each unit increase in x. 'b' is the y-intercept—the value of y when x = 0, where the graph crosses the y-axis. For example, in y = 5x + 20, slope is 5 (y increases by 5 per unit of x) and y-intercept is 20.
How do I know if a real-world problem is linear growth or linear decay?+
Check whether the quantity increases or decreases by a constant amount at regular intervals. If it increases, it is linear growth (positive slope). If it decreases, it is linear decay (negative slope). Example: A plant growing 0.5 feet per month is linear growth; water draining 15 litres per hour is linear decay.
Can a constant polynomial like 7 have a zero?+
No. A constant polynomial has degree 0 and is always the same value, never zero (unless the constant itself is 0). For example, p(x) = 7 means p(x) is always 7, regardless of x, so there is no value of x that makes p(x) = 0.
What does it mean when two linear graphs are parallel?+
Two linear graphs are parallel if they have the same slope but different y-intercepts. This means they never intersect. For example, y = 3x + 5 and y = 3x + 1 are parallel because both have slope 3. They run side-by-side without meeting.
How many points do I need to graph a linear polynomial?+
You need at least two points to draw a straight line. A common method is to find the y-intercept (when x = 0) and one other point (e.g., when x = 1 or x = 2). Plot these two points on graph paper and draw a straight line through them. The line extends infinitely in both directions.
Why is NCERT Class 9 Mathematics Chapter 2 important for board exams?+
Chapter 2 builds the foundation for linear equations, coordinate geometry, and graphing in later chapters. CBSE board exams regularly include questions on finding zeros, forming linear models from word problems, and graphing. Mastery of linear polynomials is essential for scoring well in algebra sections and for higher mathematics in Classes 10, 11, and 12.
How can CBSETUTOR.ai help with practicing linear polynomials?+
CBSETUTOR.ai offers 24×7 AI-powered tutoring where you can upload photos of worksheet problems, NCERT exercises, or word problems and receive instant step-by-step solutions. At a flat ₹999/month for all classes (6-12) and all subjects, with a 3-day free trial, it is an affordable way to get personalized doubt-solving and unlimited practice for Chapter 2 and beyond.

Related resources

Important Questions: CBSE Class 9 Mathematics Chapter 2 PolynomialsClass 9 Mathematics Chapter 2 Polynomials — Formulas & Key PointsCBSE Class 9 Mathematics Chapter 2 Polynomials Worksheet with AnswersNCERT Solutions for CBSE Class 9 Mathematics Chapter 2: PolynomialsCBSE Class 9 Mathematics Chapter 1 Number Systems — NotesCBSE Class 9 Mathematics — Number Systems: complete chapter guideNCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems – Complete GuideCBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

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