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In this question, i2 = −1.
(a)Write the complex number 2 + 3i4 − 5i in the form a + bi, where a, b ∈ ℝ.
(a)
(b)Use de Moivre’s theorem and the expression (cosθ + i sinθ)2 to prove that the following is true, for any angle θ:
cos 2θ = cos2θ − sin2θ
(b)
(c)Use de Moivre’s theorem to find two values of z for which:
z6 = −64i
Give each answer in the form c + di, where c, d ∈ ℝ. Leave c and d in surd form.
(c)
In this question, i2 = −1.
(a)Find the two solutions of the following equation in z, where z is a complex number. Give each answer in the form a + bi, where a, b ∈ ℝ. z2 + 12z + 261 = 0
(a)
(b)Use de Moivre's theorem to write (1 − √3 i)9 in the form a + bi, where a, b ∈ ℝ.
(b)
(c)The point w = −2 + 2i is shown in the Argand diagram below.
(i)Plot and label the complex number u = 4(cos π6 + i sin π6) on the diagram, as accurately as possible.
(c)(i)
(ii)The complex number o is 0 + 0i. Find the size of the angle ∠wou. Give your answer in radians.
(c)(ii)
In this question, i2 = −1.
(a)The complex number z1 = 1 + i is a root of the equation z2 + (3 − 2i)z + p = 0. Find the value of p, where p = a + bi, with a, b ∈ ℤ.
(a)
(b)Use De Moivre's theorem to find the values of w for which w2 = −1 + √3 i. Give each value of w in the form a + bi, with a, b ∈ ℝ.
(b)
(c)The Argand diagram below shows the complex number u = a + bi, where a, b ∈ ℝ.
(i)Write the complex numbers iu and iu in their simplest form, in terms of a and b, where iu is the complex conjugate of iu.
(c)(i)
(ii)Plot and label the complex numbers iu and iu on the diagram above, as accurately as possible.
(c)(ii)
(iii)State a transformation, or series of transformations, that would send u to iu. Do not include a translation in your answer.
(c)(iii)
(a)z = 6 + 2i, where i2 = −1.
(i)Show that z − iz = 8 − 4i.
(a)(i)
(ii)Show that |z|2 + |iz|2 = |z − iz|2.
(a)(ii)
(iii)The circle c passes through the points z, iz, and 0 (not to scale). z and iz are endpoints of a diameter of the circle. Find the area of the circle c in terms of π.
(a)(iii)
(b)(√3 − i)9 can be written in the form a + ib, where a, b ∈ ℤ and i2 = −1. Use de Moivre's theorem to find the value of a and the value of b.
(b)
(a)(4 − 2i)(2 + 4i) = 0 + ki, where k ∈ ℤ, and i2 = −1. Find the value of k.
(a)
(b)Find √(−5 + 12i). Give both of your answers in the form a + bi, where a, b ∈ ℝ.
(b)
(c)Use de Moivre's theorem to find the three roots of z3 = −8. Give each answer in the form a + bi, where a, b ∈ ℝ, and i2 = −1.
(c)
(a)Find the two complex numbers z1 and z2 that satisfy the following simultaneous equations, where i2 = −1:
iz1 = −4 + 3i
3z1 − z2 = 11 + 17i
Write your answers in the form a + bi where a, b ∈ ℤ.
(a)
(b)The complex numbers 3 + 2i and 5 − i are the first two terms of a geometric sequence.
(i)Find r, the common ratio of the sequence. Write your answer in the form a + bi where a, b ∈ ℤ.
(b)(i)
(ii)Use de Moivre's Theorem to find T9, the ninth term of the sequence. Write your answer in the form a + bi, where a, b ∈ ℤ.
(b)(ii)
(a)3 + 2i is a root of z2 + pz + q = 0, where p, q ∈ ℝ, and i2 = −1. Find the value of p and the value of q.
(a)
(b)(i)v = 2 − 2√3 i. Write v in the polar form r(cos θ + i sin θ), where r ∈ ℝ and 0 ≤ θ ≤ 2π.
(b)(i)
(ii)Use your answer to part (b)(i) to find the two possible values of w, where w2 = v. Give your answers in the form a + ib, where a, b ∈ ℝ.
(b)(ii)
(a)Prove, using induction, that if n is a positive integer then (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), where i2 = −1 (de Moivre's Theorem).
(a)
(b)Hence, or otherwise, find (−12 + √32 i)3 in its simplest form.
(b)
z = −√3 + i, where i2 = −1.
(a)Use de Moivre's Theorem to write z4 in the form a + b√c i, where a, b, c ∈ ℤ.
(a)
(b)The complex number w is such that |w| = 3 and w makes an angle of 30° with the positive sense of the real axis. If t = zw, write t in its simplest form.
(b)
(a)(−4 + 3i) is one root of the equation az2 + bz + c = 0, where a, b, c ∈ ℝ, and i2 = −1. Write the other root (the conjugate).
(a)
(b)Use de Moivre's Theorem to express (1 + i)8 in its simplest form.
(b)
(c)(1 + i) is a root of the equation z2 + (−2 + i)z + 3 − i = 0. Find its other root in the form m + ni, where m, n ∈ ℝ, and i2 = −1.
(c)
(a)The complex numbers z1, z2 and z3 are such that 2z1 = 1z2 + 1z3, z2 = 2 + 3i and z3 = 3 − 2i, where i2 = −1. Write z1 in the form a + bi, where a, b ∈ ℤ.
(a)
(b)Let ω be a complex number such that ωn = 1, ω ≠ 1, and S = 1 + ω + ω2 + … + ωn−1 (a root of unity). Use the formula for the sum of a finite geometric series to write the value of S in its simplest form.
(b)
Let z1 = 1 + 2i, where i2 = −1.
(a)The complex number z1 is a root of the equation 2z3 − 7z2 + 16z − 15 = 0. Find the other two roots of the equation.
(a)
(b)(i)Let w = z1 · z̄1, where z̄1 is the conjugate of z1. Plot z1, z̄1 and w on an Argand diagram and label each point.
(b)(i)
(ii)Find the measure of the acute angle ∠z1wz̄1, formed by joining z1 to w to z̄1 on the diagram. Give your answer correct to the nearest degree.
(b)(ii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.