2026
MATHEMATICS II
MOCK EXAMS
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Mock Exam 1 — Mathematics for Business and Economics II ..................................................................... 3
Tapas (7/20) ......................................................................................................................................... 4
El Plato Principal (13/20) ....................................................................................................................... 5
Mock Exam 2 — Mathematics for Business and Economics II ..................................................................... 6
Tapas (7/20) ......................................................................................................................................... 7
El Plato Principal (13/20) ....................................................................................................................... 8
Mock Exam 3 — Mathematics for Business and Economics II ..................................................................... 9
Tapas (7/20) ....................................................................................................................................... 10
El Plato Principal (13/20) ..................................................................................................................... 11
Mock Exam 4 — Mathematics for Business and Economics II ................................................................... 12
Tapas (7/20) ....................................................................................................................................... 13
El Plato Principal (13/20) ..................................................................................................................... 14
Mock Exam 5 — Mathematics for Business and Economics II ................................................................... 16
Tapas (7/20) ....................................................................................................................................... 17
El Plato Principal (13/20) ..................................................................................................................... 18
Mock Exam 6 — Mathematics for Business and Economics II ................................................................... 19
Tapas (3/20) ....................................................................................................................................... 20
El Plato Principal (14/20) ..................................................................................................................... 21
Postres (3/20) ..................................................................................................................................... 22
Mock Exam 7 — Mathematics for Business and Economics II ................................................................... 23
Tapas (3/20) ....................................................................................................................................... 24
El Plato Principal (14/20) ..................................................................................................................... 25
Postres (3/20) ..................................................................................................................................... 26
Mock Exam 8 — Mathematics for Business and Economics II ................................................................... 27
Tapas (3/20) ....................................................................................................................................... 28
El Plato Principal (14/20) ..................................................................................................................... 29
Postres (3/20) ..................................................................................................................................... 30
Mock Exam 9 — Mathematics for Business and Economics II ................................................................... 31
Tapas (3/20) ....................................................................................................................................... 32
1
, El Plato Principal (14/20) ..................................................................................................................... 33
Postres (3/20) ..................................................................................................................................... 34
Mock Exam 10 — Mathematics for Business and Economics II ................................................................. 35
Tapas (3/20) ....................................................................................................................................... 36
El Plato Principal (14/20) ..................................................................................................................... 37
Postres (3/20) ..................................................................................................................................... 38
2
,Mock Exam 1 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 1 of 10 in the practice series. Good luck!
3
,Tapas (7/20)
Q1.1. We first ask you to define your own homomorphism 𝑡, which maps vectors from ℝ! to ℝ! . Make
sure that the dimension of the null space (nullity) of your homomorphism 𝑡 equals 1.
(a) Define 𝑡 explicitly by giving an expression for 𝑡(𝑥, 𝑦, 𝑧)" .
(b) Compute the range space and its dimension (rank).
(c) Verify that the rank–nullity theorem holds for your map.
(d) Construct a basis of the null space and a basis of the range space.
(e) If you represent 𝑡 by a matrix 𝑇, how many rows and columns does 𝑇 have?
(f) What can you say about det𝑇 in the standard basis of the domain and co-domain?
Q1.2. (a) Write down a non-singular 3 × 3 matrix 𝐴 with exactly one real non-zero eigenvalue 𝜆# = 𝑎
(where 𝑎 is your day of birth). (b) Write down a non-singular 3 × 3 matrix 𝐶 with three distinct, non-zero,
real eigenvalues, all of absolute value strictly smaller than 1. (c) Compute the eigenvectors of your
matrix 𝐶. (d) What can you say about 𝐶 #$$ as the exponent grows?
Q1.3. Prove the following lemma. In any vector space 𝑉, for every 𝑣⃗ ∈ 𝑉 and every scalar 𝑟 ∈ ℝ we have
=⃗, 𝑟 ⋅ =0⃗ = =0⃗, and (−1) ⋅ 𝑣⃗ is the additive inverse of 𝑣⃗.
