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MTH1020 - Analysis of change - S2 2025

MTH1020 - Analysis of change - S2 2025

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Consider the function f(t) = 2t^3 which gives the distance of a particle from the origin after time t.

Choose the following statements that are correct.

The instantaneous velocity of the particle between times t=a and t=b can be found by finding the gradient of the chord between the two points (a, 2a^3) and (b, 2b^3).

The average velocity of the particle between times t=a and t=b can be found by finding the gradient of the chord between the two points (a, 2a^3) and (b, 2b^3).

The average velocity of the particle at time t=a can be found by finding the gradient of the tangent line at the point (a, 2a^3).

The instantaneous velocity of the particle at time t=a can be found by finding the gradient of the tangent line at the point (a, 2a^3).

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If $w_1=2\,\text{cis}\left(\dfrac{\pi}{6}\right)$ w_1=2\,\text{cis}\left(\dfrac{\pi}{6}\right) is one solution to $w^2 = 4\,\text{cis}\left(\dfrac{\pi}{3}\right)$ w^2 = 4\,\text{cis}\left(\dfrac{\pi}{3}\right) then the other solution will be:

$w_2=2\,\text{cis}\left(-\dfrac{\pi}{6}\right)$ w_2=2\,\text{cis}\left(-\dfrac{\pi}{6}\right)

$w_2=2\,\text{cis}\left(\dfrac{5\pi}{6}\right)$ w_2=2\,\text{cis}\left(\dfrac{5\pi}{6}\right)

$w_2=2\,\text{cis}\left(-\dfrac{5\pi}{6}\right)$ w_2=2\,\text{cis}\left(-\dfrac{5\pi}{6}\right)

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Consider the complex number w^2 = 1-i. The solution/s w can be found to be:

$w=\sqrt[4]{2}\,\text{cis}\left(-\dfrac{\pi}{8}\right)$ w=\sqrt[4]{2}\,\text{cis}\left(-\dfrac{\pi}{8}\right) and $w=\sqrt[4]{2}\,\text{cis}\left(\dfrac{7\pi}{8}\right)$ w=\sqrt[4]{2}\,\text{cis}\left(\dfrac{7\pi}{8}\right).

$w=\sqrt[4]{2}\,\text{cis}\left(\dfrac{\pi}{8}\right)$ w=\sqrt[4]{2}\,\text{cis}\left(\dfrac{\pi}{8}\right) and $w=\sqrt[4]{2}\,\text{cis}\left(-\dfrac{7\pi}{8}\right)$ w=\sqrt[4]{2}\,\text{cis}\left(-\dfrac{7\pi}{8}\right).

$w=\sqrt{2}\,\text{cis}\left(-\dfrac{\pi}{8}\right)$ w=\sqrt{2}\,\text{cis}\left(-\dfrac{\pi}{8}\right) only.

$w=\sqrt{2}\,\text{cis}\left(\dfrac{\pi}{8}\right)$ w=\sqrt{2}\,\text{cis}\left(\dfrac{\pi}{8}\right) and $w=\sqrt{2}\,\text{cis}\left(\dfrac{7\pi}{8}\right)$ w=\sqrt{2}\,\text{cis}\left(\dfrac{7\pi}{8}\right).

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Consider the diagram of the powers, z^n, of the complex number $z=r\,\text{cis}(\theta)$ z=r\,\text{cis}(\theta) shown below.

In the diagram z is in the first quadrant, and successive powers of z, labelled z_1, z_2, z_3 and z_4, are positioned anticlockwise from z, on a circle centred at the origin, with radius r.

Image failed to load: Powers of complex numbers on the complex plane

Using this diagram, choose the following statements that are correct.

As the powers of z follow an anticlockwise path, we can say that the values of n are positive integer values.

We cannot determine if the integer values of n are positive or negative.

We cannot determine r from the diagram.

We can see that r=1 as the magnitude of the powers of z is neither increasing nor decreasing.

