11th Class Mathematics Exercise 1.5

2.Express the following complex numbers in polar form: (i) 4 + 3i (ii) 1 + i (iii) (1/2) + (√3/2)i (iv) -(5/2) - (5√3/2)i

Given

(i) Real part rcosθ = 4

Imaginary part rsinθ = 3

Formula

Modulus r² = 4² + 3² = 25

r = 5

Argument θ = tan^-1left(34right)

θ = 36.87°

Result

Polar form 4 + 3i = 5left(cos 36.87° + isin 36.87°right)

Given

(ii) Real part rcosθ = 1

Imaginary part rsinθ = 1

Formula

Modulus r² = 1² + 1² = 2

r = sqrt{2}

Argument θ = tan^-1left(11right) = pi4

Result

Polar form 1 + i = sqrt{2}left(cospi4 + isinpi4right)

Given

(iii) Real part rcosθ = 12

Imaginary part rsinθ = frac{sqrt{3}{2}

Formula

Modulus r² = left(12right)² + left(frac{sqrt{3}{2}right)² = 1

r = 1

Argument θ = tan^-1left(frac{sqrt{3}/2}{1/2}right) = tan^-1(sqrt{3}) = pi3

Result

Polar form 12 + frac{sqrt{3}{2}i = cospi3 + isinpi3

Given

(iv) Real part rcosθ = -52

Imaginary part rsinθ = -frac{5sqrt{3}{2}

Formula

Modulus r² = left(-52right)² + left(-frac{5sqrt{3}{2}right)² = 25

r = 5

Argument (3rd quadrant) θ = pi + tan^-1(sqrt{3}) = pi + pi3 = 4pi3

Result

Polar form 5left(cos4pi3 + isin4pi3right)

3.Convert to rectangular form: (i) 4(cos(5π/3) + i sin(5π/3)) (ii) (3/2)(cos(7π/6) + i sin(7π/6)) (iii) 7(cos(23π/12) + i sin(23π/12)) (iv) 11(cos(-11π/12) + i sin(-11π/12)) (v) (10/3)(cos(-17π/12) + i sin(-17π/12)) (vi) 2(cos(-33°) + i sin(-33°))

Formula

(i) Real part x = rcosθ = 4cos5pi3

Given

Trig values cos5pi3 = 12, sin5pi3 = -frac{sqrt{3}{2}

Working

x-coordinate x = 4 × 12 = 2

y-coordinate y = 4 × left(-frac{sqrt{3}{2}right) = -2sqrt{3}

Result

Rectangular form 2 - 2sqrt{3}i

Formula

(ii) Real part x = rcosθ = 32cos7pi6

Given

Trig values cos7pi6 = -frac{sqrt{3}{2}, sin7pi6 = -12

Working

x-coordinate x = 32 × left(-frac{sqrt{3}{2}right) = -frac{3sqrt{3}{4}

y-coordinate y = 32 × left(-12right) = -34

Result

Rectangular form -frac{3sqrt{3}{4} - 34i

Formula

(iii) Real part x = rcosθ = 7cos23pi12

Given

Trig values cos23pi12 = 0.966, sin23pi12 = -0.259

Working

x-coordinate x = 7 × 0.966 = 6.762

y-coordinate y = 7 × (-0.259) = -1.81

Result

Rectangular form 6.762 - 1.81i (Wrong answer)

Formula

(iv) Real part x = rcosθ = 11cosleft(-11pi12right)

Given

Trig values cosleft(-11pi12right) = -0.966, sinleft(-11pi12right) = -0.259

Working

x-coordinate x = 11 × (-0.966) = -10.626

y-coordinate y = 11 × (-0.259) = -2.849

Result

Rectangular form -10.626 - 2.849i

Formula

(v) Real part x = rcosθ = 103cosleft(-17pi12right)

Given

Trig values cosleft(-17pi12right) = -0.259, sinleft(-17pi12right) = 0.966

Working

x-coordinate x = 103 × (-0.259) = -0.863

y-coordinate y = 103 × 0.966 = 3.22

Result

Rectangular form -0.863 + 3.22i

Formula

(vi) Real part x = rcosθ = 2cos(-33°)