0 ⋅ 𝑣⃗ = 0
Q1.4. Consider a contour plot of 𝑧 = 𝑓(𝑥, 𝑦) whose contours form concentric ellipses centred at (1,2),
with 𝑓 increasing outward, and a contour plot of 𝑧 = 𝑔(𝑥, 𝑦) whose contours are parallel lines of slope
+1 with 𝑔 increasing as one moves in the direction (1,1). A blue point is marked at the origin (0,0) on
both plots. Determine for the marked point the sign (< 0, > 0, = 0) of:
(a) 𝑓%
(b) 𝑓&
(c) 𝑔%
(d) 𝑔&
Q1.5. (a) Come up with a first-order, separable, ordinary differential equation. (b) How many boundary
conditions do you need to find one unique solution to your differential equation?
Q1.6. Consider the linear map 𝑓: ℝ! → ℝ' for which 𝑓(1,0,0)" = (2, −1)" , 𝑓(0,1,0)" = (0,3)" ,
𝑓(0,0,1)" = (−1,1)" . Determine 𝑓(𝑎, 𝑏, 𝑐)" .
Q1.7. Consider the homomorphism ℎ: ℝ( → ℝ' given by ℎ(𝑥, 𝑦, 𝑧, 𝑤)" = (𝑥 − 𝑦, 𝑧 + 𝑤)" .
(a) Find the null space of ℎ and its nullity.
(b) Build a basis for the null space.
(c) Verify the rank–nullity theorem.
(d) If ℎ is represented by a matrix 𝐻, how many rows and columns does 𝐻 have?
4
,El Plato Principal (13/20)
Q1.8. (a) Describe the plane in ℝ! that contains your 'birthday point' (2,3, 𝑎)" and the line {(2,4,5)" +
𝑣(2,5, 𝑎 + 5)" : 𝑣 ∈ ℝ}. (b) What is the geometric shape of the intersection between this plane and the
𝑥𝑦-plane?
Q1.9. Consider the two ordered sets 𝐵 = ⟨(𝑎, 𝑏, 𝑐)" , (1,0,2)" , (0,0,1)" ⟩ and 𝐷 =
⟨(𝑎, 0,0)" , (0, 𝑏, 0)" , (0,0, 𝑐)" ⟩ in ℝ! (with your personal 𝑎, 𝑏, 𝑐).
(a) Show that 𝐵 and 𝐷 span the same vector space.
(b) Find the change-of-basis matrix from 𝐵 to 𝐷.
(c) Consider the map 𝑡: ℝ! → ℝ! given by 𝑡(𝑥, 𝑦, 𝑧)" = (𝑥 + 𝑦, 𝑏(𝑥 − 𝑦), 𝑐𝑧)" . Write down the matrix 𝑇 that
represents 𝑡 in the standard basis.
(d) Find the matrix 𝑇R that represents 𝑡 with respect to 𝐵 in both domain and co-domain.
4 0 0
Q1.10. Consider the map 𝑔: ℝ! → ℝ! represented in the standard basis by 𝐺 = T1 3 −1U.
2 0 2
(a) Find the eigenvalues of 𝐺.
(b) For each eigenvalue, give a basis of the corresponding eigenspace.
(c) Decide whether 𝐺 is diagonalisable. If so, give the matrices 𝑃 and 𝐷 such that 𝐺 = 𝑃𝐷𝑃)# .
(d) Describe the behaviour of 𝐺 * as 𝑛 → ∞ on a vector along an eigenvector of the dominant eigenvalue.
Q1.11. Find the critical point(s) of 𝑓(𝑥, 𝑦) = 𝑥 ' + 4𝑥𝑦 + 5𝑦 ' − 6𝑥 − 14𝑦 + 9 and classify each as a local
minimum, maximum or saddle point using the Hessian.
