Hint
Use the stated units and precision. Use a common denominator for addition, multiply whole factors for a product, and invert the full divisor for a quotient.
Worked solution
(a)
Multiply each numerator by its missing denominator factor. Their sum is \(8 x - 36\). Neither original denominator may vanish.
Answer: \(\displaystyle \frac{8 x - 36}{\left(x - 6\right) \left(x - 4\right)},\quad x\ne 4,6\)
(b)
Multiply \(\displaystyle \frac{2}{x - 4}\) by \(\displaystyle \frac{x}{6} - 1\). The nonzero constant numerator of B gives no further zeros, but B must itself remain defined.
Answer: \(\displaystyle \frac{\frac{x}{3} - 2}{x - 4}\), \(x\ne 4,6\)
(c)
At this permitted input, direct substitution gives \(\displaystyle \frac{2}{3}\) + \(6\) = \(\displaystyle \frac{20}{3}\).
Answer: \(\displaystyle \frac{20}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Form the complete common denominator and scale both numerators correctly.
- 1 mark: Combine the numerator and retain both original exclusions.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Invert the whole second fraction and multiply, preserving the original domain.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Substitute into both original fractions at a permitted input and add exactly.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 7 marks.