2026
MATHEMATICS II
MOCK EXAM SOLUTIONS
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Mock Exam 1 — Solutions ........................................................................................................................ 6
Q1.1 ................................................................................................................................................. 6
Q1.2 ................................................................................................................................................. 6
Q1.3 ................................................................................................................................................. 7
Q1.4 ................................................................................................................................................. 7
Q1.5 ................................................................................................................................................. 7
Q1.6 ................................................................................................................................................. 8
Q1.7 ................................................................................................................................................. 8
Q1.8 ................................................................................................................................................. 8
Q1.9 ................................................................................................................................................. 9
Q1.10 ............................................................................................................................................... 9
Q1.11 ............................................................................................................................................. 10
Q1.12 ............................................................................................................................................. 11
Q1.13 ............................................................................................................................................. 11
Mock Exam 2 — Solutions ...................................................................................................................... 12
Q2.1 ............................................................................................................................................... 12
Q2.2 ............................................................................................................................................... 12
Q2.3 ............................................................................................................................................... 12
Q2.4 ............................................................................................................................................... 13
Q2.5 ............................................................................................................................................... 13
Q2.6 ............................................................................................................................................... 13
Q2.7 ............................................................................................................................................... 13
Q2.8 ............................................................................................................................................... 14
Q2.9 ............................................................................................................................................... 14
Q2.10 ............................................................................................................................................. 15
Q2.11 ............................................................................................................................................. 15
Q2.12 ............................................................................................................................................. 16
Q2.13 ............................................................................................................................................. 16
Mock Exam 3 — Solutions ...................................................................................................................... 18
1
, Q3.1 ............................................................................................................................................... 18
Q3.2 ............................................................................................................................................... 18
Q3.3 ............................................................................................................................................... 18
Q3.4 ............................................................................................................................................... 19
Q3.5 ............................................................................................................................................... 19
Q3.6 ............................................................................................................................................... 19
Q3.7 ............................................................................................................................................... 19
Q3.8 ............................................................................................................................................... 20
Q3.9 ............................................................................................................................................... 20
Q3.10 ............................................................................................................................................. 21
Q3.11 ............................................................................................................................................. 22
Q3.12 ............................................................................................................................................. 23
Q3.13 ............................................................................................................................................. 23
Mock Exam 4 — Solutions ...................................................................................................................... 24
Q4.1 ............................................................................................................................................... 24
Q4.2 ............................................................................................................................................... 24
Q4.3 ............................................................................................................................................... 24
Q4.4 ............................................................................................................................................... 25
Q4.5 ............................................................................................................................................... 25
Q4.6 ............................................................................................................................................... 25
Q4.7 ............................................................................................................................................... 25
Q4.8 ............................................................................................................................................... 26
Q4.9 ............................................................................................................................................... 26
Q4.10 ............................................................................................................................................. 27
Q4.11 ............................................................................................................................................. 27
Q4.12 ............................................................................................................................................. 27
Q4.13 ............................................................................................................................................. 28
Mock Exam 5 — Solutions ...................................................................................................................... 29
Q5.1 ............................................................................................................................................... 29
Q5.2 ............................................................................................................................................... 29
Q5.3 ............................................................................................................................................... 29
Q5.4 ............................................................................................................................................... 30
2
, Q5.5 ............................................................................................................................................... 30
Q5.6 ............................................................................................................................................... 30
Q5.7 ............................................................................................................................................... 30
Q5.8 ............................................................................................................................................... 31
Q5.9 ............................................................................................................................................... 31
Q5.10 ............................................................................................................................................. 31