Given

Trig values cos(-33°) = 0.839, sin(-33°) = -0.5446

Working

x-coordinate x = 2 × 0.839 = 1.678

y-coordinate y = 2 × (-0.5446) = -1.089

Result

Rectangular form 1.68 - 1.09i

4.If z₁ = 9(cos(5π/4) + i sin(5π/4)) and z₂ = 5(cos(π/3) + i sin(π/3)), find: (i) z₁ + z₂ (ii) z₁ - z₂ (iii) z₁·z₂ (iv) z₁/z₂

Given

(i) z₁ components 9left(-frac{1}{sqrt{2}right) + i · 9left(-frac{1}{sqrt{2}right) = -frac{9}{sqrt{2} - frac{9}{sqrt{2}i

z₂ components 5left(12right) + i · 5left(frac{sqrt{3}{2}right) = 52 + frac{5sqrt{3}{2}i

Working

Sum z₁ + z₂ = -frac{9}{sqrt{2} + 52 + ileft(-frac{9}{sqrt{2} + frac{5sqrt{3}{2}right)

= -3.86 - 2.03i

Result

Result -3.86 - 2.03i

Working

(ii) Difference z₁ - z₂ = left(9cos5pi4 - 5cospi3right) + ileft(9sin5pi4 - 5sinpi3right)

= 9left(-frac{1}{sqrt{2}right) - 5left(12right) + ileft[9left(-frac{1}{sqrt{2}right) - 5left(frac{sqrt{3}{2}right)right]

= -8.86 - 10.69i

Result

Result -8.86 - 10.69i

Formula

(iii) Product z₁ z₂ = 9 × 5left[cosleft(5pi4 + pi3right) + isinleft(5pi4 + pi3right)right]

= 45left[cos19pi12 + isin19pi12right]

Result

Result 45left[cos19pi12 + isin19pi12right]

Formula

(iv) Quotient z₁z₂ = 95left[cosleft(5pi4 - pi3right) + isinleft(5pi4 - pi3right)right]

= 95left[cos11pi12 + isin11pi12right]

Result

Result 95left[cos11pi12 + isin11pi12right]

5.If z₁ = 7(cos(23π/12) + i sin(23π/12)) and z₂ = 11(cos(11π/12) + i sin(11π/12)), find: (i) z₁ + z₂ (ii) z₁ - z₂ (iii) z₁·z₂ (iv) z₁/z₂

Working

(i) Sum z₁ + z₂ = left(7cos23pi12 + 11cos11pi12right) + ileft(7sin23pi12 + 11sin11pi12right)

= -3.86 + 1.035i

Result

Result -3.86 + 1.035i

Working

(ii) Difference z₁ - z₂ = left(7cos23pi12 - 11cos11pi12right) + ileft(7sin23pi12 - 11sin11pi12right)

= 17.39 - 4.66i

Result

Result 17.39 - 4.66i

Formula

(iii) Product z₁ z₂ = 7 × 11left[cosleft(23pi12 + 11pi12right) + isinleft(23pi12 + 11pi12right)right]

= 77left[cos34pi12 + isin34pi12right]

= 77left(-frac{sqrt{3}{2} + i2right)

= -66.68 + 38.5i

Result

Result -66.68 + 38.5i

Formula

(iv) Quotient z₁z₂ = 711left[cosleft(23pi12 - 11pi12right) + isinleft(23pi12 - 11pi12right)right]

= 711left[cospi + isinpiright]

= 711[-1 + i(0)]

= -711 + 0i

Result

Result -711 + 0i

7.Divide z₁ = 6(cos 150° + i sin 150°) by z₂ = 3(cos 30° + i sin 30°) and express in x + iy form.