As the powers of z follow an anticlockwise path, we can say that the values of n are negative integer values.

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Consider DeMoivre's theorem in exponential form:

If $z=re^{i\theta}$ z=re^{i\theta} then $z^{-n}=r^{-n}e^{-in\theta}$ z^{-n}=r^{-n}e^{-in\theta} for $n = 1,2,3,\ldots$ n = 1,2,3,\ldots.

If we let n=0 which of the following statements will be true?

For $z=re^{i\theta}$ z=re^{i\theta} when n=0 we have $z^0 = r^0e^{0i\theta}=1+i$ z^0 = r^0e^{0i\theta}=1+i.

We cannot use DeMoivre's theorem for n=0 as it only applies when n is a positive or negative integer.

For $z=re^{i\theta}$ z=re^{i\theta} when n=0 we have $z^0 = r^0e^{0i\theta}=0$ z^0 = r^0e^{0i\theta}=0.

For $z=re^{i\theta}$ z=re^{i\theta} when n=0 we have $z^0 = r^0e^{0i\theta}=1$ z^0 = r^0e^{0i\theta}=1.

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Consider the complex number z=-1+i. Convert z to exponential form and choose from below, the correct response/s.

$z^2=2e^{\frac{3\pi}{8}i}$ z^2=2e^{\frac{3\pi}{8}i}

$z^4=4e^{-\frac{3\pi}{4}i}$ z^4=4e^{-\frac{3\pi}{4}i}.

$z^3 = 2\sqrt{2}e^{-\frac{\pi}{4}i}$ z^3 = 2\sqrt{2}e^{-\frac{\pi}{4}i}

$z=\sqrt{2}e^{\frac{3\pi}{4}i}$ z=\sqrt{2}e^{\frac{3\pi}{4}i}

$z=\sqrt{2}e^{-\frac{\pi}{4}i}$ z=\sqrt{2}e^{-\frac{\pi}{4}i}

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Consider the complex number z=-1+i. Choose from below, the correct response/s.

In polar form $z^3=2\sqrt{2}\,\text{cis}\left(\dfrac{\pi}{4}\right)$ z^3=2\sqrt{2}\,\text{cis}\left(\dfrac{\pi}{4}\right).

In Cartesian form z^3 = 2+2i

In Cartesian form z^4=-4

In Cartesian form z^3 = 2-2i

In polar form $z=\sqrt{2}\,\text{cis}\left(-\dfrac{3\pi}{4}\right)$ z=\sqrt{2}\,\text{cis}\left(-\dfrac{3\pi}{4}\right).

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Consider the complex number z=1-i. Choose from below, the correct response/s.

In polar form $z^2=2\,\text{cis}\left(\dfrac{\pi}{2}\right)$ z^2=2\,\text{cis}\left(\dfrac{\pi}{2}\right).

In polar form $z^3=2\sqrt{2}\,\text{cis}\left(-\dfrac{3\pi}{4}\right)$ z^3=2\sqrt{2}\,\text{cis}\left(-\dfrac{3\pi}{4}\right).

In Cartesian form z^4 = -4

In Cartesian form z^4 = 4

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Consider the complex number $z=2\,\text{cis}\left(-\dfrac{\pi}{3}\right)$ z=2\,\text{cis}\left(-\dfrac{\pi}{3}\right). Choose from below, the correct response/s.

$z^4=16\,\text{cis}\left(\dfrac{2\pi}{3}\right)$ z^4=16\,\text{cis}\left(\dfrac{2\pi}{3}\right)

$z^3=8\,\text{cis}(\pi)$ z^3=8\,\text{cis}(\pi)

$z^4=16\,\text{cis}\left(-\dfrac{2\pi}{3}\right)$ z^4=16\,\text{cis}\left(-\dfrac{2\pi}{3}\right)

$z^2=4\,\text{cis}\left(-\dfrac{2\pi}{3}\right)$ z^2=4\,\text{cis}\left(-\dfrac{2\pi}{3}\right)

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