Q1.12. Evaluate ∬+ 3 𝑥𝑦 |𝑦 )' | 𝑑𝑥 𝑑𝑦 where 𝑅 is the region bounded by 𝑥𝑦 = 1, 𝑥𝑦 = 2, 𝑦 = 4 and 𝑦 = 9,
using the substitution 𝑥 = 𝑣⁄6𝑢' , 𝑦 = 𝑢' .
(a) Compute the Jacobian matrix and its determinant.
(b) Write down the new boundaries of integration.
(c) Sketch the new region of integration in the 𝑢𝑣-plane (describe it in words).
(d) Write the full integral in the new coordinates.
(e) Evaluate the transformed integral.
! /!
Q1.13. Consider the Bernoulli equation 𝑦′ + 𝑥 , 𝑦 = 𝑒 % 𝑦 ' where 𝛼 ∈ ℝ is a parameter.
(a) State the order of this differential equation.
(b) Perform the Bernoulli substitution 𝑧 = 𝑦 )# to obtain a linear ODE in 𝑧.
(c) Find the homogeneous solution 𝑧. (𝛼).
! /!
(d) Determine 𝛼 given that a particular solution is 𝑧/ = −𝑥 𝑒 % .
5
,Mock Exam 2 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 2 of 10 in the practice series. Good luck!
6
,Tapas (7/20)
Q2.1. Consider the homomorphism ℎ: ℝ! → ℝ' given by ℎ(𝑥, 𝑦, 𝑧)" = (𝑥 + 2𝑦 − 𝑧, 3𝑥 + 6𝑦 − 3𝑧)" .
(a) Verify that ℎ is a homomorphism.
(b) Find the null space and the nullity of ℎ.
(c) Construct a basis of the null space.
(d) Verify the rank–nullity theorem.
(e) State the shape of the matrix representation 𝐻 of ℎ.
2 1 1
Q2.2. Consider the matrix 𝐻 = T−1 3 0U which represents a map ℎ: ℝ! → ℝ! in the standard basis.
4 5 2
Decide whether ℎ is an isomorphism, using one property of 𝐻.
Q2.3. Prove the following lemma: the kernel of a linear map ℎ: 𝑉 → 𝑊 is a subspace of 𝑉.
Q2.4. Consider a contour plot of 𝑧 = 𝑓(𝑥, 𝑦) with circular contours centred at (−2,0), 𝑓 increasing
outward, and a contour plot of 𝑧 = 𝑔(𝑥, 𝑦) whose contours form a saddle pattern with the saddle at the
origin: 𝑔(𝑥, 𝑦) = 𝑥 ' − 𝑦 ' . A green point is marked at (0,0). Determine the sign (< 0, > 0, = 0) of:
(a) 𝑓%
(b) 𝑓&
(c) 𝑔%
(d) 𝑔&
Q2.5. (a) Come up with a first-order, non-linear, ordinary Bernoulli differential equation. (b) How many
boundary conditions guarantee a unique solution?
Q2.6. Consider the vectors 𝑣⃗ = (√3, 3)" and 𝑤
==⃗ = (0, 𝑥)" . For which values of 𝑥 is the angle between 𝑣⃗
==⃗ equal to 60∘ ? (Hint: cos60∘ = 1/2.)
and 𝑤
Q2.7. Consider a subset 𝑆 = {𝑠⃗# , … , 𝑠⃗1 } of an 𝑁-dimensional vector space 𝑉, with 𝑚 < 𝑁. Encircle the
correct statements:
(A) 𝑆 is potentially a basis of 𝑉.
(B) The elements of 𝑆 are potentially linearly independent.
(C) The elements of 𝑆 are potentially linearly dependent.
(D) The span of 𝑆 is a subspace of 𝑉.
(E) If we delete some elements of 𝑆, we can always construct a linearly independent set.
(F) If we add new extra elements to 𝑆, we can always construct a basis of 𝑉.
