De Moivre’s theorem

Given that, for n

show that

Figure 1 Figure 2

Figure 1 shows the central vertical cross-section of a solid wooden ornament.

Figure 2 shows the curve with equation

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation y = and the y-axis.

The ornament is modelled by the solid of revolution formed when R is rotated 360° about the y-axis. The units are centimetres.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of wood needed to make the ornament, according to the model. Give your answer to 2 significant figures.

[ Solutions based entirely on calculator technology are not acceptable.]

Given that

• the density of the wood is 0.85 g/cm³

• the mass of the ornament is 6 grams

comment on the suitability of the model.

The complex number z = e^iθ, where θ is real.

Show that

where n is a positive integer.

Show that

Hence, making your reasoning clear, determine all the solutions of

cos 5 θ + 5 cos 3 θ + 12 cos θ = 0

in the interval 0 θ < 2π

The complex number z = e^iθ, where θ is real.

Show that

where n is a positive integer.

Show that

Hence, making your reasoning clear, determine all the solutions of

cos 5 θ + 5 cos 3 θ + 12 cos θ = 0

in the interval 0 θ < 2π

Given

show that

x = –4 x = 4

Figure 1 Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Given

show that

x = –4 x = 4

Figure 1
Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Given that

write down the exact value of

|z₁z₂|

arg(z₁z₂)

Given that w = z₁z₂ and that arg(wⁿ) = 0, where n ⁺

determine

the smallest positive value of n

the corresponding value of |wⁿ|

Given that

write down the exact value of

|z₁z₂|

determine

the smallest positive value of n

the corresponding value of |wⁿ|

Given that |z| < 1, write down the sum of the infinite series

Given that z = cosθ + i sinθ),

use the answer to part (a), and de Moivre’s theorem or otherwise, to prove that

show that the sum of the infinite series 1 + z + z² + z³ + … cannot be purely imaginary, giving a reason for your answer.

Given that |z| < 1, write down the sum of the infinite series

Given that z = cosθ + i sinθ),

use the answer to part (a), and de Moivre’s theorem or otherwise, to prove that

show that the sum of the infinite series 1 + z + z² + z³ + … cannot be purely imaginary, giving a reason for your answer.

Use de Moivre’s theorem to prove that

Hence find the distinct roots of the equation

giving your answer to 3 decimal places where appropriate.

Use de Moivre’s theorem to prove that

Hence find the distinct roots of the equation

giving your answer to 3 decimal places where appropriate.