class 10 maths chapter 2 : Class 10 Maths Chapter 2 solutions

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Chapter 2 Math Exercises

Chapter 2

EXERCISE 2.1

1. 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.

(i), (ii), (iii), (iv), (v), (vi)

Solution:

The number of zeroes of a polynomial p(x) is the number of times its graph intersects the x-axis (where y=p(x)=0).

Since the specific graphs (Fig. 2.10) are not provided, I will explain the general approach and assume typical cases for polynomial graphs:

  • (i): Assume the graph intersects the x-axis at 0 points Zeroes = 0.
  • (ii): Assume the graph intersects the x-axis at 1 point Zeroes = 1.
  • (iii): Assume the graph intersects the x-axis at 2 points. Zeroes = 2.
  • (iv): Assume the graph intersects the x-axis at 3 points Zeroes = 3.
  • (v): Assume the graph intersects the x-axis at 4 points Zeroes = 4.
  • (vi): Assume the graph intersects the x-axis at 2 points Zeroes = 2.

Explanation (Step-by-Step):

  • Step 1: Identify where the graph of y=p(x) crosses or touches the x-axis.
  • Step 2: Count each intersection as a zero. A touch counts as one zero unless specified otherwise.
  • Step 3: The number of intersections equals the number of real zeroes.

EXERCISE 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x^2-2x-8

Solution:

  • Find zeroes: Solve \( x^2-2x-8=0 \).
  • Factorize: \( x^2-4x+2x-8 = x(x-4)+2(x-4)=(x-4)(x+2)=0 \).
  • Zeroes: \( x=4,-2 \).
  • Verification:
  • For a quadratic \( ax^2+bx+c \), sum of zeroes = \(-b/a\), product of zeroes = \(c/a\).
  • Here, \( a=1,b=-2,c=-8 \).
  • Sum of zeroes: \( 4+(-2)=2 \). Check: \(-b/a=-(-2)/1=2\). Matches.
  • Product of zeroes: \( 4 \times (-2)=-8 \). Check: \( c/a=(-8)/1=-8 \). Matches.

(ii) 4s^2-4s+1

Solution:

  • Find zeroes: Solve \( 4s^2-4s+1=0 \).
  • Factorize: \( 4s^2-2s-2s+1 = 2s(2s-1)-1(2s-1) = (2s-1)(2s-1)=(2s-1)^2=0 \).
  • Zero: \( s=1/2 \) (repeated root).
  • Verification:
  • \( a=4,b=-4,c=1 \).
  • Sum of zeroes: \( 1/2+1/2=1 \). Check: \(-b/a=-(-4)/4=1\). Matches.
  • Product of zeroes: \( 1/2 \times 1/2=1/4 \). Check: \( c/a=1/4 \). Matches.

(iii) 6x^2-3-7x

Solution:

  • Rewrite: \( 6x^2-7x-3=0 \).
  • Find zeroes:
  • Factorize: \( 6x^2-9x+2x-3 = 3x(2x-3)+1(2x-3) = (2x-3)(3x+1)=0 \).
  • Zeroes: \( x=3/2,-1/3 \).
  • Verification:
  • \( a=6,b=-7,c=-3 \).
  • Sum of zeroes: \( 3/2+(-1/3)=3/2-1/3=(9-2)/6=7/6 \). Check: \(-b/a=-(-7)/6=7/6\). Matches.
  • Product of zeroes: \( 3/2 \times (-1/3)=-1/2 \). Check: \( c/a=(-3)/6=-1/2 \). Matches.

(iv) 4u^2+8u

Solution:

  • Rewrite: \( 4u^2+8u=4u(u+2)=0 \).
  • Find zeroes: \( 4u=0 \) or \( u+2=0 \).
  • Zeroes: \( u=0,-2 \).
  • Verification:
  • Standard form: \( 4u^2+8u+0 \), so \( a=4,b=8,c=0 \).
  • Sum of zeroes: \( 0+(-2)=-2 \). Check: \(-b/a=-8/4=-2\). Matches.
  • Product of zeroes: \( 0 \times (-2)=0 \). Check: \( c/a=0/4=0 \). Matches.