7
,El Plato Principal (13/20)
Q2.8. (a) Describe the hyperplane (a 3-dimensional subspace of ℝ( ) that contains the two points
(1,2,3,4)" and (2,4,6,9)" and the line {(0,1,1,2)" + 𝑡(1,1,2,3)" : 𝑡 ∈ ℝ}. (b) What is the intersection
between this hyperplane and the 𝑥𝑦-plane in ℝ( (i.e. {(𝑥, 𝑦, 0,0)})?
Q2.9. Consider the set of vectors {(1,2,5)" , (4,1, −1)" , (−2,3,11)" } in ℝ! .
(a) Which subspace 𝑆 of ℝ! does this set span?
(b) Construct a basis for 𝑆.
(c) Give an implicit equation describing 𝑆.
Q2.10. Consider the bases 𝐵 = ⟨(1,0,0)" , (1,1,0)" , (1,1,1)" ⟩ and 𝐷 = ⟨(1,1,1)" , (0,1,1)" , (0,0,1)" ⟩ of ℝ! ,
and the linear map 𝑡: ℝ! → ℝ! given by 𝑡(𝑥, 𝑦, 𝑧)" = (2𝑥 − 𝑦, 𝑦 + 𝑧, 𝑥 + 𝑧)" .
(a) Verify that 𝐵 and 𝐷 are bases of ℝ! .
(b) Compute the change-of-basis matrices 𝑃2→4 and 𝑃4→2 .
(c) Write down the matrix 𝑇 of 𝑡 in the standard basis.
(d) Compute the matrix 𝑇R = Rep2,2 (𝑡) of 𝑡 in the basis 𝐵.
Q2.11. Your company is launching a product. Each social-media post costs 20 euros and each TV spot
costs 100 euros. The total marketing budget is 2000 euros. The expected revenue is 𝑅 = 4𝑐 ' 𝑡, where 𝑐 is
the number of social-media posts and 𝑡 is the number of TV spots. Using Lagrange multipliers, maximise
𝑅 subject to 20𝑐 + 100𝑡 = 2000. State (𝑐 ∗ , 𝑡 ∗ , 𝜆∗ ) and the maximal revenue. (You do not have to verify
second-order conditions.)
Q2.12. Solve the integral ∬4 sin (𝑥 ' + 𝑦 ' ) 𝑑𝐴, where 𝐷 is the region in the first quadrant bounded by 𝑥 ' +
𝑦 ' = 𝜋 and 𝑥 ' + 𝑦 ' = 2𝜋, using polar coordinates.
(a) State the Jacobian of the polar substitution.
(b) Write the new bounds of integration for 𝜃 and 𝑟.
(c) Rewrite the full integral in polar coordinates.
(d) Evaluate the integral.
Q2.13. You take a loan of 𝑃$ euros that you pay back at a constant monthly rate 𝑀, with the bank
78
charging continuous interest at rate 𝑟. The outstanding amount 𝑃(𝑡) satisfies 79 = −𝑀 + 𝑟 𝑃.
(a) Classify this ODE (order, linear/non-linear, homogeneous/heterogeneous, ordinary/partial).
(b) Solve the ODE in terms of 𝑀, 𝑟, and an integration constant.
(c) Determine the integration constant from 𝑃(𝑇) = 0 (the loan is paid off at time 𝑇).
(d) From 𝑃(0) = 𝑃$ , derive a relation between 𝑀, 𝑟, 𝑇 and 𝑃$ .
8
, Mock Exam 3 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 3 of 10 in the practice series. Good luck!
