What are suitable parametric equations for this plane curve?
Exactly one option must be correct)

*Choice (a) is incorrect*

This is an ellipse with centre $\left(1,2\right)$
and with axes of length $4$
in the $x$–direction
and $6$ in the
$y$–direction.

*Choice (b) is correct!*

The curve is an ellipse centre (1,2) with axes length 6 in the
$x$ direction
and 4 in the $y$
direction.

*Choice (c) is incorrect*

This is an ellipse
with centre $\left(-1,-2\right)$ and
with axes of length $6$
in the $x$–direction
and $4$ in the
$y$–direction.

*Choice (d) is incorrect*

This is an ellipse
with centre $\left(-1,-2\right)$ and
with axes of length $4$
in the $x$–direction
and $6$ in the
$y$–direction.

Which of the following could be the parametric equations
for the following space curve? (You may assume that
$t\ge 0$ and that the scales
are the same on the $x,\phantom{\rule{0.3em}{0ex}}y$
and $z$
axes.)
Exactly one option must be correct)

*Choice (a) is incorrect*

This curve has the following graph.

*Choice (b) is incorrect*

In this curve the
$z$ values increase
as the square of $t$,
making the spiral much ‘steeper’ than the curve shown in the question. The
parametric equations given in this option correspond to the following graph.

*Choice (c) is incorrect*

This
curve has the following graph.

*Choice (d) is correct!*

The curve circles around the $z$-axis
and the distance of the curve from the
$z$-axis increases
linearly with $t$
(and hence $z$).

Which of the following could be the parametric equations for the following space
curve?
(Hint: look at the sign of $y$
as $t$
changes.)
Exactly one option must be correct)

*Choice (a) is correct!*

As $t$
increases from large negative values through zero to large positive values,
$y$ starts
out negative then becomes positive, then negative and finally positive again. This fits
the curve shown. It is actually quite hard to “see” where in space this curve
lies, from the diagram provided, so don’t worry if you didn’t get this one
correct.

*Choice (b) is incorrect*

The
$cost$ and
$sint$ terms make this curve
wind around the $z$–axis,
and the $z={t}^{3}$
term makes the height of the curve above the
$xy$–plane
increase rapidly.

*Choice (c) is incorrect*

This is a “half”
straight line because $z=y=x+1$,
for all $t$. (It is only
half a line because $x\ge -1$,
$y\ge 0$ and
$z\ge 0$ since
${t}^{2}\ge 0$ for all
$t$.)
(The box is there to help visualize the line).

*Choice (d) is incorrect*

Let $f\left(x,y\right)={x}^{2}+{y}^{2}-1$. Which of the following
could be the graph of $z=f\left(x,y\right)$?
Exactly one option must be correct)

*Choice (a) is incorrect*

This cannot be the graph of $f\left(x,y\right)$.
Look again at the formula for $f\left(x,y\right)$!
It tells us that $f\left(x,y\right)\ge -1$ for all
$x,y$. In fact, this option
is the graph of $z=1-\left({x}^{2}+{y}^{2}\right)$.

*Choice (b) is incorrect*

This is the graph of a regular cone (it has “straight sides”), whereas
$f\left(x,y\right)$ is a paraboloid. In fact,
this is the graph of $z=\sqrt{{x}^{2}+{y}^{2}-1}$.

*Choice (c) is incorrect*

Even though this is the graph of a paraboloid it cannot be the graph of
$z=f\left(x,y\right)$ since
$f\left(0,0\right)=-1$ whereas this graph
goes through $\left(0,0,0\right)$. In fact,
this is the graph of $z={x}^{2}+{y}^{2}$.

*Choice (d) is correct!*

This is the only graph listed which is paraboloid passing through
$\left(0,0,-1\right)$.

Which option could be the equation of the following surface?
Exactly one option must be correct)

*Choice (a) is incorrect*

The function $f\left(x,y\right)=4-\left({x}^{2}+{y}^{2}\right)$
is an inverted paraboloid. It has the following graph.

*Choice (b) is incorrect*

This function does
not depend on $x$.
Its graph is:

*Choice (c) is incorrect*

This surface lies
entirely below the $xy$–plane
so it cannot be the function pictured. Its graph is

*Choice (d) is correct!*

This graph has a parabolic cross-section
$z=-{x}^{2}$ which does not depend
on the value of $y$.

Which of the following equations could describe this surface?
Exactly one option must be correct)

*Choice (a) is incorrect*

This surface looks like an egg carton, as shown below. The surface in the question can’t be given by
the equation $z=cosxcosy$ because
in the plane $x=y$, for
example, we have $z={cos}^{2}x$
and so $z\ge 0$,
which is not the case.

*Choice (b) is incorrect*

As
$sinxsiny=cos\left(x+\frac{\pi}{2}\right)cos\left(y+\frac{\pi}{2}\right)$
this surface is just a shifted version of the surface
$z=cosxcosy$.

*Choice (c) is incorrect*

As
$cosxsiny=cos\left(x\right)cos\left(y+\frac{\pi}{2}\right)$
this surface is just another shifted version of the surface
$z=cosxcosy$.

*Choice (d) is incorrect*

This function is a surface of revolution, so it could be the graph in the
question. However, the ripples in this function get closer together as
$\sqrt{{x}^{2}+{y}^{2}}$ gets
large, so it cannot be the right answer.