(v) t^2-15

Solution:

  • Rewrite: \( t^2-15=0 \).
  • Find zeroes: \( t^2=15 \), so \( t=\pm \sqrt{15} \).
  • Verification:
  • Standard form: \( t^2+0t-15 \), so \( a=1,b=0,c=-15 \).
  • Sum of zeroes: \( \sqrt{15}+(-\sqrt{15})=0 \). Check: \(-b/a=-0/1=0\). Matches.
  • Product of zeroes: \( \sqrt{15} \times (-\sqrt{15})=-15 \). Check: \( c/a=(-15)/1=-15 \). Matches.

(vi) 3x^2-x-4

Solution:

  • Find zeroes: Solve \( 3x^2-x-4=0 \).
  • Factorize: \( 3x^2-4x+3x-4 = x(3x-4)+1(3x-4) = (3x-4)(x+1)=0 \).
  • Zeroes: \( x=4/3,-1 \).
  • Verification:
  • \( a=3,b=-1,c=-4 \).
  • Sum of zeroes: \( 4/3+(-1)=4/3-3/3=1/3 \). Check: \(-b/a=-(-1)/3=1/3\). Matches.
  • Product of zeroes: \( 4/3 \times (-1)=-4/3 \). Check: \( c/a=(-4)/3=-4/3 \). Matches.

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

For a quadratic \( x^2-(\text{sum of zeroes})x+(\text{product of zeroes}) \).

(i) Sum = 1/4, Product = -1

Solution:

  • Polynomial: \( x^2-(1/4)x+(-1)=x^2-1/4 x-1 \).
  • To avoid fractions, multiply by 4: \( 4x^2-x-4 \).
  • Answer: \( 4x^2-x-4 \).

(ii) Sum = \sqrt{2}, Product = 1/3

Solution:

  • Polynomial: \( x^2-(\sqrt{2})x+1/3 \).
  • Multiply by 3: \( 3x^2-3\sqrt{2}x+1 \).
  • Answer: \( 3x^2-3\sqrt{2}x+1 \).

(iii) Sum = 0, Product = \sqrt{5}

Solution:

  • Polynomial: \( x^2-(0)x+\sqrt{5}=x^2+\sqrt{5} \).
  • Answer: \( x^2+\sqrt{5} \).

Explanation:

  • Step 1: Substitute sum = 0, product = \( \sqrt{5} \).
  • Step 2: Simplify.

(iv) Sum = 1, Product = 1

Solution:

  • Polynomial: \( x^2-(1)x+1=x^2-x+1 \).
  • Answer: \( x^2-x+1 \).

(v) Sum = -1/4, Product = 1/4

Solution:

  • Polynomial: \( x^2-(-1/4)x+1/4=x^2+1/4 x+1/4 \).
  • Multiply by 4: \( 4x^2+x+1 \).
  • Answer: \( 4x^2+x+1 \).

(vi) Sum = 4, Product = 1

Solution:

  • Polynomial: \( x^2-(4)x+1=x^2-4x+1 \).
  • Answer: \( x^2-4x+1 \).

Class 10 Maths Chapter 2 Solutions: Your Guide to Polynomials and Board Exam Success

Welcome, Class 10 students! If you’re diving into the NCERT Class 10 Maths book, Chapter 2, “Polynomials,” is an exciting and important step in your journey. Mastering class 10 Maths Chapter 2 solutions is crucial for understanding the CBSE syllabus and excelling in your board exams. This chapter builds key skills for the Class 10 Maths syllabus, and with the right approach, it’s easy and fun! In this article, we’ll explore class 10 Maths Chapter 2 solutions in simple language, break down the concepts, share tips on how students can approach it, and show how www.growupncert.com can help with exam preparation. Let’s get started and make Polynomials your strength!