9
MATHEMATICS II
MOCK EXAMS
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Mock Exam 1 — Mathematics for Business and Economics II ..................................................................... 3
Tapas (7/20) ......................................................................................................................................... 4
El Plato Principal (13/20) ....................................................................................................................... 5
Mock Exam 2 — Mathematics for Business and Economics II ..................................................................... 6
Tapas (7/20) ......................................................................................................................................... 7
El Plato Principal (13/20) ....................................................................................................................... 8
Mock Exam 3 — Mathematics for Business and Economics II ..................................................................... 9
Tapas (7/20) ....................................................................................................................................... 10
El Plato Principal (13/20) ..................................................................................................................... 11
Mock Exam 4 — Mathematics for Business and Economics II ................................................................... 12
Tapas (7/20) ....................................................................................................................................... 13
El Plato Principal (13/20) ..................................................................................................................... 14
Mock Exam 5 — Mathematics for Business and Economics II ................................................................... 16
Tapas (7/20) ....................................................................................................................................... 17
El Plato Principal (13/20) ..................................................................................................................... 18
Mock Exam 6 — Mathematics for Business and Economics II ................................................................... 19
Tapas (3/20) ....................................................................................................................................... 20
El Plato Principal (14/20) ..................................................................................................................... 21
Postres (3/20) ..................................................................................................................................... 22
Mock Exam 7 — Mathematics for Business and Economics II ................................................................... 23
Tapas (3/20) ....................................................................................................................................... 24
El Plato Principal (14/20) ..................................................................................................................... 25
Postres (3/20) ..................................................................................................................................... 26
Mock Exam 8 — Mathematics for Business and Economics II ................................................................... 27
Tapas (3/20) ....................................................................................................................................... 28
El Plato Principal (14/20) ..................................................................................................................... 29
Postres (3/20) ..................................................................................................................................... 30
Mock Exam 9 — Mathematics for Business and Economics II ................................................................... 31
Tapas (3/20) ....................................................................................................................................... 32
1
, El Plato Principal (14/20) ..................................................................................................................... 33
Postres (3/20) ..................................................................................................................................... 34
Mock Exam 10 — Mathematics for Business and Economics II ................................................................. 35
Tapas (3/20) ....................................................................................................................................... 36
El Plato Principal (14/20) ..................................................................................................................... 37
Postres (3/20) ..................................................................................................................................... 38
2
,Mock Exam 1 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 1 of 10 in the practice series. Good luck!
3
,Tapas (7/20)
Q1.1. We first ask you to define your own homomorphism 𝑡, which maps vectors from ℝ! to ℝ! . Make
sure that the dimension of the null space (nullity) of your homomorphism 𝑡 equals 1.
(a) Define 𝑡 explicitly by giving an expression for 𝑡(𝑥, 𝑦, 𝑧)" .
(b) Compute the range space and its dimension (rank).
(c) Verify that the rank–nullity theorem holds for your map.
(d) Construct a basis of the null space and a basis of the range space.
(e) If you represent 𝑡 by a matrix 𝑇, how many rows and columns does 𝑇 have?
(f) What can you say about det𝑇 in the standard basis of the domain and co-domain?
Q1.2. (a) Write down a non-singular 3 × 3 matrix 𝐴 with exactly one real non-zero eigenvalue 𝜆# = 𝑎
(where 𝑎 is your day of birth). (b) Write down a non-singular 3 × 3 matrix 𝐶 with three distinct, non-zero,
real eigenvalues, all of absolute value strictly smaller than 1. (c) Compute the eigenvectors of your
matrix 𝐶. (d) What can you say about 𝐶 #$$ as the exponent grows?
Q1.3. Prove the following lemma. In any vector space 𝑉, for every 𝑣⃗ ∈ 𝑉 and every scalar 𝑟 ∈ ℝ we have
=⃗, 𝑟 ⋅ =0⃗ = =0⃗, and (−1) ⋅ 𝑣⃗ is the additive inverse of 𝑣⃗.
