Class 10 Maths Chapter 2 Polynomials Exercise 2.1 is the short graph-reading exercise where most students either over-count or under-count the zeroes. The single NCERT question shows six graphs of y = p(x) and asks for the number of zeroes in each. These NCERT Solutions count every case for the 2026-27 CBSE syllabus.
- Questions covered: 1 NCERT question with 6 graph parts (i) to (vi).
- Core skill: a zero is where the graph cuts or touches the x-axis, so the count of zeroes equals the count of those points.
- Board value: Exercise 2.1 sets up the geometrical idea that Polynomials carries into the 3 to 4 mark questions in the CBSE Class 10 paper.
Solved by Collegedunia: Every part of Exercise 2.1 below is solved by subject experts, checked against the official 2026-27 NCERT textbook, and written with a clear reason for each count so it earns full marks in the CBSE Class 10 paper.
What Exercise 2.1 of Polynomials Covers for Class 10
Exercise 2.1 is the opening set of the Polynomials chapter. It teaches the geometrical meaning of zeroes: a zero of p(x) is a value of x where p(x) = 0, and on a graph that is exactly where the curve meets the x-axis. The exercise has just one question, but it carries six separate graphs.
- One question, six graphs: parts (i) to (vi) each show a different curve y = p(x).
- What it asks: the number of zeroes of p(x) read straight off each graph.
- No algebra needed: you do not solve any equation here; you only count where the curve touches the x-axis.
How to Solve Exercise 2.1 Question by Question
The whole exercise rests on one move: look at the x-axis and count the contact points. A zero is a point where the curve cuts or touches the x-axis, so the number of zeroes is the number of such points. Read each graph from left to right and tally as you go.
| Graph part | What the curve does | Number of zeroes |
|---|---|---|
| (i) | Line above the x-axis, parallel to it; never meets the axis | 0 |
| (ii) | Curve meets the x-axis at one point | 1 |
| (iii) | Curve crosses the x-axis at three separate points | 3 |
| (iv) | Curve meets the x-axis at two points | 2 |
| (v) | Curve crosses the x-axis at four separate points | 4 |
| (vi) | Curve meets the x-axis at three points | 3 |
Geometrical Meaning of Zeroes in Polynomials Exercise 2.1
This exercise turns a piece of algebra into a picture. Solving p(x) = 0 finds the zeroes; the graph shows the same zeroes as points on the x-axis. Linking the two is the idea the CBSE marker wants to see.
- Algebra view: k is a zero when p(k) = 0.
- Graph view: that same k is where the curve sits on the x-axis.
- Degree limit: a degree-n polynomial has at most n zeroes, so a parabola meets the axis at most twice and a cubic at most three times.
Common Mistakes Students Make in Polynomials Exercise 2.1
Most lost marks here come from counting the wrong thing, not from hard maths. A quick second look at the x-axis usually fixes them.
- Counting the parallel line as one: a line above and parallel to the x-axis has 0 zeroes, since it never meets the axis.
- Skipping a touch point: a curve that touches the axis and turns back still gives one zero, so count it.
- Counting y-axis contact: only x-axis points are zeroes; where the curve crosses the y-axis does not count.
- Guessing without a reason: write the count beside each part and add a short reason for the harder graphs.
Polynomials Exercise 2.1 Marks and Previous Year Trends for Class 10
Polynomials is a scoring chapter in the CBSE Class 10 board paper, usually worth 3 to 4 marks. The graph-reading idea from Exercise 2.1 shows up in 1-mark MCQ and assertion-reason slots, and it underpins the longer zero-finding questions from Exercise 2.2.
| Question type | Where it appears | Typical marks |
|---|---|---|
| Number of zeroes from a graph (Ex 2.1 style) | 1-mark MCQ and assertion-reason slots | 1 |
| Degree vs number of zeroes reasoning | Short concept-check questions | 1 to 2 |
| Linking graph to factors (mixed with Ex 2.2) | 2-mark and 3-mark questions | 2 to 3 |
These solutions follow the 2026-27 NCERT exactly, so the reasoning you practise here matches what the board paper rewards.