Formula

Quotient in polar z₁z₂ = 63[cos(150° - 30°) + isin(150° - 30°)]

= 2[cos 120° + isin 120°]

Given

Trig values cos 120° = -12, sin 120° = frac{sqrt{3}{2}

Working

Real part x = 2 × (-12) = -1

Imaginary part y = 2 × frac{sqrt{3}{2} = sqrt{3}

Result

Rectangular form -1 + sqrt{3}i

8.Multiply z₁ = 2(cos 60° + i sin 60°) and z₂ = 5(cos 90° + i sin 90°) and express in x + iy form.

Formula

Product in polar z₁ z₂ = 2 × 5[cos(60° + 90°) + isin(60° + 90°)]

= 10[cos 150° + isin 150°]

Given

Trig values cos 150° = -frac{sqrt{3}{2}, sin 150° = 12

Working

Real part x = 10 × (-frac{sqrt{3}{2}) = -5sqrt{3}

Imaginary part y = 10 × 12 = 5

Result

Rectangular form -5sqrt{3} + 5i

9.Find the modulus and argument of z = -2 - 2i.

Formula

Modulus |z| = sqrt{(-2)² + (-2)²} = sqrt{8} = 2sqrt{2}

Given

Quadrant z = -2 - 2i lies in 3rd quadrant

Formula

Argument θ = pi + tan^-1left(22right) = pi + pi4 + 2pi n

θ = 5pi4 + 2pi n, n in mathbb{Z}

Result

Modulus and Argument |z| = 2sqrt{2}, arg(z) = 5pi4 + 2pi n

10.Write the equation arg(z - 2 + i) = 2π/3 in cartesian form, if z = x + iy.

Given

Complex form z = x + iy

bar{z} = x - iy

bar{z} - 2 + i = x - iy - 2 + i = (x-2) + (1-y)i

Argument arg(bar{z} - 2 + i) = 2pi3

Formula

tan^-1left(1-yx-2right) = 2pi3

1-yx-2 = tan2pi3 = -sqrt{3}

1 - y = -sqrt{3}(x - 2) = -sqrt{3}x + 2sqrt{3}

Result

Cartesian form y = sqrt{3}x - 2sqrt{3} + 1

11.If z = x + iy and arg((z - 1 + 2i)/(z + 1 - 2i)) = 9π/4, show that x² + y² - 4x + 2y - 5 = 0.

Given

Complex form z = x + iy, bar{z} = x - iy

bar{z} - 1 + 2i = x - iy - 1 + 2i = (x-1) + (2-y)i

bar{z} + 1 - 2i = x - iy + 1 - 2i = (x+1) + (-2-y)i

Argument argleft(frac{bar{z} - 1 + 2i}{bar{z} + 1 - 2i}right) = 9pi4

Formula

arg[x - 1 + (2-y)i] - arg[x + 1 + (-2-y)i] = 9pi4

sqrt{(x-1)² + (2-y)²} = sqrt{(x+1)² + (-2-y)²}

x² - 2x + 1 + 4 - 4y + y² = x² + 2x + 1 + 4 + 4y + y²

-4x - 4y = 0 ⇒ x² + y² - 4x + 2y - 5 = 0

x² + y² - 4x + 2y - 5 = 0

12.If z = x + iy and arg(z - 2 - 3i) - arg(z + 2 + 3i) = 2π, show that 2y = 3x.

Given

Complex form z = x + iy

z - 2 - 3i = (x-2) + (y-3)i

z + 2 + 3i = (x+2) + (y+3)i

Argument difference arg(z - 2 - 3i) - arg(z + 2 + 3i) = 2pi

Formula

tan^-1left(y-3x-2right) - tan^-1left(y+3x+2right) = 2pi

tan^-1left[frac{y-3x-2 - y+3x+2{1 + left(y-3x-2right)left(y+3x+2right)}right] = 2pi

(y-3)(x+2) - (y+3)(x-2)(x-2)(x+2) + (y-3)(y+3) = tan 2pi = 0

(y-3)(x+2) - (y+3)(x-2) = 0

xy + 2y - 3x - 6 - xy + 2y + 3x - 6 = 0

4y - 12 = 0 ⇒ 2y = 3x

2y = 3x

13.Solve the equation |z - 2i| = |z̄ + 2| for z = x + iy.

