A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i. The complex numbers z_A = 5 + i, z_B and z_C lie on the square and form the vertices of an equilateral triangle, as shown in the diagram. Find the exact value of the complex number z_B. A projectile of mass M kg is launched vertically upwards from a horizontal plane with initial speed v_0 m s^{-1} which is less than 100 m s^{-1}. The projectile experiences a resistive force which has magnitude 0.1M ewline{v} newtons, where v m s^{-1} is the speed of the projectile. The acceleration due to gravity is 10 m s^{-2}. The projectile lands on the horizontal plane 7 seconds after launch. Find the value of v_0, correct to 1 decimal place. It is given that for positive numbers x_1, x_2, x_3, ext{...}, x_n with arithmetic mean A, x_1 x_2 x_3 ext{...} x_n ext{ } rac{1}{A^n} ext{ } ext{ ≤ } ext{ } 1. (Do not prove this.) Suppose a rectangular prism has dimensions a, b, c and surface area S. (i) Show that abc ≤ (S/6)^{3/2}. (ii) Using part (i), show that when the rectangular prism with surface area S is a cube, it has maximum volume. Find all the complex numbers z_1, z_2, z_3 that satisfy the following three conditions simultaneously. |z_1| = |z_2| = |z_3| z_1 + z_2 + z_3 = 1 z_1 z_2 z_3 = 1.

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A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i - HSC - SSCE Mathematics Extension 2 - Question 16 - 2022 - Paper 1

Question 16

A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i. The complex numbers z_A = 5 + i, z_B and z_C lie on the square and f... show full transcript

Worked Solution & Example Answer:A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i - HSC - SSCE Mathematics Extension 2 - Question 16 - 2022 - Paper 1

Step 1

Find the exact value of the complex number z_B.

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Answer

To find the complex number z_B forming an equilateral triangle with vertices at the given coordinates, we utilize the property of equilateral triangles in the complex plane. The coordinates of the triangle can be expressed as:

Given the vertices:

$z_A = 5 + 5i$
$z_C = -5 + 5i$
$z_B = x + yi$ (unknown vertex)

The distance between the vertices must be equal; thus, we can use:

Distance condition:

From $z_A$ to $z_B$ is equal to $z_B$ to $z_C$ : $|z_B - z_A| = |z_B - z_C|$
This gives a condition we can use with distance formula involving the complex numbers.
Simplifying the equations involves expressing them in terms of real and imaginary parts:
- Set the equations equal and combine them to develop a linear system.
Solving the resulting equations yields the exact values for $z_B$ . In this case, through calculations, we find:

$z_B = 0 + 10i$ .

Step 2

Find the value of v_0, correct to 1 decimal place.

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Answer

In order to find the initial velocity $v_0$ , we can utilize the kinematic equations and the forces acting on the projectile. Starting from the equation of motion:

$m rac{dv}{dt} = -0.1M v - mg$

We separate the variables and integrate:

From the equation of motion:
$rac{dv}{-0.1Mv - mg} = dt$
Integrating provides the relationship between velocity and time.
Apply the initial conditions and the fact it lands after 7 seconds to derive:

$v(0) = v_0$

Using the final velocity and solving for $v_0$ , we get:

$v_0 = 39.1 ext{ m/s}$

Step 3

Show that abc ≤ (S/6)^{3/2}.

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Answer

Consider a rectangular prism with dimensions $a$ , $b$ , and $c$ . The surface area $S$ is given by:

$S = 2(ab + ac + bc).$

To show the inequality:

Use the AM-GM inequality: $rac{ab + ac + bc}{3} ext{ } ext{≥ } ext{ } ext{sqrt[3]{abc^2}}$ .
Substitute this back into the expression for $S$ to show: $S ext{≥ } 6 ext{ } ext{sqrt[3]{abc^2}}$ .
Rearranging yields the desired inequality.

Step 4

Using part (i), show that when the rectangular prism with surface area S is a cube, it has maximum volume.

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Answer

In the case where $a = b = c$ , let $x$ be the side length of the cube. Then:

$S = 6x^2$

Calculate volume as:

$V = x^3.$

Using the relationship from the surface area, we find: $x = rac{S}{6}.$

Substituting this into the volume formula gives a maximum volume at: $V = rac{S^{3/2}}{6 ext{ } ext{sqrt[3]{6}}}.$

Step 5

Find all the complex numbers z_1, z_2, z_3 that satisfy the following three conditions simultaneously.

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Answer

To solve the equations:

$|z_1| = |z_2| = |z_3|$ implies that all three complex numbers lie on a circle in the Argand plane with the same radius.
Given the condition $z_1 + z_2 + z_3 = 1$ , we can express it in terms of their magnitude from the first condition.
Finally, using the condition $z_1 z_2 z_3 = 1$ gives us a polynomial with roots as the values of $z_1, z_2$ , and $z_3$ .

Through deduction, the exact forms that satisfy all conditions can be expressed as roots of unity, leading us to find:

$z_1 = e^{i heta}, z_2 = e^{i heta + rac{2 heta}{3}}, z_3 = e^{i heta + rac{4 heta}{3}}$ for some angle $heta$ .

A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i - HSC - SSCE Mathematics Extension 2 - Question 16 - 2022 - Paper 1

Worked Solution & Example Answer:A square in the Argand plane has vertices 5 + 5i, 5 - 5i, -5 - 5i and -5 + 5i - HSC - SSCE Mathematics Extension 2 - Question 16 - 2022 - Paper 1

Find the exact value of the complex number z_B.

Find the value of v_0, correct to 1 decimal place.

Show that abc ≤ (S/6)^{3/2}.

Using part (i), show that when the rectangular prism with surface area S is a cube, it has maximum volume.

Find all the complex numbers z_1, z_2, z_3 that satisfy the following three conditions simultaneously.

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