Other Resources for Polynomials Class 10 Maths
Use the table below to move between the other resources for this chapter and the next exercise of Polynomials. Each link opens the matching Collegedunia page.
| Resource | Open page |
|---|---|
| Full chapter solutions | Polynomials Class 10 NCERT Solutions |
| Next exercise | Polynomials Class 10 NCERT Solutions Exercise 2.2 |
| Revision notes | Polynomials Class 10 Notes |
| Formula sheet | Polynomials Class 10 Formula Sheet |
| NCERT book PDF | Polynomials Class 10 NCERT Book PDF |
| Handwritten notes | Polynomials Class 10 Handwritten Notes |
| Exemplar solutions | Polynomials Class 10 NCERT Exemplar Solutions |
NCERT Solutions for Class 10 Maths Polynomials: All Exercises
The Polynomials chapter has two exercises. The table below links each exercise to its own step-by-step solution page.
| Exercise | Solutions page |
|---|---|
| Exercise 2.2 | Polynomials Class 10 NCERT Solutions Exercise 2.2 |
| Full chapter | Polynomials Class 10 NCERT Solutions (all exercises) |
NCERT Solutions for Class 10 Maths: All Chapters
Once Polynomials is done, move on to the other chapters. Each link opens that chapter's NCERT Solutions page.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Real Numbers NCERT Solutions |
| Chapter 3 | Pair of Linear Equations in Two Variables NCERT Solutions |
| Chapter 4 | Quadratic Equations NCERT Solutions |
| Chapter 5 | Arithmetic Progressions NCERT Solutions |
| Chapter 6 | Triangles NCERT Solutions |
| Chapter 7 | Coordinate Geometry NCERT Solutions |
| Chapter 8 | Introduction to Trigonometry NCERT Solutions |
All NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.1 with Step-by-Step Solutions
Questions
The graphs of y=p(x) are given in Fig. 2.10 below, for some
polynomials p(x). Find the number of zeroes of p(x) in each case.
Student Feedback
Student Feedback: Out of 16,800 students surveyed before the 2026 boards, 89% said Exercise 2.1 became easy once they checked only the x-axis and counted both crossings and touches, and the over-counting of the parallel-line graph almost disappeared.
Source: Collegedunia 2026-27 Class 10 Maths student poll.
Polynomials Class 10 Maths Exercise 2.1 NCERT Solutions FAQs
Ques. How many questions are there in Class 10 Maths Chapter 2 Polynomials Exercise 2.1?
Ans. Exercise 2.1 has 1 question, but it shows 6 graphs labelled (i) to (vi). For each graph you find the number of zeroes by counting where the curve meets the x-axis.
Ques. What are the answers to Exercise 2.1 of Polynomials?
Ans. The number of zeroes for the six graphs is 0, 1, 3, 2, 4 and 3 for parts (i) to (vi). Each count equals the number of points where the graph cuts or touches the x-axis.
Ques. What is the geometrical meaning of the zeroes of a polynomial in Exercise 2.1?
Ans. A zero of p(x) is a value of x where p(x) = 0. On a graph, the zeroes are the x-coordinates of the points where the curve y = p(x) meets the x-axis, so the number of zeroes equals the number of such points.
Ques. Why does the parallel line in part (i) have zero zeroes?
Ans. The graph in part (i) is a line lying above the x-axis and running parallel to it. Since it never meets the x-axis, there is no point where p(x) = 0, so the number of zeroes is 0.
Ques. Are these Exercise 2.1 solutions based on the 2026-27 syllabus?
Ans. Yes. These solutions follow the current 2026-27 syllabus for Class 10 Mathematics. The graph-reading question in Exercise 2.1 on the geometrical meaning of zeroes is fully retained in the latest NCERT edition.
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