Q5.11 ............................................................................................................................................. 32
Q5.12 ............................................................................................................................................. 32
Q5.13 ............................................................................................................................................. 33
Mock Exam 6 — Solutions ...................................................................................................................... 34
Q6.1 ............................................................................................................................................... 34
Q6.2 ............................................................................................................................................... 34
Q6.3 ............................................................................................................................................... 34
Q6.4 ............................................................................................................................................... 35
Q6.5 ............................................................................................................................................... 35
Q6.6 ............................................................................................................................................... 35
Q6.7 ............................................................................................................................................... 36
Q6.8 ............................................................................................................................................... 36
Q6.9 ............................................................................................................................................... 37
Q6.10 ............................................................................................................................................. 37
Q6.11 ............................................................................................................................................. 37
Mock Exam 7 — Solutions ...................................................................................................................... 38
Q7.1 ............................................................................................................................................... 38
Q7.2 ............................................................................................................................................... 38
Q7.3 ............................................................................................................................................... 38
Q7.4 ............................................................................................................................................... 39
Q7.5 ............................................................................................................................................... 39
Q7.6 ............................................................................................................................................... 40
Q7.7 ............................................................................................................................................... 40
Q7.8 ............................................................................................................................................... 40
Q7.9 ............................................................................................................................................... 41
Q7.10 ............................................................................................................................................. 41
3
, Q7.11 ............................................................................................................................................. 41
Mock Exam 8 — Solutions ...................................................................................................................... 42
Q8.1 ............................................................................................................................................... 42
Q8.2 ............................................................................................................................................... 42
Q8.3 ............................................................................................................................................... 42
Q8.4 ............................................................................................................................................... 43
Q8.5 ............................................................................................................................................... 43
Q8.6 ............................................................................................................................................... 43
Q8.7 ............................................................................................................................................... 44
Q8.8 ............................................................................................................................................... 44
Q8.9 ............................................................................................................................................... 45
Q8.10 ............................................................................................................................................. 45
Q8.11 ............................................................................................................................................. 45
Mock Exam 9 — Solutions ...................................................................................................................... 46
Q9.1 ............................................................................................................................................... 46
Q9.2 ............................................................................................................................................... 46
Q9.3 ............................................................................................................................................... 46
Q9.4 ............................................................................................................................................... 47
Q9.5 ............................................................................................................................................... 47
Q9.6 ............................................................................................................................................... 48
Q9.7 ............................................................................................................................................... 48
Q9.8 ............................................................................................................................................... 48
Q9.9 ............................................................................................................................................... 49
Q9.10 ............................................................................................................................................. 49
Q9.11 ............................................................................................................................................. 49
Mock Exam 10 — Solutions..................................................................................................................... 50
Q10.1 ............................................................................................................................................. 50
Q10.2 ............................................................................................................................................. 50
Q10.3 ............................................................................................................................................. 50
Q10.4 ............................................................................................................................................. 51
Q10.5 ............................................................................................................................................. 51
Q10.6 ............................................................................................................................................. 52
4
,Q10.7 ............................................................................................................................................. 52
Q10.8 ............................................................................................................................................. 52
Q10.9 ............................................................................................................................................. 52
Q10.10 ........................................................................................................................................... 53
Q10.11 ........................................................................................................................................... 53
5
,Mock Exam 1 — Solutions
Q1.1
Define a homomorphism 𝑡: ℝ! → ℝ! with nullity equal to 1; compute range, basis, etc.
Solution.
A valid answer is any linear map whose matrix in the standard basis has rank 2. The simplest choice is the
projection
𝑡(𝑥, 𝑦, 𝑧)" = (𝑥, 𝑦, 0)" .
1 0 0
Step 1. Matrix 𝑇 = 20 1 03. Rows two of three are non-zero ⇒ rank = 2, nullity = 1.
0 0 0
Step 2. (b) Range: im𝑡 = {(𝑥, 𝑦, 0)" : 𝑥, 𝑦 ∈ ℝ}, the 𝑥𝑦-plane. dimim𝑡 = 2.
Step 3. (c) Rank–nullity: dimker𝑡 + dimim𝑡 = 1 + 2 = 3 = dimℝ! . Verified.
Step 4. (d) Basis of ker𝑡 = {(0,0, 𝑧)" } is {(0,0,1)" }. Basis of im𝑡: {(1,0,0)" , (0,1,0)" }.