Understanding Chapter 2: Polynomials in the CBSE Syllabus

What Is Chapter 2 About?

Chapter 2, “Polynomials,” in the NCERT Class 10 Maths book introduces you to expressions like x² + 3x + 2. A polynomial is a mathematical expression with variables (like x) and constants (like 2, 3) combined using addition, subtraction, and multiplication. The Class 10 Maths syllabus includes this chapter to teach you how to work with these expressions, find their roots, and understand their behavior. Class 10 Maths Chapter 2 solutions guide you through solving problems step by step, making it a key part of your board exam prep.

Key Concepts in Polynomials

Here’s a simple look at the main ideas:

  • Types of Polynomials: Based on degree (highest power of the variable):
    • Linear: Degree 1, e.g., 2x + 3.
    • Quadratic: Degree 2, e.g., x² – 5x + 6.
    • Cubic: Degree 3, e.g., x³ + 2x² – 1.
  • Zeros of a Polynomial: The values of x where the polynomial equals zero. For example, for x² – 4, the zeros are x = 2 and x = –2 because (2)² – 4 = 0 and (–2)² – 4 = 0.
  • Division Algorithm: If you divide a polynomial p(x) by another g(x), you get p(x) = g(x) × q(x) + r(x), where q(x) is the quotient and r(x) is the remainder.
  • Relationship Between Zeros and Coefficients: For a quadratic polynomial ax² + bx + c, the sum of zeros is –b/a, and the product of zeros is c/a.

Why It Matters for Board Exams

Polynomials are a vital part of the CBSE syllabus, often carrying 6-8 marks in board exams. You’ll see questions on finding zeros, verifying the division algorithm, or relating zeros to coefficients. Class 10 Maths Chapter 2 solutions help you tackle these, building skills for later chapters like Quadratic Equations. With good exam preparation, you’ll score well and gain confidence!

NCERT Exercises in Chapter 2

The NCERT book has three exercises:

  • Exercise 2.1: Focuses on understanding polynomials and their degrees.
  • Exercise 2.2: Deals with finding zeros and the relationship between zeros and coefficients.
  • Exercise 2.3: Covers the division algorithm and related problems. Solutions for Class 10 Maths Chapter 2 make these exercises easy to master.

Class 10 Maths Chapter 2 Solutions: Sample Problems

Let’s look at some sample class 10 Maths Chapter 2 solutions to see how to solve problems.

Exercise 2.1, Question 1

Problem: The graphs of y = p(x) are given below. Find the number of zeros of p(x). (Assume a graph shows a quadratic polynomial crossing the x-axis twice.) Solution:

  • Zeros are points where the graph of p(x) crosses the x-axis (where p(x) = 0).
  • If the graph crosses the x-axis twice, the polynomial has 2 zeros.
  • Answer: The number of zeros is 2.

Exercise 2.2, Question 2

Problem: Find the zeros of the quadratic polynomial x² – 3x – 4 and verify the relationship between zeros and coefficients. Solution:

  • Step 1: Find the Zeros
    • Polynomial: x² – 3x – 4.
    • Factorize: x² – 3x – 4 = (x – 4)(x + 1).
    • Set to zero: (x – 4)(x + 1) = 0.
    • Solve: x – 4 = 0 or x + 1 = 0.
    • Zeros: x = 4 and x = –1.
  • Step 2: Verify Relationship
    • For a quadratic ax² + bx + c, here a = 1, b = –3, c = –4.
    • Sum of zeros = –b/a = –(–3)/1 = 3.
    • Check: 4 + (–1) = 3, matches!
    • Product of zeros = c/a = –4/1 = –4.
    • Check: 4 × (–1) = –4, matches!
  • Answer: Zeros are 4 and –1, and the relationship is verified.

