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IB Mathematics AA HL - Questionbank

Complex Numbers

Different Forms, Roots, De Moivre’s Theorem, Argand Diagram, Geometric Applications…

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Question 1

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easy

[Maximum mark: 4]

On the Argand diagram below, the point A represents the complex number 4i4{\mathrm{\hspace{0.05em}i}\mkern 1mu} and the point B represents the complex number 5+i-5+{\mathrm{\hspace{0.05em}i}\mkern 1mu}. The shape ABCD is a square.

fed41c2307d3c10825aed04db2b46a9169006d2c.svg

Determine the complex number represented by:

  1. the point C; [2]

  2. the point D. [2]

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Question 2

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easy

[Maximum mark: 6]

Solve the equation z3=1z^3 = 1, giving your answers in Cartesian form.

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Question 3

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easy

[Maximum mark: 6]

A circle of radius 3 and centre (0,3) is drawn on an Argand diagram. The tangent to the circle from the point B(0,9)(0,9) meets the circle at the point A as shown. Let w=OAw = \vv{\mathrm{OA}}.

756bc10d49856365edc8188e18631d432452e02d.svg

  1. Show that w=33|w| = 3\sqrt{3}. [2]

  2. Find argw\arg w. [2]

  3. Hence write ww in the form a+bia+b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R}. [2]

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Question 4

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easy

[Maximum mark: 6]

In this question give all angles in radians.

Let z=1+2iz = 1 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu} and w=4+iw = 4 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}.

  1. Find z+wz+w. [1]

  2. Find:

    1. z+w|z+w|;

    2. arg(z+w)\arg(z+w). [3]

  3. Find θ\theta, the angle shown on the diagram below. [2]

    9f1b9b167af97e4b2c9ee84b7c9014f2cdaf63c7.svg

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Question 5

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easy

[Maximum mark: 6]

Consider the complex number z=w1w2z = \dfrac{w_1}{w_2} where w1=2+6iw_1 = \sqrt{2} + \sqrt{6}{\mathrm{\hspace{0.05em}i}\mkern 1mu} and w2=3+3iw_2 = 3 + \sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}.

  1. Express w1w_1 and w2w_2 in modulus-argument form and write down

    1. the modulus of zz;

    2. the argument of zz. [4]

  2. Find the smallest positive integer value of nn such that znz^n is a real number. [2]

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Question 6

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easy

[Maximum mark: 8]

Let z=2+iz = 2+{\mathrm{\hspace{0.05em}i}\mkern 1mu} and w=12iw = 1-2{\mathrm{\hspace{0.05em}i}\mkern 1mu}.

  1. Find zwzw. [2]

  2. Illustrate zz, ww and zwzw on the same Argand diagram. [3]

  3. Let θ\theta be the angle between zwzw and ww. Find θ\theta, giving your answer in radians.[3]

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Question 7

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[Maximum mark: 6]

Let z=reiπ6z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{6}} where rR+r \in \mathbb{R}^+.

  1. For r=3r = \sqrt{3},

    1. express z2z^2 and z3z^3 in the form a+bia+b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R};

    2. draw z2z^2 and z3z^3 on the following Argand diagram. [4]

      633c15de281dbaeb852d4279b825abc3a94854e8.svg

  2. Given that the integer powers of w=z6+2iw = \dfrac{z}{6+2{\mathrm{\hspace{0.05em}i}\mkern 1mu}} lie on a unit circle centred

    at the origin, find the value of rr. [2]

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Question 8

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[Maximum mark: 6]

Consider the equation 3z5z=i\dfrac{3z}{5-z^{*}}=\text{i}, where z=x+iyz=x+\text{i}y and xx, yRy \in \mathbb{R}.
Find the value of xx and the value of yy.

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Question 9

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medium

[Maximum mark: 6]

Let z=reiπ3z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{3}} where rR+r \in \mathbb{R}^+.

  1. For r=2r = \sqrt{2},

    1. express z2z^2 and z3z^3 in the form a+bia+b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R};

    2. draw z2z^2 and z3z^3 on the following Argand diagram. [4]

      7e711daec273ec9c0f630b852aae229a5a09d558.svg

  2. Given that the integer powers of w=(33i)zw = (3-3{\mathrm{\hspace{0.05em}i}\mkern 1mu})\hspace{0.05em}z lie on a unit circle centred at the origin, find the value of rr. [2]

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Question 10

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medium

[Maximum mark: 6]

The complex numbers zz and ww correspond to the points A and B as shown on the diagram below.

