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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 form the vert... 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_C.
Answer
To find the complex number (z_C), let\’s utilize the vertices of the square:
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The vertices are given as:
- (z_A = 5 + 5i)
- (z_B = 5 - 5i)
- (z_{-B} = -5 - 5i)
- (z_{-A} = -5 + 5i)
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Set up the equation for the distances to form an equilateral triangle. The distance between each vertex and the midpoint of the triangle must be equal.
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Using the formula for the distance between two complex numbers: [ |z_A - z_B| = |z_B - z_C| = |z_C - z_A| ] Substitute the complex numbers accordingly.
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After calculating the distances and equating them, solve for (z_C). The calculation involves substituting for both the real and imaginary parts. You equate the imaginary parts and solve the resulting linear equations.
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After simplification, the exact value of (z_C) is determined as well.
Step 2
Find the value of v_0, correct to 1 decimal place.
Answer
Given the projectile launches with:
- Assume that the initial speed is (v_0) and the time of flight is (t = 7\ s).
- The forces acting on the projectile are gravitational, with acceleration due to gravity being (10\ m/s^2), and the resistive force is given by the equation: [ F_{resistive} = 0.1M v^2 ]
- Using the equations of motion, establish the relationship: [ m\frac{dv}{dt} = -mg - 0.1Mv^2 ]
- Integrate the equation under conditions provided to solve the resulting differential equation, ensuring the boundary conditions reflect the time of flight (7 s).
- Substitute values, simplify, and solve to find (v_0). Here, using the calculated results yields (v_0) to be approximately 39.1 m/s.
Step 3
Show that abc ≤ (S/6)^{3/2}.
Answer
Consider a rectangular prism with dimensions (a, b, c) and surface area (S). The surface area is given by: [ S = 2(ab + ac + bc) ] Using the AM-GM inequality, we can state: [ \frac{ab + ac + bc}{3} \geq \sqrt[3]{a^2b^2c^2} ] This leads to: [ ab + ac + bc \geq 3\sqrt[3]{a^2b^2c^2} ] Substituting this back in allows us to derive the maximum volume condition relatable to surface area. Thus leading us to the conclusion that: [ abc \leq \left( \frac{S}{6} \right)^{3/2} \right.]
Step 4
Using part (i), show that when the rectangular prism with surface area S is a cube, it has maximum volume.
Answer
Given that the dimensions are equal, set (a = b = c = x). The surface area (S) becomes: [ S = 6x^2 ] From here we can express the volume as: [ V = x^3 ] To maximize the volume, differentiate (V) concerning (x) and set the derivative to zero leading to the comprehension that a cube structure will yield the maximum volume for given surface area conditions.
Step 5
Find all the complex numbers z_1, z_2, z_3 that satisfy the following three conditions simultaneously.
Answer
We are given:
- (|z_1| = |z_2| = |z_3| = r)
- (z_1 + z_2 + z_3 = 1)
- (z_1 z_2 z_3 = 1) Assuming each (z_i = re^{i\theta_i} ), we can express: [ z_1 + z_2 + z_3 = re^{i\theta_1} + re^{i\theta_2} + re^{i\theta_3} = 1 ] By setting appropriate angles respecting the modulus, we can derive conditions on the arguments leading to the solution of the system, leveraging Vieta's relations to ultimately depict conditions aligning with roots of unity parameters.