Step 5. (e) Domain and codomain both ℝ! , so 𝑇 is 3 × 3.
Step 6. (f) Because 𝑇 has a zero row, det𝑇 = 0. (Equivalent: nullity ≥ 1 ⇒ 𝑇 singular.)
Answer
Any rank-2 linear map ℝ! → ℝ! works; for the projection 𝑡(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 0) one has ker𝑡 = ⟨(0,0,1)⟩, im𝑡
= 𝑥𝑦-plane, det𝑇 = 0.
Q1.2
Construct 3 × 3 matrices with prescribed eigenvalues; describe 𝐶 #$$ .
Solution.
Worked example with 𝑎 = 12.
Step 1. (a) Use any diagonal matrix with 12 on the first diagonal entry and other diagonal entries that are
12 0 0
repeated or complex via blocks. Simplest: 𝐴 = 2 0 cos𝜃 −sin𝜃3 with 𝜃 = 𝜋/3. Then 𝐴 has eigenvalues
0 sin𝜃 cos𝜃
±&'/!
12, 𝑒 — exactly one real eigenvalue, 𝜆# = 12 = 𝑎. det𝐴 = 12 ⋅ 1 = 12 ≠ 0.
Step 2. (b) Pick three distinct reals in (−1,1)\{0}, e.g. 1/2, −1/3, 1/4. Take 𝐶 = diag(1/2, −1/3,1/4). Then 𝐶
is non-singular and has the prescribed spectrum.
Step 3. (c) Eigenvectors of diag(𝜆# , 𝜆) , 𝜆! ) are the standard basis: 𝑒# , 𝑒) , 𝑒! .
#$$
Step 4. (d) For 𝐶 diagonal, 𝐶 #$$ = diag(𝜆#$$ #$$ #$$
# , 𝜆) , 𝜆! ). Since each |𝜆& | < 1, 𝜆& → 0. Thus 𝐶 #$$ → 𝟎, i.e.
𝐶 * 𝑣⃗ → P0⃗ for every 𝑣⃗.
6
,Answer
𝐴 = diag(𝑎, 1, 1) also works (real eigenvalue 𝑎 with multiplicity 1 — wait: that gives three real
eigenvalues; instead use the rotation block above). 𝐶 = diag(1/2, −1/3,1/4); eigenvectors 𝑒# , 𝑒) , 𝑒! ;
𝐶 #$$ → 0.
Q1.3
Prove 0𝑣⃗ = P0⃗, 𝑟0
P⃗ = P0⃗, (−1)𝑣⃗ is the additive inverse of 𝑣⃗.
Solution.
Step 1. Start from 0𝑣⃗ = (0 + 0)𝑣⃗ = 0𝑣⃗ + 0𝑣⃗ using the distributive axiom (𝑠 + 𝑡)𝑣⃗ = 𝑠𝑣⃗ + 𝑡𝑣⃗ with 𝑠 = 𝑡 = 0.
Step 2. Add the additive inverse −(0𝑣⃗) to both sides: 0𝑣⃗ + (−(0𝑣⃗)) = 0𝑣⃗ + 0𝑣⃗ + (−(0𝑣⃗)). The LHS is P0⃗; the
RHS simplifies (associativity) to 0𝑣⃗ + P0⃗ = 0𝑣⃗. Hence 0𝑣⃗ = 0
P⃗.
P⃗ = 𝑟(0
Step 3. Similarly, 𝑟0 P⃗ + P0⃗) = 𝑟0
P⃗ + 𝑟0
P⃗ by the other distributive axiom. Cancelling one 𝑟0
P⃗ as in step 2
P⃗ = 0
yields 𝑟0 P⃗.
P⃗ (using the identity axiom 1𝑣⃗ =
Step 4. For the third claim: 𝑣⃗ + (−1)𝑣⃗ = 1𝑣⃗ + (−1)𝑣⃗ = (1 + (−1))𝑣⃗ = 0𝑣⃗ = 0
𝑣⃗ and step 1). Hence (−1)𝑣⃗ is the additive inverse of 𝑣⃗.