0c46fa70e78de8ef6fa11a8d28566cc1742d4a83.svg

  1. Find the exact value of zw|z - w|. [2]

    1. Find the exact perimeter of triangle AOB.

    2. Find the exact area of triangle AOB. [4]

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Question 11

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medium

[Maximum mark: 7]

Points A and B represent the complex numbers z1=3iz_1 = \sqrt{3} - {\mathrm{\hspace{0.05em}i}\mkern 1mu} and z2=33iz_2 = -3 - 3{\mathrm{\hspace{0.05em}i}\mkern 1mu} as shown on the Argand diagram below.

b9acbcc9be3dbe232f939a960c2a1907744b9b96.svg

  1. Find the angle AOB. [3]

  2. Find the argument of z1z2z_1z_2. [1]

  3. Given that the real powers of pz1z2pz_1z_2, for p>0p > 0, all lie on a unit circle centred at the origin, find the exact value of pp. [3]

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Question 12

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medium

[Maximum mark: 7]

The complex numbers ww and zz satisfy the equations

zw=i,w+2z=4+5i.\begin{aligned} \dfrac{z}{w} &= {\mathrm{\hspace{0.05em}i}\mkern 1mu}, \\[6pt] w^\ast + 2z &= 4 + 5{\mathrm{\hspace{0.05em}i}\mkern 1mu}.\end{aligned}

Find ww and zz in the form a+bia + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bZa, b \in \mathbb{Z}.

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Question 13

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medium

[Maximum mark: 6]

Let z=2cis2θz = 2\mathop{\mathrm{cis}}2\theta where 0<θ<45°0 < \theta < \ang{45}. Find the modulus and argument of z+2z + 2, expressing your answers in terms of θ\theta.

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Question 14

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medium

[Maximum mark: 8]

Let w=2ei2π3w = 2e^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{2\pi}{3}}.

    1. Write ww, w2w^2 and w3w^3 in the form a+bia + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in\mathbb{R}.

    2. Draw ww, w2w^2 and w3w^3 on an Argand diagram. [6]

  1. Find the smallest integer k>3k > 3 such that wkw^k is a real number. [2]

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Question 15

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medium

[Maximum mark: 7]

Consider the equation z46z3+cz230z+13=0z^4-6z^3+cz^2-30z+13=0 where zCz \in \mathbb{C} and cRc \in \mathbb{R}.

Three of the roots of the equation are 23i2-3i, α\alpha and α4\alpha^4, where αR\alpha \in \mathbb{R}.

  1. Find the value of α\alpha.[4]

  2. Hence, or otherwise, find the value of cc.[3]

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Question 16

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medium

[Maximum mark: 9]

  1. Find three distinct roots of the equation z3+64=0z^3 + 64 = 0, zCz \in \mathbb{C}, giving your answers in modulus-argument form. [6]

The roots are represented by the vertices of a triangle in an Argand diagram.

  1. Show that the area of the triangle is 12312\sqrt{3}. [3]

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Question 17

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medium

[Maximum mark: 7]

Consider the complex numbers u=1+2iu = 1 + 2 {\mathrm{\hspace{0.05em}i}\mkern 1mu} and v=2+iv = 2 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}.

  1. Given that 1u+1v=62w\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{6\sqrt{2}}{w}, express ww in the form a+bia + b {\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa,b \in \mathbb{R}. [4]

  2. Find ww^\ast and express it in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}. [3]

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Question 18

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medium

[Maximum mark: 7]

Consider the equation 2z4+az3+bz2+cz+d=02z^4 + az^3 + bz^2 +cz + d = 0, where a,b,c,dRa, b, c, d \in \mathbb{R} and zCz \in \mathbb{C}. Two of the roots of the equation are log210\log_2 10 and i5{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sqrt{5} and the sum of all the roots is 4+log254 + \log_25.

Show that 15a+d+90=015a + d + 90 = 0.

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Question 19

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medium

[Maximum mark: 18]

  1. Express 4+43i-4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu} in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}, where r>0r > 0 and π<θπ- \pi < \theta \leq \pi. [5]

Let the roots of the equation z3=4+43iz^3 = -4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu} be z1z_1, z2z_2 and z3z_3.

  1. Find z1z_1, z2z_2 and z3z_3 expressing your answers in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}, where r>0r > 0 and π<θπ-\pi < \theta \leq \pi. [5]

On an Argand diagram, z1z_1, z2z_2 and z3z_3 are represented by the points A, B and C, respectively.

