Hint
Equating y-values gives a quadratic whose discriminant depends on c. Recover y after finding the repeated x-value.
Worked solution
(a)
Intersections satisfy \(\displaystyle x^{2}-5x+1-c=0\). Tangency requires \(\displaystyle 25-4(1-c)=21+4c=0\), so \(\displaystyle c= - \frac{21}{4}\).
Answer: \(\displaystyle c=-\frac{21}{4}\).
(b)
The repeated root is \(\displaystyle x= \frac{5}{2}\). On the tangent, \(\displaystyle y= 2(\frac{5}{2})- \frac{21}{4}= - \frac{1}{4}\). The point is \(\displaystyle (\frac{5}{2}, - \frac{1}{4})\).
Answer: \(\displaystyle (\frac52,-\frac14)\).
(c)
The intersection equation is \(\displaystyle x^{2}-5x+6=(x-2)(x-3)=0\). The points are (2,−1) and (3,1). There are two distinct roots (equivalently, discriminant 1>0), so the line is a secant.
Answer: \((2,-1)\), \((3,1)\); two distinct intersection roots.
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: Obtain the correct intersection equation.
- 1 mark: Set its discriminant equal to zero.
- 1 mark: Solve for \(\displaystyle c= - \frac{21}{4}\).
Alternative methods: Completing the square or matching the derivative to 2 is accepted.
Common error: Using the discriminant of the curve before subtracting the line.
(b) · 2 marks
- 1 mark: Find \(\displaystyle x= \frac{5}{2}\).
- 1 mark: Substitute to give the paired coordinate \(\displaystyle (\frac{5}{2}, - \frac{1}{4})\).
Alternative methods: Substitution into the curve is accepted.
Common error: Using the intercept c as the y-coordinate of contact.
(c) · 3 marks
- 1 mark: Solve to obtain \(\displaystyle x=2\) and 3.
- 1 mark: Give both paired points.
- 1 mark: Explain using two distinct roots or a positive discriminant why it is not tangent.
Alternative methods: Any correct intersection method with a valid distinction from a repeated root is accepted.
Common error: Assuming parallel lines must have the same number of intersections.
Total: 8 marks.