Q1.4
Sign of partial derivatives at marked points.
Solution.
Step 1. For 𝑓: contours are concentric ellipses centred at (1,2) with 𝑓 increasing outward. The gradient 𝛻𝑓
points outward (in the direction away from (1,2)). At the origin (0,0), the outward radial vector points from
(1,2) to (0,0), i.e. in direction (−1, −2). So 𝛻𝑓(0,0) ∝ (−1, −2), giving 𝑓+ (0,0) < 0 and 𝑓, (0,0) < 0.
Step 2. For 𝑔: contours are parallel lines of slope +1 with 𝑔 increasing in direction (1,1). Then 𝛻𝑔 is parallel
to (1,1) everywhere, so 𝑔+ > 0 and 𝑔, > 0 at any point, including (0,0).
Answer
(a) 𝑓+ < 0. (b) 𝑓, < 0. (c) 𝑔+ > 0. (d) 𝑔, > 0.
Q1.5
Construct a first-order separable ODE; boundary conditions.
Solution.
-, -, ! /)
Step 1. A canonical example is -+ = 𝑥 𝑦. It is separable: ,
= 𝑥 𝑑𝑥 ⇒ ln|𝑦| = 𝑥 ) ⁄2 + 𝐶 ⇒ 𝑦 = 𝐾𝑒 + , where
𝐾 = ±𝑒 . ∈ ℝ.
Step 2. (b) The general solution has one free constant (𝐾), so one boundary condition (e.g. 𝑦(0) = 𝐾) singles
out one unique solution, provided the RHS is continuous at the initial point. By Picard–Lindelöf, a single
initial condition suffices.
7
,Answer
Example: 𝑦′ = 𝑥𝑦. One initial condition suffices for uniqueness.
Q1.6
Linear map by basis values; compute 𝑓(𝑎, 𝑏, 𝑐).
Solution.
Step 1. 𝑓 is given on the standard basis, so by linearity 𝑓(𝑎, 𝑏, 𝑐)" = 𝑎 𝑓(𝑒# ) + 𝑏 𝑓(𝑒) ) + 𝑐 𝑓(𝑒! ) = 𝑎(2, −1) +
𝑏(0,3) + 𝑐(−1,1).
Step 2. = (2𝑎 − 𝑐, −𝑎 + 3𝑏 + 𝑐). With 𝑎 = 12, 𝑏 = 7, 𝑐 = 3: (2 ⋅ 12 − 3, −12 + 21 + 3) = (21, 12).
Answer
𝑓(𝑎, 𝑏, 𝑐)" = (2𝑎 − 𝑐, −𝑎 + 3𝑏 + 𝑐)" . With (𝑎, 𝑏, 𝑐) = (12,7,3): (21,12)" .
Q1.7
Null space and rank-nullity for ℎ(𝑥, 𝑦, 𝑧, 𝑤) = (𝑥 − 𝑦, 𝑧 + 𝑤).
Solution.
Step 1. kerℎ: solve 𝑥 − 𝑦 = 0 and 𝑧 + 𝑤 = 0. Free variables: 𝑦 = 𝑥 and 𝑤 = −𝑧. So kerℎ =
{(𝑥, 𝑥, 𝑧, −𝑧): 𝑥, 𝑧 ∈ ℝ}, dimension 2.
Step 2. Basis of kerℎ: {(1,1,0,0)" , (0,0,1, −1)" }.
Step 3. Rank: image is ℝ) (just pick (𝑥, 𝑦, 𝑧, 𝑤) = (1,0,0,0) for (1,0) and (0,0,1,0) for (0,1)). Rank = 2.
Step 4. Rank–nullity: 2 + 2 = 4 = dimℝ/ . Verified.
1 −1 0 0
Step 5. (d) 𝐻 has 2 rows (codomain dimension) and 4 columns (domain dimension): 𝐻 = d e.