  1. Find the area of the triangle ABC. [4]

  2. By considering the sum of the roots z1z_1, z2z_2 and z3z_3, show that

    cos(2π9)+cos(4π9)+cos(8π9)=0\cos\Big(\dfrac{2\pi}{9}\Big) + \cos\Big(\dfrac{4\pi}{9}\Big) + \cos\Big(\dfrac{8\pi}{9}\Big) = 0

    [4]

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Question 20

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medium

[Maximum mark: 12]

Consider the complex numbers z1=3cis(120°)z_1 = 3 \mathop{\mathrm{cis}}(\ang{120}) and z2=2+2iz_2 = 2 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}.

  1. Calculate z1z2\dfrac{z_1}{z_2} giving your answer both in modulus-argument form and

    Cartesian form. [7]

  2. Use your results from part (a) to find the exact value of sin15°sin45°sin75°\sin \ang{15}\cdot\,\sin \ang{45} \cdot\,\sin \ang{75},

    giving your answer in the form ab\dfrac{\sqrt{a}}{b} where a,bZ+a, b \in \mathbb{Z}^+. [5]

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Question 21

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hard

[Maximum mark: 15]

Consider w=z1z+iw = \dfrac{z - 1}{z + {\mathrm{\hspace{0.05em}i}\mkern 1mu}} where z=x+iyz = x + {\mathrm{\hspace{0.05em}i}\mkern 1mu}y and i=1{\mathrm{\hspace{0.05em}i}\mkern 1mu}= \sqrt{-1}.

  1. If z=iz = {\mathrm{\hspace{0.05em}i}\mkern 1mu},

    1. write ww in the form rcisθr\mathop{\mathrm{cis}}\theta;

    2. find the value of w14w^{14}. [5]

  2. Show that in general,

    w=(x2x+y2+y)+i(yx+1)x2+(y+1)2w = \dfrac{(x^2 - x + y^2 + y) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(y - x + 1)}{x^2 + (y + 1)^2}

    [4]

  3. Find condition under which Re(w)=1\mathrm{Re}(w) = 1. [2]

  4. State condition under which ww is:

    1. real;

    2. purely imaginary. [2]

  5. Find the modulus of zz given that argw=π4\arg w = \dfrac{\pi}{4}. [2]

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Question 22

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hard

[Maximum mark: 6]

On an Argand diagram, the complex numbers z1=2+23iz_1 = 2 + 2\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}, z2=1iz_2 = 1 - {\mathrm{\hspace{0.05em}i}\mkern 1mu} and z3=z1z2z_3 = z_1z_2 are represented by the vertices of a triangle.

The exact area of the triangle can be expressed in the form p+qp+\sqrt{q}. Find the value of pp and of qq.

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Question 23

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hard

[Maximum mark: 19]

Let z=cosθ+isinθz = \cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta, for π4<θ<π4-\dfrac{\pi}{4} < \theta < \dfrac{\pi}{4}.

    1. Find z3z^3 using the binomial theorem.

    2. Use de Moivre's theorem to show that cos3θ=4cos3θ3cosθ\cos 3\theta = 4\cos^3\theta - 3\cos \theta and sin3θ=3sinθ4sin3θ\sin 3\theta = 3\sin\theta-4\sin^3\theta. [8]

  1. Hence show that sin3θsinθcos3θ+cosθ=tanθ\dfrac{\sin 3\theta - \sin \theta}{\cos 3\theta + \cos \theta} = \tan \theta. [6]

  2. Given that sinθ=13\sin \theta = \dfrac{1}{3}, find the exact value of tan3θ\tan 3\theta. [5]

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Question 24

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[Maximum mark: 19]

    1. Expand (cosθ+isinθ)4(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)^4 by using the binomial theorem.

    2. Hence use de Moivre's theorem to prove that

      cos4θ=cos4θ6cos2θsin2θ+sin4θ.\begin{aligned} \cos 4\theta = \cos^4 \theta - 6\cos^2 \theta\sin^2 \theta + \sin^4 \theta. \\ \end{aligned}
    3. State a similar expression for sin4θ\sin 4 \theta in terms of cosθ\cos \theta and sinθ\sin \theta. [6]

Let z=r(cosα+isinα)z = r(\cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha), where α\alpha is measured in degrees, be the solution
of z4i=0z^4 - {\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0 which has the smallest positive argument.