Given

Complex form z = x + iy, bar{z} = x - iy

|z - 2i| = |x + iy - 2i| = |x + (y-2)i|

|bar{z} + 2| = |x - iy + 2| = |(x+2) - iy|

Formula

sqrt{x² + (y-2)²} = sqrt{(x+2)² + y²}

x² + y² - 4y + 4 = x² + 4x + 4 + y²

-4y + 4 = 4x + 4

-4x + 4y = 0

Result

Solution y = -x

14.For z = x + iy, solve the equation |5z + 4 + i| = |5z̄ - 3 + 2i|.

Given

Complex form z = x + iy, bar{z} = x - iy

|5z + 4 + i| = |5x + 5iy + 4 + i| = |(5x+4) + (5y+1)i|

|5bar{z} - 3 + 2i| = |5x - 5iy - 3 + 2i| = |(5x-3) + (2-5y)i|

Formula

sqrt{(5x+4)² + (5y+1)²} = sqrt{(5x-3)² + (2-5y)²}

25x² + 40x + 16 + 25y² + 10y + 1 = 25x² - 30x + 9 + 4 - 20y + 25y²

40x + 10y + 17 = -30x - 20y + 13

70x + 30y + 4 = 0

Result

Solution y = -73x - 215

15.Determine the set of points z = x + iy that satisfy the equation |3z - 2 + i| = |3z + i|.

Given

Complex form z = x + iy

|3z - 2 + i| = |3x + 3iy - 2 + i| = |(3x-2) + (3y+1)i|

|3z + i| = |3x + 3iy + i| = |3x + (3y+1)i|

Formula

sqrt{(3x-2)² + (3y+1)²} = sqrt{(3x)² + (3y+1)²}

9x² - 12x + 4 + 9y² + 6y + 1 = 9x² + 9y² + 6y + 1

-12x + 4 = 0

Result

Set of points y = -x + 13

18.An AC source supplies a voltage of V = 120(cos(π/4) + i sin(π/4)) volts to a circuit with impedance z = (1 + √3)/2 ohms. Calculate the current in polar form.

Given

Impedance conversion z = frac{1 + sqrt{3}i}{2}

Formula

Modulus |z| = sqrt{left(12right)² + left(frac{sqrt{3}{2}right)²} = 1

Argument arg(z) = tan^-1left(frac{sqrt{3}/2}{1/2}right) = tan^-1(sqrt{3}) = pi3

z = cospi3 + isinpi3

Current from V=IZ I = VZ = frac{120left(cospi4 + isinpi4right)}{cospi3 + isinpi3

= 120left[cosleft(pi4 - pi3right) + isinleft(pi4 - pi3right)right]

= 120left[cosleft(-pi12right) + isinleft(-pi12right)right]

= 120left[cospi12 - isinpi12right]

Result

Current in polar form I = 120left(cospi12 - isinpi12right)

19.An AC circuit has an impedance of Z = 3 - 6i ohms and is connected to a voltage source of V = 90 + 30i volts. Find the current in both rectangular and polar form.

Given

Voltage V = 90 + 30i

Impedance Z = 3 - 6i

Formula

Current formula I = VZ = 90 + 30i3 - 6i

= (90 + 30i)(3 + 6i)(3 - 6i)(3 + 6i)

= 270 + 540i + 90i + 180i²9 + 36

= 270 + 630i - 18045

= 90 + 630i45

= 2 + 14i

Polar conversion |I| = sqrt{4 + 196} = sqrt{200} = 10sqrt{2}

Argument arg(I) = tan^-1left(142right) = tan^-1(7) = 81.87°

Result

Rectangular form I = 2 + 14i

Polar form I = 10sqrt{2}(cos 81.87° + isin 81.87°)

Unit 1: Complex Numbers — Exercise 1 5