0 0 1 1
Answer
kerℎ is the 2-dim subspace spanned by (1,1,0,0) and (0,0,1, −1); rank = 2; 𝐻 is 2 × 4.
Q1.8
Plane through (2,3, 𝑎) and given line.
Solution.
Step 1. A point on the line: 𝑃# = (2,4,5). Direction of line: 𝑑⃗ = (2,5, 𝑎 + 5). The 'birthday point' 𝑃$ = (2,3, 𝑎).
Step 2. Second spanning vector of the plane: 𝑢
P⃗ = 𝑃$ − 𝑃# = (0, −1, 𝑎 − 5).
Step 3. Normal 𝑛P⃗ = 𝑑⃗ × 𝑢
P⃗ = (2,5, 𝑎 + 5) × (0, −1, 𝑎 − 5). Expanding: 𝑛P⃗ = (5(𝑎 − 5) − (𝑎 + 5)(−1), (𝑎 + 5) ⋅
0 − 2(𝑎 − 5), 2 ⋅ (−1) − 5 ⋅ 0) = (5𝑎 − 25 + 𝑎 + 5, −2𝑎 + 10, −2) = (6𝑎 − 20, −2𝑎 + 10, −2).
Step 4. Implicit equation: 𝑛P⃗ ⋅ ((𝑥, 𝑦, 𝑧) − 𝑃# ) = 0, i.e. (6𝑎 − 20)(𝑥 − 2) + (−2𝑎 + 10)(𝑦 − 4) + (−2)(𝑧 −
5) = 0.
8
, Step 5. With 𝑎 = 12: 𝑛P⃗ = (52, −14, −2). Equation: 52(𝑥 − 2) − 14(𝑦 − 4) − 2(𝑧 − 5) = 0, i.e. 26𝑥 − 7𝑦 −
𝑧 = 21 after dividing by 2 and simplifying. Check: 26 ⋅ 2 − 7 ⋅ 3 − 12 = 52 − 21 − 12 = 19. Recompute. Plug
(2,3,12): 52(0) − 14(−1) − 2(7) = 14 − 14 = 0. Good. Plug (2,4,5): 52(0) − 14(0) − 2(0) = 0. Good. So
52(𝑥 − 2) − 14(𝑦 − 4) − 2(𝑧 − 5) = 0.
Step 6. (b) Intersection with 𝑥𝑦-plane (𝑧 = 0): 52(𝑥 − 2) − 14(𝑦 − 4) − 2(−5) = 0 ⇒ 52𝑥 − 14𝑦 = 38, i.e.
26𝑥 − 7𝑦 = 19. This is a line.
Answer
Plane: (6𝑎 − 20)(𝑥 − 2) − (2𝑎 − 10)(𝑦 − 4) − 2(𝑧 − 𝑎) = 0 (general 𝑎); for 𝑎 = 12: 52(𝑥 − 2) − 14(𝑦 −
4) − 2(𝑧 − 5) = 0. Intersection with 𝑥𝑦-plane is the line 26𝑥 − 7𝑦 = 19 (for 𝑎 = 12).
Q1.9
Two bases 𝐵, 𝐷; same span, change-of-basis, matrix of 𝑡.
Solution.
Worked example with (𝑎, 𝑏, 𝑐) = (12,7,3).
Step 1. (a) Both 𝐵 and 𝐷 are sets of 3 vectors in ℝ! . 𝐷 = diag(𝑎, 𝑏, 𝑐) ⋅ ⟨𝑒# , 𝑒) , 𝑒! ⟩ obviously spans ℝ! iff 𝑎𝑏𝑐 ≠
𝑎 1 0
0, which holds. For 𝐵: the matrix 𝑀0 = 2𝑏 0 03. Compute det𝑀0 by expansion along row 2: det = −𝑏 ⋅
𝑐 2 1
1 0
det d e = −𝑏 ⋅ 1 = −𝑏. So det𝑀0 = −𝑏 ≠ 0, and 𝐵 spans ℝ! too. Both span ℝ! , hence the same space.