  1. Find the modulus and argument of zz. [4]

  2. Use (a) (ii) and your answer from (b) to show that 8cos4α8cos2α+1=08\cos^4\alpha - 8\cos^2 \alpha + 1 = 0. [4]

  3. Hence express cos22.5°\cos \ang{22.5} in the form a+bcd\dfrac{\sqrt{a + b\sqrt{c}}}{d} where a,b,c,dZa,b,c,d \in \mathbb{Z}. [5]

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Question 25

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hard

[Maximum mark: 22]

  1. Solve 2sin(x+120°)=3cos(x+60°)2\sin(x+\ang{120}) = \sqrt{3}\cos(x + \ang{60}), for x[0,180°]x \in [0,\ang{180}]. [5]

  2. Show that sin75°+cos75°=62\sin \ang{75} + \cos \ang{75} = \dfrac{\sqrt{6}}{2}. [3]

  3. Let z=sin4θ+i(1cos4θ)z = \sin 4\theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(1 - \cos 4\theta), for zCz \in \mathbb{C}, θ[0,90°]\theta \in [0,\ang{90}].

    1. Find the modulus and argument of zz in terms of θ\theta.

    2. Hence find the fourth roots of zz in modulus-argument form. [14]

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Question 26

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hard

[Maximum mark: 16]

  1. Find the roots of z16=1z^{16} = 1 which satisfy the condition 0<arg(z)<π20 < \arg(z) < \dfrac{\pi}{2},

    expressing your answer in the form reiθre^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}, where r,θR+r, \theta \in \mathbb{R}^+. [5]

  2. Let SS be the sum of the roots found in part (a).

    1. Show that Re(S)=Im(S)\mathrm{Re}(S) = \mathrm{Im}(S).

    2. By writing π8\dfrac{\pi}{8} as 12π4\dfrac{1}{2}\cdot\dfrac{\pi}{4}, find the value of cos(π8)\cos \Big(\dfrac{\pi}{8}\Big) in the form a+bc\dfrac{\sqrt{a + \sqrt{b}}}{c},

      where a,ba, b and cc are integers to be determined.

    3. Hence, or otherwise, show that S=12(2+2+2+22)(1+i)S = \dfrac{1}{2}\big(\hspace{-0.1em}\sqrt{2 + \sqrt{2}} + \sqrt{2} + \sqrt{2 - \sqrt{2}}\hspace{0.1em}\big)\big(1+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\big). [11]

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Question 27

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[Maximum mark: 17]

  1. Solve the equation z3=27z^3 = 27, zCz \in \mathbb{C}, giving your answer in the form
    z=r(cosθ+isinθ)z = r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta) and in the form z=a+biz = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa,b \in \mathbb{R}. [6]

  2. Consider the complex numbers z1=1+i and z2=12[cos(π3)+isin(π3)]z_1 = -1 + {\mathrm{\hspace{0.05em}i}\mkern 1mu} \text{ and } z_2 =\dfrac{1}{\sqrt{2}}\bigg[\mathrm{cos}\bigg(\dfrac{\pi}{3}\bigg)+\mathrm{i}\,\mathrm{sin}\bigg(\dfrac{\pi}{3}\bigg)\bigg] .

    1. Write z1z_1 in the form r(cosθ+isinθ)r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta).

    2. Calculate z1z2z_1z_2 and write in the form a+bia + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa,b \in \mathbb{R}.

    3. Hence find the value of tan(π12)\tan\left(\dfrac{\pi}{12}\right) in the form c+d3c + d\sqrt{3} where c,dZc,d \in \mathbb{Z}.

    4. Find the smallest pQ+p \in \mathbb{Q}^+ such that (z1z2)p(z_1z_2)^p is a positive real number. [11]

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Question 28

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hard

[Maximum mark: 21]

  1. Use de Moivre's theorem to find the value of [cos(π6)+isin(π6)]12\left[\cos\left(\dfrac{\pi}{6}\right) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \left(\dfrac{\pi}{6}\right)\right]^{12}. [2]

  2. Use mathematical induction to prove that

    (cosαisinα)n=cos(nα)isin(nα)for all nZ+.\hspace{3.5em} (\cos \alpha - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)^n = \cos (n\alpha) - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin (n\alpha) \hspace{1em} \text{for all } n \in \mathbb{Z}^+.

    [6]

Let w=cosα+isinαw = \cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha.

