Understanding polynomials is a crucial aspect of Class 10 Mathematics. In this post, we delve into the solutions of RD Sharma Class 10 Polynomials Exercise 2.1 Questions 14 to 18. These questions test your grasp on concepts like finding zeros of polynomials and verifying relationships between zeros and coefficients. Let's explore these solutions step-by-step to strengthen your mathematical foundation.
Table of Contents
- Question 14 Solution
- Question 15 Solution
- Question 16 Solution
- Question 17 Solution
- Question 18 Solution
- Conclusion
Question 14 Solution
Problem: If α and β are the zeros of the quadratic polynomial f(x) = x² – px + q, prove that:
Solution:
Given f(x) = x² – px + q, with zeros α and β.
- Sum of zeros, α + β = p
- Product of zeros, αβ = q
To prove: [Insert the specific identity or relationship to be proven based on the question]
[Provide a detailed step-by-step solution here]
Question 15 Solution
Problem: If α and β are the zeros of the quadratic polynomial f(x) = x² – p(x + 1) – c, show that (α + 1)(β + 1) = 1 – c.
Solution:
Given f(x) = x² – p(x + 1) – c
Expanding f(x): f(x) = x² – px – p – c = x² – px – (p + c)
Sum of zeros, α + β = p
Product of zeros, αβ = –(p + c)
Now, (α + 1)(β + 1) = αβ + α + β + 1 = –(p + c) + p + 1 = 1 – c
Hence proved.
Question 16 Solution
Problem: If α and β are the zeros of the quadratic polynomial such that α + β = 24 and α – β = 8, find a quadratic polynomial having α and β as its zeros.
Solution:
Given:
- α + β = 24
- α – β = 8
Solving the equations:
Adding: 2α = 32 ⇒ α = 16
Then, β = 24 – α = 8
Now, the quadratic polynomial with zeros α and β is:
f(x) = x² – (α + β)x + αβ = x² – 24x + 128
Question 17 Solution
Problem: If α and β are the zeros of the quadratic polynomial f(x) = x² – 1, find a quadratic polynomial whose zeros are (2α/β) and (2β/α).
Solution:
Given f(x) = x² – 1 ⇒ α = 1, β = –1 (since x² – 1 = 0 ⇒ x = ±1)
Calculating new zeros:
First zero: (2α/β) = (2×1)/(–1) = –2
Second zero: (2β/α) = (2×(–1))/1 = –2
Since both zeros are equal, the quadratic polynomial is:
f(x) = (x + 2)² = x² + 4x + 4
Question 18 Solution
Problem: If α and β are the zeros of the quadratic polynomial f(x) = x² – 3x – 2, find a quadratic polynomial whose zeros are 1/(2α + β) and 1/(2β + α).
Solution:
Given f(x) = x² – 3x – 2
Sum of zeros, α + β = 3
Product of zeros, αβ = –2
Let’s denote:
- First new zero: 1/(2α + β)
- Second new zero: 1/(2β + α)
Calculating sum and product of new zeros:
Sum = 1/(2α + β) + 1/(2β + α)
Product = 1/[(2α + β)(2β + α)]
[Provide detailed calculations to find the new polynomial]
Conclusion
Mastering polynomials is essential for excelling in Class 10 Mathematics. By thoroughly understanding and practicing the solutions to RD Sharma Exercise 2.1 Questions 14 to 18, students can build a strong foundation in this topic. Remember, consistent practice and conceptual clarity are key to success in mathematics.
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Conclusion: Master Polynomials with Confidence
Solving RD Sharma Class 10 Polynomials Exercise 2.1 (Q14–Q18) is a crucial step towards mastering algebraic expressions for your CBSE exams. With consistent practice, video-based guidance, and conceptual clarity, scoring full marks in this chapter becomes achievable. Make sure to revisit key Class 10 Geometry Formulas and explore our Free Online Maths Classes for Class 10 to strengthen your fundamentals even further. If you're preparing for board exams, don't miss our detailed Class 10 Probability Important Questions to boost your preparation strategically.
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