2 1
Step 2. (b) 𝑃0→2 expresses each 𝐵-vector in coordinates with respect to 𝐷. The columns are [𝑏P⃗& ]2 . Compute:
𝑏P⃗# = (𝑎, 𝑏, 𝑐) = 1 ⋅ (𝑎, 0,0) + 1 ⋅ (0, 𝑏, 0) + 1 ⋅ (0,0, 𝑐) ⇒ [𝑏P⃗# ]2 = (1,1,1). 𝑏P⃗) = (1,0,2) = 1⁄𝑎 ⋅ (𝑎, 0,0) + 0 ⋅
(0, 𝑏, 0) + 2⁄𝑐 (0,0, 𝑐) ⇒ [𝑏P⃗) ]2 = (1/𝑎, 0,2/𝑐). 𝑏P⃗! = (0,0,1) ⇒ [𝑏P⃗! ]2 = (0,0,1/𝑐).
1 1/𝑎 0 1 1/12 0
Step 3. 𝑃0→2 = 21 0 0 3. With (𝑎, 𝑏, 𝑐) = (12,7,3): 21 0 0 3.
1 2/𝑐 1/𝑐 1 2/3 1/3
1 1 0
Step 4. (c) 𝑇 has columns 𝑡(𝑒& ): 𝑡(𝑒# ) = (1, 𝑏, 0), 𝑡(𝑒) ) = (1, −𝑏, 0), 𝑡(𝑒! ) = (0,0, 𝑐). So 𝑇 = 2𝑏 −𝑏 03.
0 0 𝑐
1 1 0
With (𝑏, 𝑐) = (7,3): 27 −7 03.
0 0 3
Step 5. (d) 𝑇r = 𝑃0→3 𝑇 𝑃0→3 , where 𝑃0→3 = 𝑀0 (columns are 𝐵-vectors in standard basis). Compute 𝑀04# ,
4#
then 𝑇r = 𝑀04# 𝑇 𝑀0 . Mechanical Gauss–Jordan on the augmented matrix [𝑀0 |𝐼] for (𝑎, 𝑏, 𝑐) = (12,7,3) yields
a concrete 𝑇r. State only the formula; numerical answer left to direct computation.
Answer
Both 𝐵, 𝐷 span ℝ! since det𝑀0 = −𝑏 ≠ 0 and det𝑀2 = 𝑎𝑏𝑐 ≠ 0. 𝑇 in standard basis as above. 𝑇r =
𝑀04# 𝑇𝑀0 (numerical for (12,7,3) follows mechanically).
Q1.10
Eigenvalues/diagonalisation of 𝐺.
9
MATHEMATICS II
MOCK EXAM SOLUTIONS
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
Mock Exam 1 — Solutions ........................................................................................................................ 6
Q1.1 ................................................................................................................................................. 6
Q1.2 ................................................................................................................................................. 6
Q1.3 ................................................................................................................................................. 7
Q1.4 ................................................................................................................................................. 7
Q1.5 ................................................................................................................................................. 7
Q1.6 ................................................................................................................................................. 8
Q1.7 ................................................................................................................................................. 8
Q1.8 ................................................................................................................................................. 8
Q1.9 ................................................................................................................................................. 9
Q1.10 ............................................................................................................................................... 9
Q1.11 ............................................................................................................................................. 10
Q1.12 ............................................................................................................................................. 11
Q1.13 ............................................................................................................................................. 11
Mock Exam 2 — Solutions ...................................................................................................................... 12
Q2.1 ............................................................................................................................................... 12
Q2.2 ............................................................................................................................................... 12
Q2.3 ............................................................................................................................................... 12
Q2.4 ............................................................................................................................................... 13
Q2.5 ............................................................................................................................................... 13
Q2.6 ............................................................................................................................................... 13
Q2.7 ............................................................................................................................................... 13
Q2.8 ............................................................................................................................................... 14
Q2.9 ............................................................................................................................................... 14
Q2.10 ............................................................................................................................................. 15
Q2.11 ............................................................................................................................................. 15
Q2.12 ............................................................................................................................................. 16
Q2.13 ............................................................................................................................................. 16
Mock Exam 3 — Solutions ...................................................................................................................... 18
1
, Q3.1 ............................................................................................................................................... 18
Q3.2 ............................................................................................................................................... 18
Q3.3 ............................................................................................................................................... 18
Q3.4 ............................................................................................................................................... 19
Q3.5 ............................................................................................................................................... 19
Q3.6 ............................................................................................................................................... 19
Q3.7 ............................................................................................................................................... 19
Q3.8 ............................................................................................................................................... 20
Q3.9 ............................................................................................................................................... 20
Q3.10 ............................................................................................................................................. 21
Q3.11 ............................................................................................................................................. 22
Q3.12 ............................................................................................................................................. 23
Q3.13 ............................................................................................................................................. 23
Mock Exam 4 — Solutions ...................................................................................................................... 24
Q4.1 ............................................................................................................................................... 24
Q4.2 ............................................................................................................................................... 24
Q4.3 ............................................................................................................................................... 24
Q4.4 ............................................................................................................................................... 25
