Hint
For part (a), match slopes rather than line intercepts. For part (c), do not divide by a zero slope.
Worked solution
(a)
The derivative is 2x−6. Parallelism requires \(\displaystyle 2x-6=4\), so \(\displaystyle x=5\). Substitution into the curve gives \(\displaystyle y=25-30+11=6\), hence (5,6).
Answer: \((5,6)\).
(b)
The normal slope is \(\displaystyle - \frac{1}{4}\) and the point is (5,6). Thus \(\displaystyle y- 6= - \frac{x- 5}{4}\), or \(\displaystyle y= - \frac{x}{4}+ \frac{29}{4}\).
Answer: \(y-6=-\tfrac14(x-5)\), equivalently \(\displaystyle y= - \frac{x}{4}+ \frac{29}{4}\).
(c)
At \(\displaystyle x=3\), the derivative 2x−6 is zero. The tangent is horizontal, \(\displaystyle y=2\); the perpendicular normal is vertical, \(\displaystyle x=3. A\) finite slope formula \(\displaystyle - \frac{1}{0}\) is not used.
Answer: Tangent \(\displaystyle y=2\); normal \(\displaystyle x=3\), because the tangent is horizontal.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Set the derivative equal to the required slope 4.
- 1 mark: Solve for \(\displaystyle x=5\).
- 1 mark: Give the point (5,6).
Alternative methods: A correct first-principles slope expression is accepted.
Common error: Solving for an intersection with the given line instead of matching tangent slopes.
(b) · 2 marks
- 1 mark: Use normal slope \(\displaystyle - \frac{1}{4}\).
- 1 mark: Give the correct line through (5,6).
Alternative methods: Any equivalent exact line equation is accepted.
Common error: Using the given line intercept −7 for the normal.
(c) · 2 marks
- 1 mark: Give the horizontal tangent \(\displaystyle y=2\).
- 1 mark: Give the vertical normal \(\displaystyle x=3\) with a perpendicular-to-horizontal explanation.
Alternative methods: A correctly labelled sketch with both equations is accepted.
Common error: Writing \(\displaystyle y=3\) for a vertical line or treating \(\displaystyle - \frac{1}{0}\) as a real slope.
Total: 7 marks.