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IB Math 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: 6]

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

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Question 2

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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 3

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[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 4

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[Maximum mark: 6]

  1. Verify that 1+2i1+2{\mathrm{\hspace{0.05em}i}\mkern 1mu} and 12i-1-2{\mathrm{\hspace{0.05em}i}\mkern 1mu} are the second roots of 3+4i-3+4{\mathrm{\hspace{0.05em}i}\mkern 1mu}. [2]

  2. Hence, find two distinct roots of the equation z23z+(3i)=0z^2 - 3z + (3-{\mathrm{\hspace{0.05em}i}\mkern 1mu}) = 0, zCz \in \mathbb{C},
    giving your answers in the form z=a+biz = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R}. [4]

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Question 5

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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 6

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[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 7

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[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 8

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[Maximum mark: 5]

Consider z=cosθ+isinθz=\cos\theta+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\theta where zCz \in \mathbb{C}, z0z\neq 0.

Show that 12z2+(12z2)=cos(2θ)\dfrac{1}{2z^2}+\left(\dfrac{1}{2z^2}\right)^\ast = \cos(2\theta).

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Question 9

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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 10

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[Maximum mark: 7]

Consider the complex numbers

z=3(cosπ6isinπ6) and w=5(coskπ12+isinkπ12)z = 3\left(\cos\dfrac{\pi}{6}-{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\dfrac{\pi}{6}\right) \enskip\text{ and }\enskip w = 5\left(\cos\dfrac{k\pi}{12}+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\dfrac{k\pi}{12}\right)

where kZ+k\in\mathbb{Z}^{+}.

  1. Express zz and ww in modulus-argument form and write down
    1. the modulus of zwzw.

    2. the argument of zwzw in terms of kk.[3]

Suppose that zwZzw\in \mathbb{Z}.

    1. Find the minimum value of kk.

    2. For the value of kk found in part (i), find the value of zwzw.[4]

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Question 11

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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 12

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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 13

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[Maximum mark: 8]

Find the six distinct roots of the equation z6+(18i)z38i=0z^6+(1-8{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^3-8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0, zCz \in \mathbb{C}, giving your answers in the form z=a+biz = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R}.

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Question 14

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medium

[Maximum mark: 7]

Consider the polynomial equation z4+10z3+50z2+130z+169=0z^4 + 10z^3 + 50z^2 + 130z + 169 = 0 where zCz \in \mathbb{C}.

Two of the roots of this equation are a+iba + {\mathrm{\hspace{0.05em}i}\mkern 1mu}b and b+iab + {\mathrm{\hspace{0.05em}i}\mkern 1mu}a where a,bZa,b \in \mathbb{Z}.

  1. Write down the other two roots in terms of aa and bb. [1]

  2. Find the possible values of aa. [6]

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Question 15

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[Maximum mark: 7]

Two distinct roots for the polynomial equation z412z3+57z2120z+100=0z^4 - 12z^3 + 57z^2 - 120z + 100 =0 are a1+bia - 1 + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} and a+12bia + 1 - 2b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where zCz \in \mathbb{C} and a,bZa,b \in \mathbb{Z}.

  1. Write down the other two roots in terms of aa and bb. [1]

  2. Find the value of aa. [2]

  3. Hence, or otherwise, find the four roots of the polynomial. [4]

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Question 16

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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 17

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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 18

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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 19

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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 20

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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 21

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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 22

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[Maximum mark: 18]

    1. Verify that 1+i1+{\mathrm{\hspace{0.05em}i}\mkern 1mu} and 1i-1-{\mathrm{\hspace{0.05em}i}\mkern 1mu} are the second roots of 2i2{\mathrm{\hspace{0.05em}i}\mkern 1mu}.

    2. Find two distinct roots of the equation z22z+(12i)=0z^2 - 2z + (1-2i) = 0, zCz \in \mathbb{C}, giving your answers in the form z=a+biz = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R}. [5]

  1. Find six distinct roots of the equation z6+(8i)z38i=0z^6+(8-{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^3-8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0, zCz \in \mathbb{C}, giving your answers in the form z=a+biz = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu} where a,bRa, b \in \mathbb{R}. [8]

On an Argand diagram, z1=3+iz_1 = \sqrt{3}+{\mathrm{\hspace{0.05em}i}\mkern 1mu}, z2=3+iz_2 = -\sqrt{3} +{\mathrm{\hspace{0.05em}i}\mkern 1mu} and z3=2iz_3 = -2{\mathrm{\hspace{0.05em}i}\mkern 1mu} are represented by the points A, B and C, respectively.

    1. Verify that z1z_1, z2z_2 and z3z_3 are the roots of the equation z38i=0z^3 - 8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0, zCz \in \mathbb{C}.

    2. Show that the area of the triangle ABC is 333\sqrt{3}. [5]

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Question 23

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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 24

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[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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The AA HL Questionbank is designed to help IB students practice AA HL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AA HL Question bank. For example, if you want to practice AA HL exam style questions that involve Complex Numbers, you can go to AA SL Topic 1 (Number & Algebra) and go to the Complex Numbers area of the question bank. On this page there is a carefully designed set of IB Math AA HL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.

The AA HL Questionbank is perfect for revising a particular topic or concept, in-depth. For example, if you wanted to improve your knowledge of Counting Principles (Combinations & Permutations), there is a set of full length IB Math AA HL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AA HL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AA HL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.

With an extensive and growing library of full length IB Math Analysis and Approaches (AA) HL exam style questions in the AA HL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AA HL Question bank, then a popular option would be to go to the AA HL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AA HL exam.

More IB Math AA HL Resources