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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 diag... 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
Answer
To locate the complex number on the diagram, note that the real part 'a' is on the x-axis and the imaginary part 'b' is on the y-axis. The fourth roots of a complex number are found by using the formula for the n-th roots:
[ z_k = r^{1/n} \left( \cos\left( \frac{\theta + 2k\pi}{n} \right) + i \sin\left( \frac{\theta + 2k\pi}{n} \right) \right) ]
where is the modulus and is the argument of the complex number. Therefore, the four roots can be indicated accordingly on the diagram.
Step 2
Use an appropriate substitution to evaluate
Answer
To evaluate the integral
[ \int_{\sqrt{10}}^{\sqrt{3}} \frac{x^3 \sqrt{x^2 - 9}}{x^2} dx, ]
we can use the substitution: let ( u = \sqrt{x^2 - 9} ). Then, we differentiate to find
[ du = \frac{x}{\sqrt{x^2 - 9}} dx \Rightarrow dx = \frac{du \sqrt{x^2 - 9}}{x} ]
Next, we change the limits of integration. For ( x = \sqrt{10} ), [ u = \sqrt{10^2 - 9} = \sqrt{91} ]
and for ( x = \sqrt{3} ), [ u = \sqrt{3^2 - 9} = \sqrt{-9} ]
Notice that ( \sqrt{-9} ) suggests our integral evaluation needs adjustments due to the limits. Evaluating the transformed expression should now give the required result accordingly.