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MTH1020 - Analysis of change - S2 2025

MTH1020 - Analysis of change - S2 2025

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Consider the diagram below showing a shaded region of the complex plane.

Image failed to load: Region of the complex plane

From the following options choose the correct set of complex numbers z that correspond to the shaded region.

${z: \text{Arg}(z)\leq \dfrac{5\pi}{6}}$ {z: \text{Arg}(z)\leq \dfrac{5\pi}{6}}

${z: \dfrac{\pi}{6} < \text{Arg}(z) < \dfrac{5\pi}{6}}$ {z: \dfrac{\pi}{6} < \text{Arg}(z) < \dfrac{5\pi}{6}}

${z: \text{Arg}(z)\geq \dfrac{\pi}{6}}$ {z: \text{Arg}(z)\geq \dfrac{\pi}{6}}

${z: \dfrac{\pi}{6}\leq \text{Arg}(z)\leq \dfrac{5\pi}{6}}$ {z: \dfrac{\pi}{6}\leq \text{Arg}(z)\leq \dfrac{5\pi}{6}}

${z: \dfrac{\pi}{6} < \text{Arg}(z) \leq \dfrac{5\pi}{6}}$ {z: \dfrac{\pi}{6} < \text{Arg}(z) \leq \dfrac{5\pi}{6}}

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Choose the correct response/s below:

The equation "y = 2x + 5" can be written as a statement that a proposition P(n) is true for all integers n.

In mathematics, a proposition P(n) is a statement that is either true or false.

In mathematics, a proposition P(n) will always be proven true.

The statement "All odd numbers are prime" can be written as a statement that a proposition P(n) is true for all integers n.

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Consider the complex number z=a+bi where a and b are non-zero real numbers.

Express the complex number $\dfrac{1}{\overline{z}}$ \dfrac{1}{\overline{z}} in terms of z only.

Choose the correct response below.

$\dfrac{1}{\overline{z}} = \dfrac{1}{|z|}$ \dfrac{1}{\overline{z}} = \dfrac{1}{|z|}

$\dfrac{1}{\overline{z}} = \dfrac{1}{|z|^2}$ \dfrac{1}{\overline{z}} = \dfrac{1}{|z|^2}

$\dfrac{1}{\overline{z}} = \dfrac{z}{|z|}$ \dfrac{1}{\overline{z}} = \dfrac{z}{|z|}

$\dfrac{1}{\overline{z}} = \dfrac{z}{|z|^2}$ \dfrac{1}{\overline{z}} = \dfrac{z}{|z|^2}

$\dfrac{1}{\overline{z}} = -\dfrac{1}{z}$ \dfrac{1}{\overline{z}} = -\dfrac{1}{z}

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Consider the diagram of the complex plane below. If the complex number $\overline{z}$ \overline{z} is in the fourth quadrant as shown, which quadrant will the complex number z be in? Choose the correct response from below.

Image failed to load: complex conjugate in fourth quadrant

the second quadrant

the fourth quadrant

the first quadrant

the third quadrant

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Consider the diagram of the complex plane below. If the complex number $\overline{z}$ \overline{z} is in the second quadrant as shown, which quadrant will the complex number z be in? Choose the correct response from below.

Image failed to load: complex conjugate

the second quadrant

the fourth quadrant

the first quadrant

the third quadrant

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Which of the following statements regarding the vector projection of $\vec{u}$ \vec{u} onto $\vec{v}$ \vec{v} is true?

The resulting answer can be found by multiplying the scalar projection of vector $\vec{u}$ \vec{u} onto vector $\vec{v}$ \vec{v} by a unit vector in the direction of $\vec{u}$ \vec{u}.

The resulting answer can be found by multiplying the scalar projection of vector $\vec{u}$ \vec{u} onto vector $\vec{v}$ \vec{v} by a unit vector in the direction of $\vec{v}$ \vec{v}.

The resulting answer will be the perpendicular component of $\vec{u}$ \vec{u} in the direction of $\vec{v}$ \vec{v}.

The resulting answer will be the perpendicular component of $\vec{v}$ \vec{v} in the direction of $\vec{u}$ \vec{u}.

The resulting answer will be a scalar quantity.

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Consider the diagram below and choose the correct statement/s:

Image failed to load: Vector diagram showing F perpendicular

$\cos( \theta) = \frac{ | \vec{F}_{\perp} | }{ | \vec{F} |}$ \cos( \theta) = \frac{ | \vec{F}_{\perp} | }{ | \vec{F} |}

$|\vec{F}_{\perp}| = |\vec{F}|\sin(\theta)$ |\vec{F}_{\perp}| = |\vec{F}|\sin(\theta)

$|\vec{F}_{\perp}| = |\vec{F}|\cos(\theta)$ |\vec{F}_{\perp}| = |\vec{F}|\cos(\theta)

$\cos( \phi) = \frac{ | \vec{F}_{\perp} | }{ | \vec{F} |}$ \cos( \phi) = \frac{ | \vec{F}_{\perp} | }{ | \vec{F} |}

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Torque is a force that rotates, or turns, an object about its pivot, or fulcrum. Some examples are:

turning of a doorknob

turning of a see saw on its fulcrum

unscrewing of a bottle top

Consider the diagram below and choose the correct statements regarding torque:

Image failed to load: torque diagram

torque is perpendicular to vector $\vec{r}$ \vec{r} only

torque is perpendicular to the vectors $\vec{r}$ \vec{r} and $\vec{F}$ \vec{F}

torque is perpendicular to vector $\vec{F}$ \vec{F} only

torque is a vector quantity

torque is a scalar quantity

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