TMUA Algebraic Techniques & Manipulation 4 — Questions and Answers
Question 1: If x - 1/x = 3, find x² + 1/x².
- 11 (Correct answer)
- 9
- 7
- 13
Correct answer: 11
Squaring x - 1/x = 3 gives x² - 2 + 1/x² = 9, so x² + 1/x² = 11.
Question 2: Find the remainder when x³ - 3x² + 5x - 2 is divided by (x - 2).
- 4 (Correct answer)
- 0
- 8
- 2
Correct answer: 4
By the remainder theorem, substitute x=2: 8 - 12 + 10 - 2 = 4.
Question 3: Solve for real x: x/(x-2) + 2/(x+1) = 1
- x = 0 or x = 3 (Correct answer)
- x = 2 or x = -1
- x = 0 or x = -3
- x = 1 or x = 4
Correct answer: x = 0 or x = 3
Multiplying through by (x-2)(x+1) and simplifying yields x²-3x=0, so x=0 or x=3.
Question 4: Simplify: (√5 + √3)² - (√5 - √3)²
- 4√15 (Correct answer)
- 16
- 8√15
- 2√15
Correct answer: 4√15
Using a²-b² = (a+b)(a-b): [(√5+√3)+(√5-√3)][(√5+√3)-(√5-√3)] = (2√5)(2√3) = 4√15.
Question 5: For what values of c does x² + 6x + c > 0 for all real x?
- c > 9 (Correct answer)
- c > 6
- c > 3
- c > 36
Correct answer: c > 9
The quadratic is always positive when discriminant < 0: 36 - 4c < 0, giving c > 9.
Question 6: Factorise: x³ + 8
- (x+2)(x²-2x+4) (Correct answer)
- (x+2)(x²+2x+4)
- (x-2)(x²+2x+4)
- (x+2)²(x-2)
Correct answer: (x+2)(x²-2x+4)
This is a sum of cubes: a³+b³ = (a+b)(a²-ab+b²) with a=x, b=2.
Question 7: If f(x) = (x+1)/(x-1), find f(f(x)).
- x (Correct answer)
- 1/x
- (x+1)/(x+2)
- 2x/(x-1)
Correct answer: x
f(f(x)) = [(x+1)/(x-1)+1]/[(x+1)/(x-1)-1] = [2x/( x-1)]/[2/(x-1)] = x.
TMUA Algebraic Techniques & Manipulation 4
If x - 1/x = 3, find x² + 1/x².