Hint
Use the stated units and precision. Multiply by the appropriate root or conjugate, retain the complete denominator and simplify.
Worked solution
(a)
\(\displaystyle \frac{1}{a+\sqrt d}=\frac{a-\sqrt d}{a^2-d}\); here the denominator is \(19\), so the exact simplified result is \(\displaystyle \frac{5}{19} - \frac{\sqrt{6}}{19}\).
Answer: \(\displaystyle \frac{5}{19} - \frac{\sqrt{6}}{19}\)
(b)
Multiply by the appropriate root or conjugate, retain the complete denominator and simplify.
\(\displaystyle - \frac{1}{19}\)
Answer: \(\displaystyle - \frac{1}{19}\)
(c)
Multiply by the appropriate root or conjugate, retain the complete denominator and simplify.
\(19\)
Answer: \(19\)
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: Multiply numerator and denominator by the nonzero conjugate.
- 1 mark: Simplify the resulting rational denominator and both numerator terms.
- 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: Multiply by the appropriate root or conjugate, retain the complete denominator and simplify.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
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: Multiply by the appropriate root or conjugate, retain the complete denominator and simplify.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
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.