DAV Maths Algebraic Identities Chapter 7 Enrichment Question Answers

Are you looking for complete, step-by-step verified solutions for DAV Class 8 Maths Chapter 7 Enrichment Questions (Algebraic Identities)? At NCERTBooksPDF, our detailed solutions provide clear proofs and step-by-step methods for reciprocal identity problems, sum-product systems, and advanced identity applications.

Key Identities Used in Enrichment Problems

Chapter 7 Enrichment Questions: Complete Solutions

Question 1: Reciprocal Powers

Problem: If x + 1/x = 5, find the value of:
(i) x² + 1/x²
(ii) x⁴ + 1/x⁴
Solution:
(i) Finding x² + 1/x²:
Squaring both sides of x + 1/x = 5:
(x + 1/x)² = 5²
x² + 2(x)(1/x) + (1/x)² = 25
x² + 2 + 1/x² = 25
x² + 1/x² = 25 – 2 = 23.

(ii) Finding x⁴ + 1/x⁴:
Squaring both sides of x² + 1/x² = 23:
(x² + 1/x²)² = 23²
(x²)² + 2(x²)(1/x²) + (1/x²)² = 529
x⁴ + 2 + 1/x⁴ = 529
x⁴ + 1/x⁴ = 529 – 2 = 527.

Question 2: Difference of Reciprocals

Problem: If x – 1/x = 7, find the value of x² + 1/x².
Solution:
Squaring both sides of x – 1/x = 7:
(x – 1/x)² = 7²
x² – 2(x)(1/x) + 1/x² = 49
x² – 2 + 1/x² = 49
x² + 1/x² = 49 + 2 = 51.

Question 3: Linear Combination System

Problem: If 2x + 3y = 12 and xy = 6, find the value of 4x² + 9y².
Solution:
Squaring both sides of 2x + 3y = 12:
(2x + 3y)² = 12²
(2x)² + 2(2x)(3y) + (3y)² = 144
4x² + 12xy + 9y² = 144
Substitute xy = 6:
4x² + 12(6) + 9y² = 144
4x² + 72 + 9y² = 144
4x² + 9y² = 144 – 72 = 72.

Question 4: Symmetric Values

Problem: If a² + b² = 29 and ab = 10, find the values of (i) a + b and (ii) a – b.
Solution:
(i) Finding a + b:
(a + b)² = a² + b² + 2ab
= 29 + 2(10) = 29 + 20 = 49
a + b = √49 = ±7 (or 7 for positive values).

(ii) Finding a – b:
(a – b)² = a² + b² – 2ab
= 29 – 2(10) = 29 – 20 = 9
a – b = √9 = ±3 (or 3 for a > b).

Question 5: Continuous Identity Multiplication

Problem: Show that: (x + 1)(x – 1)(x² + 1)(x⁴ + 1) = x⁸ – 1.
Proof:
Step 1: (x + 1)(x – 1) = x² – 1.
Step 2: (x² – 1)(x² + 1) = (x²)² – 1² = x⁴ – 1.
Step 3: (x⁴ – 1)(x⁴ + 1) = (x⁴)² – 1² = x⁸ – 1.
Hence Proved!

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