Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet. On the diagram provided in the writing booklet, indicate the locations of all of the fourth roots of the complex number $a + ib$. (b) Use an appropriate substitution to evaluate $$\int_{\sqrt{10}}^{\sqrt{3}} \frac{x^3 \sqrt{x^2 - 9}}{x^2} dx.$$ Question 13 continues on page 10

SSCE HSC Mathematics Extension 2 Roots of complex numbers: Use the Question 13 Writing Book

Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2021 - Paper 1

Question 13

Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet. On the di... show full transcript

Worked Solution & Example Answer:Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2021 - Paper 1

Step 1

The location of the complex number $a + ib$ and its fourth roots

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Answer

To indicate the locations of all of the fourth roots of the complex number $a + ib$ , we can start by determining the modulus and argument of $a + ib$ . The modulus is given by:

$|a + ib| = \sqrt{a^2 + b^2}$

The argument (angle) is given by:

$\theta = \tan^{-1}\left(\frac{b}{a}\right)$

The fourth roots can then be found using the formula for the roots of a complex number:

$z_k = |z|^{1/n} \left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right)\right)$

For $n = 4$ , where $k = 0, 1, 2, 3$ .

This will yield four points representing the fourth roots on the Argand diagram.

Step 2

Use an appropriate substitution to evaluate

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Answer

Let us evaluate the integral:

$\int_{\sqrt{10}}^{\sqrt{3}} \frac{x^3 \sqrt{x^2 - 9}}{x^2} dx$ .

First, we simplify the integral:

$\int_{\sqrt{10}}^{\sqrt{3}} x \sqrt{x^2 - 9} \, dx$ .

Next, we use the substitution:

Let: $u = \sqrt{x^2 - 9}$ Then: $du = \frac{x}{\sqrt{x^2 - 9}} \, dx \rightarrow \ dx = \frac{du \sqrt{x^2 - 9}}{x}$ .

To find the limits:

When $x = \sqrt{10}$ , $u = \sqrt{10^2 - 9} = \sqrt{91}$ .
When $x = \sqrt{3}$ , $u = \sqrt{3^2 - 9} = \sqrt{-9}$ , which is not valid, indicating a possible missing part in the equation's domain.

In later steps, we would integrate with the correct upper and lower limits for valid $u$ based on the actual conversion of $x$ to $u$ and then evaluate the integral accordingly.

Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2021 - Paper 1

Worked Solution & Example Answer:Use the Question 13 Writing Booklet (a) The location of the complex number $a + ib$ is shown on the diagram on page 1 of the Question 13 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2021 - Paper 1

The location of the complex number $a + ib$ and its fourth roots

Use an appropriate substitution to evaluate

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