Exercise 2.3, Question 1

Problem: Divide the polynomial p(x) = x³ – 3x² + 5x – 3 by g(x) = x² – 2 and find the quotient and remainder. Solution:

  • Use the division algorithm: p(x) = g(x) × q(x) + r(x).
  • Divide x³ – 3x² + 5x – 3 by x² – 2.
  • Step 1: Divide the first term: x³ ÷ x² = x.
  • Step 2: Multiply: x × (x² – 2) = x³ – 2x.
  • Step 3: Subtract: (x³ – 3x² + 5x – 3) – (x³ – 2x) = –3x² + 7x – 3.
  • Step 4: Divide the next term: –3x² ÷ x² = –3.
  • Step 5: Multiply: –3 × (x² – 2) = –3x² + 6.
  • Step 6: Subtract: (–3x² + 7x – 3) – (–3x² + 6) = 7x – 9.
  • Step 7: Stop, as the degree of 7x – 9 (1) is less than the degree of x² – 2 (2).
  • Result: Quotient q(x) = x – 3, Remainder r(x) = 7x – 9.
  • Answer: Quotient is x – 3, remainder is 7x – 9.

These class 10 Maths Chapter 2 solutions show clear steps, perfect for board exam practice.

How Students Can Approach Class 10 Maths Chapter 2 Solutions

Mastering class 10 Maths Chapter 2 solutions for board exams is simple with a smart plan. Here are easy tips for exam preparation within the CBSE syllabus.

1. Know the Class 10 Maths Syllabus

  • What to Do: Check the CBSE syllabus for Class 10 Maths. Chapter 2, Polynomials, is worth 6-8 marks in board exams.
  • How It Helps: Focuses your study on key areas like zeros and division.
  • Tip: Get the syllabus from the CBSE website or your teacher.

2. Read the NCERT Chapter

  • What to Do: Start with the NCERT Class 10 Maths book. Read about polynomials, degrees, and zeros.
  • How It Helps: Builds a clear understanding for solving problems.
  • Tip: Study examples before exercises.

3. Learn Key Concepts

  • What to Do: Focus on types of polynomials, finding zeros, and the division algorithm.
  • How It Helps: Makes class 10 Maths Chapter 2 solutions easier.
  • Tip: Note key ideas: e.g., sum of zeros = –b/a for ax² + bx + c.

4. Solve NCERT Exercises

  • What to Do: Work through Exercises 2.1, 2.2, and 2.3. Start with simple questions, then try harder ones.
  • How It Helps: Practice builds skills for board exams.
  • Tip: Solve all problems—don’t skip any!

5. Practice Step-by-Step

  • What to Do: Break problems into steps. For example, to find zeros, factorize, set to zero, and solve.
  • How It Helps: Reduces mistakes and boosts confidence.
  • Tip: Write steps neatly for the exam.

6. Memorize Relationships

  • What to Do: Learn rules like sum of zeros = –b/a and product of zeros = c/a for quadratics.
  • How It Helps: Speeds up solving in board exams.
  • Tip: Make a formula card and revise daily.

7. Use Graphs

  • What to Do: Sketch graphs of polynomials to find zeros (where the graph crosses the x-axis).
  • How It Helps: Visuals clarify concepts, especially for Exercise 2.1.
  • Tip: Use graph paper for accuracy.

8. Practice Previous Years’ Papers

  • What to Do: Solve past CBSE board exam papers for Polynomials questions.
  • How It Helps: Shows question types and helps with time management.
  • Tip: Practice 5-10 questions in 20-30 minutes.

9. Check Solutions for Class 10 Maths Chapter 2

  • What to Do: After solving, compare with class 10 Maths Chapter 2 solutions.
  • How It Helps: Corrects errors and teaches better methods.
  • Tip: Note mistakes and retry problems.

