TMUA Algebraic Techniques & Manipulation 2 — Questions and Answers
Question 1: Simplify: (x² - 9) / (x² - x - 6)
- (x+3)/(x+2) (Correct answer)
- (x-3)/(x-2)
- (x+3)/(x-2)
- (x-3)/(x+2)
Correct answer: (x+3)/(x+2)
Factor numerator as (x-3)(x+3) and denominator as (x-3)(x+2), then cancel (x-3) to get (x+3)/(x+2).
Question 2: If p and q are roots of x² - 5x + 3 = 0, find p² + q².
- 19 (Correct answer)
- 22
- 13
- 25
Correct answer: 19
Using p+q=5 and pq=3, p²+q² = (p+q)² - 2pq = 25 - 6 = 19.
Question 3: Solve for x: |2x - 3| = 7
- x = 5 or x = -2 (Correct answer)
- x = 5 or x = 2
- x = -2 or x = -5
- x = 5 only
Correct answer: x = 5 or x = -2
Setting 2x-3=7 gives x=5; setting 2x-3=-7 gives x=-2.
Question 4: Expand and simplify: (2x + 3)² - (2x - 3)²
- 24x (Correct answer)
- 8x² + 18
- 24
- 8x
Correct answer: 24x
Using difference of squares: [(2x+3)+(2x-3)][(2x+3)-(2x-3)] = (4x)(6) = 24x.
Question 5: Find all values of k such that x² + kx + 9 = 0 has exactly one real solution.
- k = ±6 (Correct answer)
- k = ±3
- k = ±9
- k = ±4
Correct answer: k = ±6
For one real solution, discriminant = 0: k² - 36 = 0, so k = ±6.
Question 6: Simplify: (2x³y²)³ / (4x²y)
- 2x⁷y⁵ (Correct answer)
- 8x⁷y⁵
- 2x⁵y⁵
- 8x⁹y⁶
Correct answer: 2x⁷y⁵
Numerator becomes 8x⁹y⁶; dividing by 4x²y gives 2x⁷y⁵.
Question 7: If f(x) = x² + 2x - 1, find the value of f(x+1) - f(x).
- 2x + 3 (Correct answer)
- 2x + 2
- 2x + 1
- x + 3
Correct answer: 2x + 3
f(x+1) = (x+1)²+2(x+1)-1 = x²+4x+2; subtracting f(x) = x²+2x-1 gives 2x+3.
TMUA Algebraic Techniques & Manipulation 2
Simplify: (x² - 9) / (x² - x - 6)