0 ⋅ 𝑣⃗ = 0
Q1.4. Consider a contour plot of 𝑧 = 𝑓(𝑥, 𝑦) whose contours form concentric ellipses centred at (1,2),
with 𝑓 increasing outward, and a contour plot of 𝑧 = 𝑔(𝑥, 𝑦) whose contours are parallel lines of slope
+1 with 𝑔 increasing as one moves in the direction (1,1). A blue point is marked at the origin (0,0) on
both plots. Determine for the marked point the sign (< 0, > 0, = 0) of:
(a) 𝑓%
(b) 𝑓&
(c) 𝑔%
(d) 𝑔&
Q1.5. (a) Come up with a first-order, separable, ordinary differential equation. (b) How many boundary
conditions do you need to find one unique solution to your differential equation?
Q1.6. Consider the linear map 𝑓: ℝ! → ℝ' for which 𝑓(1,0,0)" = (2, −1)" , 𝑓(0,1,0)" = (0,3)" ,
𝑓(0,0,1)" = (−1,1)" . Determine 𝑓(𝑎, 𝑏, 𝑐)" .
Q1.7. Consider the homomorphism ℎ: ℝ( → ℝ' given by ℎ(𝑥, 𝑦, 𝑧, 𝑤)" = (𝑥 − 𝑦, 𝑧 + 𝑤)" .
(a) Find the null space of ℎ and its nullity.
(b) Build a basis for the null space.
(c) Verify the rank–nullity theorem.
(d) If ℎ is represented by a matrix 𝐻, how many rows and columns does 𝐻 have?
4
,El Plato Principal (13/20)
Q1.8. (a) Describe the plane in ℝ! that contains your 'birthday point' (2,3, 𝑎)" and the line {(2,4,5)" +
𝑣(2,5, 𝑎 + 5)" : 𝑣 ∈ ℝ}. (b) What is the geometric shape of the intersection between this plane and the
𝑥𝑦-plane?
Q1.9. Consider the two ordered sets 𝐵 = ⟨(𝑎, 𝑏, 𝑐)" , (1,0,2)" , (0,0,1)" ⟩ and 𝐷 =
⟨(𝑎, 0,0)" , (0, 𝑏, 0)" , (0,0, 𝑐)" ⟩ in ℝ! (with your personal 𝑎, 𝑏, 𝑐).
(a) Show that 𝐵 and 𝐷 span the same vector space.
(b) Find the change-of-basis matrix from 𝐵 to 𝐷.
(c) Consider the map 𝑡: ℝ! → ℝ! given by 𝑡(𝑥, 𝑦, 𝑧)" = (𝑥 + 𝑦, 𝑏(𝑥 − 𝑦), 𝑐𝑧)" . Write down the matrix 𝑇 that
represents 𝑡 in the standard basis.
(d) Find the matrix 𝑇R that represents 𝑡 with respect to 𝐵 in both domain and co-domain.
4 0 0
Q1.10. Consider the map 𝑔: ℝ! → ℝ! represented in the standard basis by 𝐺 = T1 3 −1U.
2 0 2
(a) Find the eigenvalues of 𝐺.
(b) For each eigenvalue, give a basis of the corresponding eigenspace.
(c) Decide whether 𝐺 is diagonalisable. If so, give the matrices 𝑃 and 𝐷 such that 𝐺 = 𝑃𝐷𝑃)# .
(d) Describe the behaviour of 𝐺 * as 𝑛 → ∞ on a vector along an eigenvector of the dominant eigenvalue.
Q1.11. Find the critical point(s) of 𝑓(𝑥, 𝑦) = 𝑥 ' + 4𝑥𝑦 + 5𝑦 ' − 6𝑥 − 14𝑦 + 9 and classify each as a local
minimum, maximum or saddle point using the Hessian.
Q1.12. Evaluate ∬+ 3 𝑥𝑦 |𝑦 )' | 𝑑𝑥 𝑑𝑦 where 𝑅 is the region bounded by 𝑥𝑦 = 1, 𝑥𝑦 = 2, 𝑦 = 4 and 𝑦 = 9,
using the substitution 𝑥 = 𝑣⁄6𝑢' , 𝑦 = 𝑢' .
