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 \(12 x - 71\). Neither original denominator may vanish.

Answer: \(\displaystyle \frac{12 x - 71}{\left(x - 8\right) \left(x - 3\right)},\quad x\ne 3,8\)

(b)

Multiply \(\displaystyle \frac{7}{x - 3}\) by \(\displaystyle \frac{x}{5} - \frac{8}{5}\). The nonzero constant numerator of B gives no further zeros, but B must itself remain defined.

Answer: \(\displaystyle \frac{\frac{7 x}{5} - \frac{56}{5}}{x - 3}\), \(x\ne 3,8\)

(c)

At this permitted input, direct substitution gives \(\displaystyle \frac{7}{6}\) + \(5\) = \(\displaystyle \frac{37}{6}\).

Answer: \(\displaystyle \frac{37}{6}\)

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.