*Choice (e) is correct!*

This is a surface of revolution rotated about the
$z$-axis and its equation
is given by either $z=cos\sqrt{{x}^{2}+{y}^{2}}$
or $z=sin\sqrt{{x}^{2}+{y}^{2}}$. When
$x=y=0$ we have
$z=1$ (from the
graph) so it is $z=cos\sqrt{{x}^{2}+{y}^{2}}$.

*Choice (f) is incorrect*

This is
very similar to the graph in the question; however, this cannot be the right function
because $z=0$
when $\left(x,y\right)=\left(0,0\right)$
which does not agree with the graph in the question.

Which of the equations below could describe the following surface?
Exactly one option must be correct)

*Choice (a) is incorrect*

This cannot be the right graph because
$z\to -\infty $ as
$\left(x,y\right)\to \left(0,0\right)$. In fact, the
function $z=log\left({x}^{2}+{y}^{2}\right)$
has the graph:

*Choice (b) is incorrect*

This function
looks like a very “steep” paraboloid of revolution. It cannot be the function in the question
because $z=0$
when $\left(x,y\right)=\left(0,0\right)$. In
fact, $z={e}^{{x}^{2}+{y}^{2}}$
has the following graph.

*Choice (c) is incorrect*

This function looks like a very “steep” inverted paraboloid of
revolution. It cannot be the function in this option because here,
$z$
is always less than 1 and becomes very large and negative as
${x}^{2}+{y}^{2}$ increases.
In fact, $z=1-{e}^{{x}^{2}+{y}^{2}}$
has the following graph.

*Choice (d) is correct!*

This is a surface of
rotation about the $z$-axis.
As ${x}^{2}+{y}^{2}\to \infty $ as
$\left(x,y\right)\to \left(0,0\right)$ it is
the only function listed which is consistent with the graph in the question.

Which one of the following shows some of the level curves for the function
$z=x\left(x+y\right)$ ?
Exactly one option must be correct)

*Choice (a) is incorrect*

These level curves are ellipses with equations of the form
$\frac{{\left(x-1\right)}^{2}}{{a}^{2}}+\frac{{\left(y-2\right)}^{2}}{{b}^{2}}=1$,
which are not the same equations as the level curves for the function with equation
$z=x\left(x+y\right)$.

*Choice (b) is correct!*

The level curve at height $z=c$
satisfies the equation $c=x\left(x+y\right)$;
that is, $y=\frac{c}{x}-x$.

*Choice (c) is incorrect*

These level curves appear to have an asymptote at
$x=2$ whereas the level
curves of $z=f\left(x,y\right)$ have
an asymptote at $x=0$.

*Choice (d) is incorrect*

These level curves are ellipses with equations of the form
$\frac{{\left(x-1\right)}^{2}}{{a}^{2}}+\frac{{\left(y-2\right)}^{2}}{{b}^{2}}=1$,
which are not the same equations as the level curves for the function with equation
$z=x\left(x+y\right)$.

Some of the level curves of an unknown function
$z=f\left(x,y\right)$ are
given below.
Which of the following functions could be
$f\left(x,y\right)$ ?
(Note: we do not know the height of these level curves.) Exactly one option must be
correct)

*Choice (a) is incorrect*

The
level curves of this function are circles centred at the origin.

*Choice (b) is incorrect*

The level curves of this function are ellipses centered at
$\left(-1,-2\right)$.

*Choice (c) is incorrect*

The level curves of this function are ellipses centered at
$\left(1,2\right)$. The
semi-major axis of each ellipse is vertical and the semi-minor axis is horizontal. That
is, the ellipses are taller than they are wide, and so this option doesn’t match the
given set of curves.

*Choice (d) is correct!*

The level curves of this function are ellipses centered at
$\left(1,2\right)$. The semi-major
axis of each ellipse is horizontal and the semi-minor axis is vertical. The ratio of these two
axes is $3:2$,
matching that shown on the set of curves.

Which of the following graphs could be part of the set of level curves for some surface
$z=f\left(x,y\right)$ ?
(Zero or more options can be correct)

For example, choice (a) should be True.

For example, choice (b) should be True.

For example, choice (c) should be True.

*There is at least one mistake.*

For example, choice (a) should be True.

For example, these could be some of the level curves of the surface given
by

$$f\left(x,y\right)=\sqrt{3{\left(x-1\right)}^{2}+2{\left(y-2\right)}^{2}}$$

.*There is at least one mistake.*

For example, choice (b) should be True.

For example, these could be some of the level curves of the surface given
by

$$f\left(x,y\right)={\left(x-1\right)}^{2}-{\left(y-2\right)}^{2}$$

.*There is at least one mistake.*

For example, choice (c) should be True.

For example, these could be some of the level curves of the surface given
by

$$f\left(x,y\right)=y-x$$

.*Correct!*

*True*For example, these could be some of the level curves of the surface given by$$f\left(x,y\right)=\sqrt{3{\left(x-1\right)}^{2}+2{\left(y-2\right)}^{2}}$$.*True*For example, these could be some of the level curves of the surface given by$$f\left(x,y\right)={\left(x-1\right)}^{2}-{\left(y-2\right)}^{2}$$.*True*For example, these could be some of the level curves of the surface given by$$f\left(x,y\right)=y-x$$.