(a) Compute the Jacobian matrix and its determinant.
(b) Write down the new boundaries of integration.
(c) Sketch the new region of integration in the 𝑢𝑣-plane (describe it in words).
(d) Write the full integral in the new coordinates.
(e) Evaluate the transformed integral.
! /!
Q1.13. Consider the Bernoulli equation 𝑦′ + 𝑥 , 𝑦 = 𝑒 % 𝑦 ' where 𝛼 ∈ ℝ is a parameter.
(a) State the order of this differential equation.
(b) Perform the Bernoulli substitution 𝑧 = 𝑦 )# to obtain a linear ODE in 𝑧.
(c) Find the homogeneous solution 𝑧. (𝛼).
! /!
(d) Determine 𝛼 given that a particular solution is 𝑧/ = −𝑥 𝑒 % .
5
,Mock Exam 2 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 2 of 10 in the practice series. Good luck!
6
,Tapas (7/20)
Q2.1. Consider the homomorphism ℎ: ℝ! → ℝ' given by ℎ(𝑥, 𝑦, 𝑧)" = (𝑥 + 2𝑦 − 𝑧, 3𝑥 + 6𝑦 − 3𝑧)" .
(a) Verify that ℎ is a homomorphism.
(b) Find the null space and the nullity of ℎ.
(c) Construct a basis of the null space.
(d) Verify the rank–nullity theorem.
(e) State the shape of the matrix representation 𝐻 of ℎ.
2 1 1
Q2.2. Consider the matrix 𝐻 = T−1 3 0U which represents a map ℎ: ℝ! → ℝ! in the standard basis.
4 5 2
Decide whether ℎ is an isomorphism, using one property of 𝐻.
Q2.3. Prove the following lemma: the kernel of a linear map ℎ: 𝑉 → 𝑊 is a subspace of 𝑉.
Q2.4. Consider a contour plot of 𝑧 = 𝑓(𝑥, 𝑦) with circular contours centred at (−2,0), 𝑓 increasing
outward, and a contour plot of 𝑧 = 𝑔(𝑥, 𝑦) whose contours form a saddle pattern with the saddle at the
origin: 𝑔(𝑥, 𝑦) = 𝑥 ' − 𝑦 ' . A green point is marked at (0,0). Determine the sign (< 0, > 0, = 0) of:
(a) 𝑓%
(b) 𝑓&
(c) 𝑔%
(d) 𝑔&
Q2.5. (a) Come up with a first-order, non-linear, ordinary Bernoulli differential equation. (b) How many
boundary conditions guarantee a unique solution?
Q2.6. Consider the vectors 𝑣⃗ = (√3, 3)" and 𝑤
==⃗ = (0, 𝑥)" . For which values of 𝑥 is the angle between 𝑣⃗
==⃗ equal to 60∘ ? (Hint: cos60∘ = 1/2.)
and 𝑤
Q2.7. Consider a subset 𝑆 = {𝑠⃗# , … , 𝑠⃗1 } of an 𝑁-dimensional vector space 𝑉, with 𝑚 < 𝑁. Encircle the
correct statements:
(A) 𝑆 is potentially a basis of 𝑉.
(B) The elements of 𝑆 are potentially linearly independent.
(C) The elements of 𝑆 are potentially linearly dependent.
(D) The span of 𝑆 is a subspace of 𝑉.
(E) If we delete some elements of 𝑆, we can always construct a linearly independent set.
(F) If we add new extra elements to 𝑆, we can always construct a basis of 𝑉.
7
,El Plato Principal (13/20)
Q2.8. (a) Describe the hyperplane (a 3-dimensional subspace of ℝ( ) that contains the two points
(1,2,3,4)" and (2,4,6,9)" and the line {(0,1,1,2)" + 𝑡(1,1,2,3)" : 𝑡 ∈ ℝ}. (b) What is the intersection
between this hyperplane and the 𝑥𝑦-plane in ℝ( (i.e. {(𝑥, 𝑦, 0,0)})?
