MATH2010 Complementary Mock B

Original, non-official practice: 4 questions, 100 marks, suggested 90 minutes. This complements Mock A on arc length, reparametrization, cylindrical/spherical coordinates and equality conditions. It uses known Weeks 1–3 topics and current course notes; it does not claim complete course or official examination coverage, and does not infer Week 4 content. The marks and timing are practice-design choices, not school examination rules. No calculator is needed for the exact answers; this is not an official calculator-policy statement. Angles are in radians.

Cylindrical coordinates are $(r,\theta,z)$ with $r\ge0$. Spherical coordinates are $(\rho,\theta,\phi)$ with $\rho\ge0$, $\phi$ measured from the positive $z$-axis, $0\le\theta<2\pi$ and $0\le\phi\le\pi$ for representative choices. On the axis or at the origin angles can be nonunique. Show reasoning, retain exact values, and give every parameter interval.

Revision route · Solutions

Q1. A helix, distance and unit speed — 30 marks, about 27 minutes

Worked solution

Let $h(t)=(2\cos t,2\sin t,t)$, $0\le t\le\pi$.

(a) [8] Find velocity and speed, and the tangent line at $t=\pi/2$.

Paired solutions

(b) [8] Find the travelled arc length and the distance between endpoints. Prove that these two numbers are different.

Paired solutions

(c) [8] Let $s$ denote distance travelled from $t=0$. Find $t$ as a function of $s$, give a unit-speed parametrization with its domain, and verify the speed directly.

Paired solutions

(d) [6] Reverse the original traversal using $q(u)=h(\pi-u)$, $0\le u\le\pi$. Find the velocity and speed of $q$, and explain why the length remains positive and unchanged.

Paired solutions

Q2. Zero velocity and tangents — 20 marks, about 17 minutes

Worked solution

Let $a(u)=(u^3,0)$ for $-1\le u\le1$ and $b(t)=(t,0)$ for $-1\le t\le1$.

(a) [6] Find their images and their speeds. Compare the velocities at parameter zero.

Paired solutions

(b) [6] Compute both travelled lengths directly. At $u=0$, explain geometrically whether the image has a tangent line, and why blindly inserting $a'(0)$ into the regular tangent-line formula does not establish it.

Paired solutions

(c) [8] The substitution $t=\phi(u)=u^3$ is increasing and one-to-one. Does it satisfy the sufficient theorem requiring a continuously differentiable monotone change of parameter with nonzero derivative everywhere? Discuss the inverse derivative at zero. Explain why failure of these hypotheses does not contradict the equality of lengths found in (b).

Paired solutions

Q3. Two radii and a cone — 30 marks, about 27 minutes

Worked solution

Use the coordinate conventions stated above. Let $P=(-1,\sqrt3,-2)$ and

$E=\{(x,y,z):1\le\sqrt{x^2+y^2+z^2}\le3,\ z\ge\sqrt{x^2+y^2}\}.$

The set $E$ includes the boundaries specified by these inequalities.

(a) [10] Give cylindrical and spherical coordinates of $P$, explain the angle choices, and verify by substitution.

Paired solutions

(b) [10] Express $E$ in spherical coordinates using ranges for all three coordinates. Derive the angular restriction without squaring away a sign condition.

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(c) [6] Express the same $E$ in cylindrical coordinates. Explain why replacing $z\ge r$ by $z^2\ge r^2$ would add unwanted points.

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(d) [4] In cylindrical coordinates, identify the geometric set $\theta=\pi/2$ with $r\ge0$ and unrestricted $z$. Is it the entire plane $x=0$?

Paired solutions

Q4. Equality conditions matter — 20 marks, about 19 minutes

Worked solution

All vectors are real Euclidean vectors.

(a) [8] For $u\ne0$, prove Cauchy–Schwarz by expanding $\|v-tu\|^2$ at the minimizing scalar $t$. State the exact equality condition, including what happens when one vector is zero.

Paired solutions

(b) [6] Let $u=(1,2,-2)$ and $v=-2u$. Compute $\langle u,v\rangle$, $\|u\|$, $\|v\|$ and $\|u+v\|$. Decide whether equality holds in the absolute-value Cauchy–Schwarz inequality and in the triangle inequality. Give the numerical comparison in each case.

Paired solutions

(c) [6] Correct this statement: “If two vectors are linearly dependent, then equality holds in the triangle inequality.” Prove the corrected condition for nonzero vectors, and include the zero-vector cases.

Paired solutions

Check before marking

Check velocity versus speed, the new parameter interval, whether the inverse has a finite derivative, the spherical polar-angle convention, and the positive-multiple condition for triangle equality. Then use the worked solutions to choose the next topic to revisit.