Exercise 9.5 - Q2
Textbook Exercise Problem
Use the identity (x + a)(x + b) = x² + (a + b)x + ab to find the product: (x + 3)(x + 7).
Step-by-Step Verified Solution:
- Step 1: Identify terms: a = 3, b = 7.
- Step 2: (a + b) = 3 + 7 = 10, and ab = 3 × 7 = 21.
- Step 3: Substitute into identity: x² + (10)x + 21.
Final Answer: x² + 10x + 21
Key Concept / Exam Strategy: The coefficient of the linear term is the sum of constants; the constant term is their product.
Generated Exercise #1
Foundation
Question: Evaluate (103)² using a suitable algebraic identity.
Core Concept Tested: Binomial square expansion on numerical constants
Mentor Hint: 103 = 100 + 3. Use (a + b)².
Step-by-Step Solution:
- Step 1: Write as (100 + 3)².
- Step 2: (100)² + 2(100)(3) + (3)².
- Step 3: 10,000 + 600 + 9 = 10,609.
Verified Answer: 10,609
Generated Exercise #2
Intermediate
Question: Evaluate 98 × 102 using the difference of squares identity.
Core Concept Tested: (a - b)(a + b) = a² - b²
Mentor Hint: 98 = 100 - 2 and 102 = 100 + 2.
Step-by-Step Solution:
- Step 1: (100 - 2)(100 + 2) = (100)² - (2)²
- Step 2: 10,000 - 4 = 9,996.
Verified Answer: 9,996
Generated Exercise #3
Intermediate
Question: If x + 1/x = 5, find the value of x² + 1/x².
Core Concept Tested: Reciprocal sum squaring identity
Mentor Hint: Square both sides of (x + 1/x). Note that 2·x·(1/x) = 2.
Step-by-Step Solution:
- Step 1: (x + 1/x)² = x² + 2 + 1/x².
- Step 2: 5² = x² + 1/x² + 2.
- Step 3: 25 - 2 = x² + 1/x² => x² + 1/x² = 23.
Verified Answer: 23
Generated Exercise #4
JEE-NEET Foundation
Question: If x - 1/x = 4, find the value of x⁴ + 1/x⁴.
Core Concept Tested: Two-stage iterative algebraic identity squaring
Mentor Hint: Square once to find x² + 1/x², then square again.
Step-by-Step Solution:
- Step 1: (x - 1/x)² = x² - 2 + 1/x² => 4² = x² + 1/x² - 2 => x² + 1/x² = 18.
- Step 2: Square both sides: (x² + 1/x²)² = x⁴ + 2 + 1/x⁴.
- Step 3: 18² = x⁴ + 1/x⁴ + 2 => 324 - 2 = x⁴ + 1/x⁴ = 322.
Verified Answer: 322
Generated Exercise #5
Advanced HOTS
Question: Factorize completely: 16x⁴ - 81y⁴.
Core Concept Tested: Repeated application of difference of two squares
Mentor Hint: Write as (4x²)² - (9y²)².
Step-by-Step Solution:
- Step 1: (4x² - 9y²)(4x² + 9y²).
- Step 2: Factorize (4x² - 9y²) further: (2x - 3y)(2x + 3y).
- Step 3: Final product = (2x - 3y)(2x + 3y)(4x² + 9y²).
Verified Answer: (2x - 3y)(2x + 3y)(4x² + 9y²)