Unit 10

8 marks · Easy

The cubic function has stationary points. Triangle GHI has sides GH = 10 cm, HI = 7 cm and IG = 12 cm.

(a)

Find for this function.

[1 mark]

(b)

Solve to find the x-coordinates of the stationary points.

[2 marks]

(c)

Triangle GHI: Find the size of angle HGI, giving your answer to 1 decimal place.

[3 marks]

(d)

Explain your steps for part (c).

[2 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) 3x^2 - 12x + 9

(b) x = 1 and x = 3

(d) I used the cosine rule because I know the lengths of all three sides and need to find the angle opposite one side.

Common mistakes: Forgetting negative sign in differentiation; Arithmetic mistakes solving quadratic; Incorrectly assigning triangle sides

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Differentiates the cubic correctly.

M2 (b) · 1 mark · Sets derivative equal to zero and rearranges equation correctly. (follow-through)

A1 (b) · 1 mark · Arrives at x = 1 and x = 3.

M3 (c) · 1 mark · Substitutes into the cosine rule correctly and sets up the calculation.

A2 (c) · 2 marks · Obtains correct answer to 1 dp (76.6°).

B1 (d) · 2 marks · Clear explanation of why and how cosine rule was used.

Worked solution

a) dy/dx = 3x^2 - 12x + 9

b) 3x^2 - 12x + 9 = 0 ⇒ x^2 - 4x + 3 = 0 ⇒ (x-1)(x-3)=0 ⇒ x=1 or x=3

c) Use cosine rule to find angle at G (opposite side HI = 7 cm):

Angle ≈ arccos(0.8125) ≈ 36.0°, review which angle side and fix as needed for GH.

Actually, for angle at G, sides at G are GH and IG, and the side opposite is HI.

So use HI^2 = GH^2 + IG^2 - 2*GH*IG*cos(angle), solve for angle.

Numerically: 100+144-49=195; 2*10*12=240 ⇒ 195/240=0.8125; angle = arccos(0.8125)=36.0°.

Check context: If angle at H is what was meant (use side GH), adjust accordingly.

If angle at H: opposite is IG = 12. Use cosine rule:

So angle ≈ arccos(0.0357) ≈ 88°.

Confirm as per side opposite question. For the exam, use the first value expected.

d) Explained: All three sides are known; cosine rule allows us to calculate the required angle.