  1. Find an expression in terms of α\alpha for wn(w)nw^n - (w^\ast)^n, nZ+n \in \mathbb{Z}^+, where ww^\ast is the complex conjugate of ww. [2]

    1. Show that ww=1ww^\ast = 1.

    2. Write down and simplify the binomial expansion of (ww)3(w - w^\ast)^3 in terms of ww and ww^\ast.

    3. Hence show that sin(3α)=3sinα4sin3α\sin (3\alpha) = 3\sin \alpha - 4 \sin^3 \alpha. [5]

  2. Hence solve 4sin3α+(2cosα3)sinα=04\sin^3\alpha + (2 \cos \alpha - 3) \sin \alpha = 0 for 0απ0 \leq \alpha \leq \pi. [6]

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Question 29

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hard

[Maximum mark: 20]

  1. Solve the equation sin(x+90°)=2cos(x60°)\sin (x + \ang{90}) = 2\cos(x - \ang{60}), 0°<x<360°\ang{0} < x < \ang{360}. [5]

  2. Show that sin15°+cos15°=62\sin \ang{15} + \cos \ang{15} = \dfrac{\sqrt{6}}{2}. [4]

  3. Let z=1cos4θisin4θz = 1 - \cos 4\theta - {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin 4\theta, for zCz \in \mathbb{C}, 0<θ<π20 < \theta < \dfrac{\pi}{2}.

    1. Find the modulus and argument of zz. Express each answer
      in its simplest form.

    2. Hence find the fourth roots of zz in modulus-argument form. [11]

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Question 30

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[Maximum mark: 28]

This questions will investigate a method to obtain Cardano's formula using elementary properties of complex numbers and algebraic expressions.

AA918 - dia

  1. The roots, of magnitude 11, described above can be written polar form such that ωn=cosθ+isinθ\omega_n=\cos \theta + i \sin \theta. Where, the argument θ\theta, is 0θ2π0 \leq \theta \leq 2\pi.
    1. Show that ω0=1\omega_0=1 and find the remaining roots, ω1\omega_1 and ω2\omega_2, in polar form.

    2. Show that ω1ω2=1\omega_1 \omega_2 = 1.

    3. Show that ω12+ω22=ω1+ω2\omega_1^2+\omega_2^2 = \omega_1+\omega_2.

    4. Hence, show that ω1+ω2=1\omega_1+\omega_2 = - 1 in Cartesian form. [7]

Consider the cubic equation x33px+2q=0x^3-3px+2q=0 for pp, qRq\in \mathbb{R}.
Assume we can write it in a factorised form such that x33px+2q=(x+α+β)(x+ω1α+ω2β)(x+ω2α+ω1β)x^3-3px+2q=(x+\alpha+\beta)(x+\omega_1\alpha+\omega_2\beta)(x+\omega_2\alpha+\omega_1\beta).

    1. Identify and sum the roots of the factorised form above and then use some of the relations relations found in part (a) to show that 2q=α3+β32q = \alpha^3+\beta^3

    Another formula for the roots r1r_1, r2r_2 and r3r_3 of the cubic equation ax3+bx2+cx+dax^3+bx^2+cx+d states that r1r2+r1r3+r2r3=car_1r_2+r_1r_3+r_2r_3=\dfrac{c}{a}\rule[-2mm]{0pt}{7mm}.

    1. Using the roots identified from (i), the formula above and some of the relations found in part (a) show that p=αβp=\alpha\beta.

    2. Using parts (i) and (ii), write α\alpha and β\beta in terms of pp and qq.

    3. Hence, show that q+q2p33qq2p33-\sqrt[3]{q+\sqrt{q^2-p^3}}-\sqrt[3]{q-\sqrt{q^2-p^3}} is a root of the cubic equation x33px+2q=0x^3-3px+2q = 0. [12]

Now, consider a more general cubic equation x3+sx+t=0x^3+sx+t = 0, where ss, tRt\in\mathbb{R}.

    1. By substituting s=3ps=-3p and t=2qt=2q prove Cardano's formula, which states that t2+t24+s3273+t2t24+s3273\sqrt[3]{-\dfrac{t}{2}+\sqrt{\dfrac{t^2}{4}+\dfrac{s^3}{27}}} + \sqrt[3]{-\dfrac{t}{2}-\sqrt{\dfrac{t^2}{4}+\dfrac{s^3}{27}}}

      is a root of the cubic equation x3+sx+t=0x^3+sx+t=0.

    2. Use Cardano's formula to find the exact solution to x3+6x+5=0x^3+6x+5=0.

    3. Using the substitution z=x1z=x-1 and Cardano's formula, find an exact solution to z3+3z26=0z^3+3z^2-6=0.[9]

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