MAU11E02 Engineering Mathematics 2 Assignment 1, 2026 | TCD Ireland

MAU11E02 Assignment 1

©Trinity College Dublin, The University of Dublin. You are not permitted to distribute or share this document with 3rd parties.

Doing so is a violation of academic integrity, see http://www.tcd.ie/academic_integrity.

This is part of your continuous assessment and contributes to your final grade.
Due date: Monday, 16/03/2026, 06:00am. No late assignments!
Upload your solutions to Blackboard as a single pdf file. No other formats!
WRITE CLEAR AND CONCISE. SCRIBBLED SOLUTIONS WILL NOT BE MARKED.

Problem 1 (25 points)

Consider the lines L1L_1L1​ and L2L_2L2​ with parametric equations

L1:x=1+3t,y=2−t,z=−3+4t.L_1 : \quad x = 1 + 3t, \quad y = 2 – t, \quad z = -3 + 4t.L1​:x=1+3t,y=2−t,z=−3+4t.
L2:x=1−s,y=−2+s,z=4+2s.L_2 : \quad x = 1 – s, \quad y = -2 + s, \quad z = 4 + 2s.L2​:x=1−s,y=−2+s,z=4+2s.

a) Are these lines L1L_1L1​ and L2L_2L2​ parallel to each other? Justify your answer.

b) Do the lines L1L_1L1​ and L2L_2L2​ intersect at some point? Justify your answer.

Problem 2 (25 points)

Given the following implicit equation

379+98x+7×2−32y+4y2=0379 + 98x + 7x^2 – 32y + 4y^2 = 0379+98x+7x2−32y+4y2=0

a) Find the center and all four vertices of the ellipse described by the equation after writing it in standard form.

b) Parameterise all points on the ellipse using a minimal number of functions $y=f(x)$.

c) The equation is fulfilled for points $(x,y)$ on the ellipse. What are the intervals $X$ and $Y$ for admissible values $x\in X$ and $y\in Y$?

Problem 3 (25 points)

Given

L1:x−4−2=y+18=z−32,andL2:x=−t,  y=−2+3t,  z=5t.L_1 : \frac{x – 4}{-2} = \frac{y + 1}{8} = \frac{z – 3}{2}, \quad \text{and} \quad L_2 : x = -t, \; y = -2 + 3t, \; z = 5t.L1​:−2x−4​=8y+1​=2z−3​,andL2​:x=−t,y=−2+3t,z=5t.

a) Find the smallest angle between the lines L1L_1L1​ and L2L_2L2​.

b) Find the shortest distance from the point $P=(2,7,5)$ on the line L1L_1L1​ to the line L2L_2L2​.

Problem 4 (25 points)

Let $T$ be the triangle with vertices $A,B,$ and $C$ given by

$A=(2,8,-2),B=(2,3,1),C=(1,0,5).$

a) Calculate the area of the triangle $T$.

b) Find the equation of the plane containing the points $A,B,$ and $C$.

c) What is the distance of this plane from the origin.