Mastering CBSE board exams requires clear concepts, consistent practice, and access to reliable study materials. When preparing for your board exams, having accurate guidance is essential, especially for foundational topics like algebra. As you begin your preparation, utilizing reliable NCERT solutions can make a significant difference in your daily practice routine.
Chapter 2 of the Class 10 mathematics textbook plays a crucial role in building your algebraic foundation. Board examiners frequently test your understanding of polynomial zeroes, their graphical representations, and the relationship between coefficients and zeroes. Scoring full marks in this chapter is entirely achievable if you understand the underlying formulas and present your steps neatly during the exam. To help you master every concept, this guide provides comprehensive ncert solutions class 10 maths chapter 2 with step-by-step working.
Understanding Polynomials in Class 10
Before diving into the textbook problems, let us quickly recap what you learned previously. A polynomial is an algebraic expression consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Depending on the highest degree of the variable, polynomials are classified as linear (degree 1), quadratic (degree 2), and cubic (degree 3). For Class 10 CBSE students, quadratic polynomials of the form $ax^2 + bx + c$ (where $a \neq 0$) form the core of the syllabus. For a broader overview of all topics covered this academic year, you can Browse all chapter notes on our platform.
Why This Chapter Matters for Board Exams
- It carries a solid weightage in the algebra section of the annual CBSE board question paper.
- Concepts like the Geometrical Meaning of Zeroes of a Polynomial often appear in 1-mark and 2-mark questions.
- Quadratic polynomial formation using sum and product of zeroes frequently features as a 3-mark or 4-mark application question.
Exercise 2.1: Geometrical Meaning of the Zeroes of a Polynomial
The graph of $y = p(x)$ for a polynomial $p(x)$ is given in various questions. The goal is to find the number of zeroes of $p(x)$ in each case.
Key Concept
The zeroes of a polynomial $p(x)$ are precisely the $x$-coordinates of the points where the graph of $y = p(x)$ intersects the $x$-axis.
Solved Examples and Textbook Questions
Question 1: Find the number of zeroes of $p(x)$, in each case, given below.
Graph (i): The line is parallel to the $x$-axis and never intersects it.
- Step-by-step working: Since the graph does not intersect the $x$-axis at any point, the number of zeroes is 0.
- Final Answer: 0
Graph (ii): The graph intersects the $x$-axis at exactly one point.
- Step-by-step working: The graph cuts the $x$-axis at one unique location, meaning the polynomial has 1 real zero.
- Final Answer: 1
Graph (iii): The graph intersects the $x$-axis at three distinct points.
- Step-by-step working: Counting the intersection points along the horizontal axis, we find three locations where $y = 0$.
- Final Answer: 3
To test your visual understanding of these graphs, you can attempt interactive Practice tests available on our portal.
Exercise 2.2: Relationship Between Zeroes and Coefficients of a Quadratic Polynomial
This exercise deals with finding the zeroes of quadratic polynomials and verifying the relationship between those zeroes and their coefficients.
Core Formulas
For a general quadratic polynomial $p(x) = ax^2 + bx + c$, let $\alpha$ and $\beta$ be its zeroes.
- Sum of zeroes ($\alpha + \beta$) = $-\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2} = -\frac{b}{a}$
- Product of zeroes ($\alpha\beta$) = $\frac{\text{Constant term}}{\text{Coefficient of } x^2} = \frac{c}{a}$
Step-by-Step Problem Solving
Question 1(i): Find the zeroes of the quadratic polynomial $x^2 - 2x - 8$ and verify the relationship between the zeroes and the coefficients.
Step 1: Factorize the polynomial. We need two numbers whose product is $-8$ and whose sum is $-2$. These numbers are $-4$ and $+2$. $x^2 - 4x + 2x - 8$ $= x(x - 4) + 2(x - 4)$ $= (x - 4)(x + 2)$
Step 2: Find the zeroes. Set $p(x) = 0$: $(x - 4)(x + 2) = 0$ This gives $x = 4$ or $x = -2$. Therefore, the zeroes are $\alpha = 4$ and $\beta = -2$.
Step 3: Verify the relationship.
- Sum of zeroes: $\alpha + \beta = 4 + (-2) = 2$ Using formula: $-\frac{b}{a} = -\frac{-2}{1} = 2$. (Verified!)
- Product of zeroes: $\alpha\beta = (4)(-2) = -8$ Using formula: $\frac{c}{a} = \frac{-8}{1} = -8$. (Verified!)
Final Answer: Zeroes are $4$ and $-2$. Both coefficient relationships are successfully verified.
Question 2(i): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively: $\frac{1}{4}, -1$.
Step 1: Identify given values. Sum of zeroes ($\alpha + \beta$) = $\frac{1}{4}$ Product of zeroes ($\alpha\beta$) = $-1$
Step 2: Apply the standard polynomial formula. A quadratic polynomial is given by $k[x^2 - (\text{Sum of zeroes})x + (\text{Product of zeroes})]$, where $k$ is a non-zero real number. $= k\left[x^2 - \left(\frac{1}{4}\right)x + (-1)\right]$ $= k\left(x^2 - \frac{1}{4}x - 1\right)$
Step 3: Simplify for standard integer coefficients. Taking $k = 4$, we get: $4x^2 - x - 4$
Final Answer: $4x^2 - x - 4$
For further curriculum details and official guidelines, you may refer directly to the NCERT Official Website to stay updated with any curriculum modifications. You can also review the CBSE Academic Website for sample question papers.
Tips to Score Full Marks in Class 10 Maths Chapter 2
Achieving a perfect score in board mathematics requires more than just knowing formulas. Follow these expert strategies during your preparation:
- Show Every Step Clearly: Board exams award step marks. Never jump directly from the polynomial equation to the final answer without showing the factorization steps.
- Double-Check Signs: Many students lose marks due to minor sign errors (e.g., confusing $-b/a$ with $b/a$). Always rewrite the values of $a, b,$ and $c$ explicitly before applying formulas.
- Practice Word Problems: Pay extra attention to application-based questions where you must frame polynomials from real-world contexts.
- Revise Regularly: Revisit these ncert solutions class 10 maths chapter 2 notes at least once every two weeks to keep your memory sharp before pre-boards.
To explore more learning resources, you can always Explore CircleOfLearning for comprehensive study guides across all subjects.
Frequently Asked Questions
What are the main topics in Class 10 Maths Chapter 2?
The chapter covers the geometrical meaning of the zeroes of a polynomial, the relationship between zeroes and coefficients of quadratic polynomials, and division algorithms for polynomials.
How many marks does Polynomials carry in the CBSE board exam?
Algebra as a unit—which includes polynomials and linear equations—carries a significant weightage in the board exam. Polynomials alone generally contribute around 3 to 5 marks through objective and short-answer questions.
Are NCERT solutions enough for board exam preparation?
Yes, mastering all questions in the NCERT textbook and exemplar is the primary requirement for scoring high marks. Once you finish NCERT, you can move on to sample papers for extra practice.
What is the formula for the sum of zeroes of a quadratic polynomial?
For a quadratic polynomial $ax^2 + bx + c$, the sum of the zeroes is given by $-\frac{b}{a}$, where $b$ is the coefficient of $x$ and $a$ is the coefficient of $x^2$.
How can I practice more questions for this chapter?
You can solve additional practice questions and take chapter-wise mock tests on educational platforms like CircleOfLearning to boost your speed and accuracy.
Mastering polynomials is a stepping stone toward conquering higher-level algebra in senior secondary classes. Keep practicing consistently, clear your doubts early, and refine your presentation skills for the upcoming board examinations.
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