Q2.9. Consider the set of vectors {(1,2,5)" , (4,1, −1)" , (−2,3,11)" } in ℝ! .
(a) Which subspace 𝑆 of ℝ! does this set span?
(b) Construct a basis for 𝑆.
(c) Give an implicit equation describing 𝑆.
Q2.10. Consider the bases 𝐵 = ⟨(1,0,0)" , (1,1,0)" , (1,1,1)" ⟩ and 𝐷 = ⟨(1,1,1)" , (0,1,1)" , (0,0,1)" ⟩ of ℝ! ,
and the linear map 𝑡: ℝ! → ℝ! given by 𝑡(𝑥, 𝑦, 𝑧)" = (2𝑥 − 𝑦, 𝑦 + 𝑧, 𝑥 + 𝑧)" .
(a) Verify that 𝐵 and 𝐷 are bases of ℝ! .
(b) Compute the change-of-basis matrices 𝑃2→4 and 𝑃4→2 .
(c) Write down the matrix 𝑇 of 𝑡 in the standard basis.
(d) Compute the matrix 𝑇R = Rep2,2 (𝑡) of 𝑡 in the basis 𝐵.
Q2.11. Your company is launching a product. Each social-media post costs 20 euros and each TV spot
costs 100 euros. The total marketing budget is 2000 euros. The expected revenue is 𝑅 = 4𝑐 ' 𝑡, where 𝑐 is
the number of social-media posts and 𝑡 is the number of TV spots. Using Lagrange multipliers, maximise
𝑅 subject to 20𝑐 + 100𝑡 = 2000. State (𝑐 ∗ , 𝑡 ∗ , 𝜆∗ ) and the maximal revenue. (You do not have to verify
second-order conditions.)
Q2.12. Solve the integral ∬4 sin (𝑥 ' + 𝑦 ' ) 𝑑𝐴, where 𝐷 is the region in the first quadrant bounded by 𝑥 ' +
𝑦 ' = 𝜋 and 𝑥 ' + 𝑦 ' = 2𝜋, using polar coordinates.
(a) State the Jacobian of the polar substitution.
(b) Write the new bounds of integration for 𝜃 and 𝑟.
(c) Rewrite the full integral in polar coordinates.
(d) Evaluate the integral.
Q2.13. You take a loan of 𝑃$ euros that you pay back at a constant monthly rate 𝑀, with the bank
78
charging continuous interest at rate 𝑟. The outstanding amount 𝑃(𝑡) satisfies 79 = −𝑀 + 𝑟 𝑃.
(a) Classify this ODE (order, linear/non-linear, homogeneous/heterogeneous, ordinary/partial).
(b) Solve the ODE in terms of 𝑀, 𝑟, and an integration constant.
(c) Determine the integration constant from 𝑃(𝑇) = 0 (the loan is paid off at time 𝑇).
(d) From 𝑃(0) = 𝑃$ , derive a relation between 𝑀, 𝑟, 𝑇 and 𝑃$ .
8
, Mock Exam 3 — Mathematics for Business and Economics II
Please read the exam instructions carefully — they are indispensable for your exam.
This is a closed-book exam. You may use a simple (non-graphing) calculator. You are not allowed to use
books, notes, your friends, the syllabus, or the internet. You are evaluated on your mathematical method,
not only your result — show every step.
Some questions are personalised with the following parameters: 𝑎 = your day of birth, 𝑏 = your month of
birth, 𝑐 = the last non-zero digit of your Student-ID. Worked example: a student born on 12 July with ID
0461230 has 𝑎 = 12, 𝑏 = 7, 𝑐 = 3.
This is Mock Exam 3 of 10 in the practice series. Good luck!
9
