Tutorial 1
MAM2084F, 2025, UCT
This tutorial is aimed at refreshing some prior knowledge needed for this course
1. Calculate the following:
a b
d
x
(a) x y
+ x y
b c
y
e
 

3
3 0 1



(b) 1 2 1
2 
1 1 1
1
0 2 −1
   
3
1



×
(c)
2 
2
1
1
2. (Hand-in problem) Solve the following differential equations
dy
(a) dx
= xex
2
y(0) = 0
dy
(b) dx
= sin2 (3x)
y(0) = 1
dy
= ex cos(x)
(c) dx
dy
x
(d) dx
= 6x2 +5x+1




1
2
3. Find a unit vector in R3 that is orthogonal to the two vectors  1  and  −1 .
1
1
4. Find all complex numbers z such that z 5 = −32. Write your answer in polar
coordinates i.e. in the form z = reiθ where 0 ≤ θ < 2π
5. (Hand-in problem) Find the unique point in R3 sitting on the intersection of the
three planes x + 2y + z = 2, 2x + y + 4z = 2 and x + y − z = 4.
6. The combustion of octane is a chemical reaction of the form
aC8 H18 + bO2 → cCO2 + dH2 O
(a) Write down 3 equations that the coefficients a, b, c and d must satisfy
(b) Solve for b, c and d in terms of a
(c) Find the smallest value for a that allows all four coefficients to be integervalued
1