NCEA Scholarship Calculus free practice questions with answers

Question 1 (8 marks) [Complex numbers and roots of unity]

(a) Solve z³ = −8 and give all three solutions exactly, in the form a + bi.

(a) Answer: z = −2, z = 1 + i√3, z = 1 − i√3

• Entry step: worth marks in the 1-4 range of the 8-point question score.
• Write −8 = 8 cis(π + 2kπ), so z = 2 cis( • k = 0: 1 + i√3; k = 1: −2; k = 2: 1 − i√3. [1 mark]

Roots of a complex number: take the root of the modulus and divide the argument (plus 2kπ) by n; the n roots are evenly spaced around a circle.

(b) Let ω = cos( cos( (b) Answer: cos( • Scholarship band (5-6): the part needs your own method and accurate algebra.

• By de Moivre ω⁵ = cis(2π) = 1, so 0 = ω⁵ − 1 = (ω − 1)(1 + ω + ω² + ω³ + ω⁴). Since ω ≠ 1, the second factor is 0. [1 mark]
• Take real parts: 1 + cos( • cos( + cos( Sums of roots of unity vanish: multiply out (ω − 1) times the geometric sum, then take real parts and use symmetry to pair the angles.

(c) Hence find the exact value of cos( (c) Answer: cos( • Outstanding band (7-8): the part needs insight and a complete, efficient argument.

• Let c = cos( • c = (−2 ± √20) ÷ 8 = • belongs to cos( Two expressions for the same unknown: a double-angle identity turns the sum from the roots of unity into a quadratic you can solve exactly; use the quadrant to reject the wrong root. π + 2kπ

3

) for k = 0, 1, 2. [1 mark]

2π

5 ) + i sin(2π

5 ). Explain why 1 + ω + ω² + ω³ + ω⁴ = 0, and use it to show that

2π

5 ) + cos(4π

5 ) = −1

2 .

2π

5 ) + cos(4π

5 ) = −1

2

2π

5 ) + cos(4π

5 ) + cos(6π

5 ) + cos(8π

5 ) = 0. [1 mark]

6π

5 ) = cos(4π

5 ) and cos(8π

5 ) = cos(2π

5 ) (symmetry about the real axis), so 1 + 2[cos(2π

5 )

4π

5 )] = 0, giving the result. [1 mark]

2π

5 ). Show how you decide between the possible values.

2π

5 ) = √5 − 1

4

2π

5 ). Then cos(4π

5 ) = 2c² − 1 (double angle). Substituting into part (b): c + 2c² − 1 = −1

2 , so 4c² + 2c − 1 = 0. [1 mark]

−1 ± √5

4

. [1 mark]

2π

5 is in the first quadrant, so c > 0 and c = √5 − 1

4

≈ 0.309. (The other root is negative and 4π

5 ).) [1 mark]