10. Revise Regularly

  • What to Do: Review Chapter 2 weekly. Re-solve tough questions like division problems.
  • How It Helps: Keeps concepts fresh for the board exam.
  • Tip: Spend 30 minutes weekly on Polynomials.

11. Clear Doubts Fast

  • What to Do: Ask your teacher or friends if the division algorithm is tricky.
  • How It Helps: Strengthens your base for exam preparation.
  • Tip: Join a study group for discussion.

12. Manage Time

  • What to Do: Solve 5-10 problems in 20-30 minutes daily to build speed.
  • How It Helps: Prepares you for the 3-hour board exam.
  • Tip: Focus on quick, accurate solving.

13. Stay Positive

  • What to Do: Believe in yourself! Practice daily and stay calm.
  • How It Helps: Reduces fear and boosts confidence.
  • Tip: Take breaks after exercises—relax and recharge!

How www.growupncert.com Helps Students

The website www.growupncert.com is a fantastic tool for Class 10 students. Here’s how it supports your exam preparation for class 10 Maths Chapter 2 solutions and the CBSE syllabus:

  • Class 10 Maths Chapter 2 Solutions: Get step-by-step answers to NCERT Exercises 2.1, 2.2, and 2.3. Stuck on finding zeros? The site explains it simply.
  • Solutions for Class 10 Maths Chapters: Find help for all chapters, building your Class 10 Maths syllabus skills.
  • Previous Years’ Question Papers: Download past CBSE board exam papers. Practice Polynomials questions to know the pattern.
  • CBSE Syllabus Alignment: The site matches the Class 10 Maths syllabus, keeping your focus on board exam topics.
  • Practice Tests: Take mock tests for Chapter 2 to check your skills and speed.
  • Simple Explanations: Concepts like zeros or division algorithm are broken down in easy language.
  • Revision Notes: Get short notes on key points, like sum and product of zeros, for quick review.
  • Doubt Support: Some sections may offer tips or ways to ask questions, helping with tricky problems.
  • How to Use It: Visit www.growupncert.com, find class 10 Maths Chapter 2 solutions, practice with past papers, and review notes. Pair this with your NCERT book for success.

Time Management for Board Exam Prep

  • Daily Study: Spend 30-60 minutes on Chapter 2. Solve 5-10 problems from class 10 Maths Chapter 2 solutions.
  • Weekly Goals: Master Polynomials in 1-2 weeks, then move to other chapters.
  • Before Exam: Revise Chapter 2 in 2-3 days. Focus on zeros and division.
  • Practice: Solve past papers in 3 hours to mimic the board exam.

Common Challenges and Solutions

  • Challenge: Confusion with finding zeros.
    • Solution: Practice factoring. Check class 10 Maths Chapter 2 solutions on www.growupncert.com.
  • Challenge: Trouble with division algorithm.
    • Solution: Follow step-by-step division. Use the website for examples.
  • Challenge: Silly mistakes.
    • Solution: Write steps clearly and verify with solutions.
  • Challenge: Exam time pressure.
    • Solution: Practice timed questions from past papers.

Benefits of Mastering Class 10 Maths Chapter 2 Solutions

  • Board Exam Success: Chapter 2 questions (6-8 marks) are common. Solutions help you nail them.
  • Strong Foundation: Prepares you for Quadratic Equations and future studies.
  • Confidence Boost: Practice and www.growupncert.com make Polynomials easy.

Final Thoughts

Class 10 Maths Chapter 2 solutions are your key to mastering Polynomials in the CBSE syllabus. This chapter, part of the Class 10 Maths syllabus, teaches you about types, zeros, and division of polynomials. For exam preparation, read the NCERT book, solve exercises, practice step by step, and use past papers. The website www.growupncert.com helps with clear class 10 Maths Chapter 2 solutions, practice tests, and notes. Study daily, revise regularly, and stay positive. With this approach, you’ll ace Chapter 2 and shine in your board exams. Start today—you’ve got this!

 

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