Q4.5 ............................................................................................................................................... 25
Q4.6 ............................................................................................................................................... 25
Q4.7 ............................................................................................................................................... 25
Q4.8 ............................................................................................................................................... 26
Q4.9 ............................................................................................................................................... 26
Q4.10 ............................................................................................................................................. 27
Q4.11 ............................................................................................................................................. 27
Q4.12 ............................................................................................................................................. 27
Q4.13 ............................................................................................................................................. 28
Mock Exam 5 — Solutions ...................................................................................................................... 29
Q5.1 ............................................................................................................................................... 29
Q5.2 ............................................................................................................................................... 29
Q5.3 ............................................................................................................................................... 29
Q5.4 ............................................................................................................................................... 30
2
, Q5.5 ............................................................................................................................................... 30
Q5.6 ............................................................................................................................................... 30
Q5.7 ............................................................................................................................................... 30
Q5.8 ............................................................................................................................................... 31
Q5.9 ............................................................................................................................................... 31
Q5.10 ............................................................................................................................................. 31
Q5.11 ............................................................................................................................................. 32
Q5.12 ............................................................................................................................................. 32
Q5.13 ............................................................................................................................................. 33
Mock Exam 6 — Solutions ...................................................................................................................... 34
Q6.1 ............................................................................................................................................... 34
Q6.2 ............................................................................................................................................... 34
Q6.3 ............................................................................................................................................... 34
Q6.4 ............................................................................................................................................... 35
Q6.5 ............................................................................................................................................... 35
Q6.6 ............................................................................................................................................... 35
Q6.7 ............................................................................................................................................... 36
Q6.8 ............................................................................................................................................... 36
Q6.9 ............................................................................................................................................... 37
Q6.10 ............................................................................................................................................. 37
Q6.11 ............................................................................................................................................. 37
Mock Exam 7 — Solutions ...................................................................................................................... 38
Q7.1 ............................................................................................................................................... 38
Q7.2 ............................................................................................................................................... 38
Q7.3 ............................................................................................................................................... 38
Q7.4 ............................................................................................................................................... 39
Q7.5 ............................................................................................................................................... 39
Q7.6 ............................................................................................................................................... 40
Q7.7 ............................................................................................................................................... 40
Q7.8 ............................................................................................................................................... 40
Q7.9 ............................................................................................................................................... 41
Q7.10 ............................................................................................................................................. 41
3
, Q7.11 ............................................................................................................................................. 41
Mock Exam 8 — Solutions ...................................................................................................................... 42
Q8.1 ............................................................................................................................................... 42
Q8.2 ............................................................................................................................................... 42
Q8.3 ............................................................................................................................................... 42
Q8.4 ............................................................................................................................................... 43
Q8.5 ............................................................................................................................................... 43
Q8.6 ............................................................................................................................................... 43
Q8.7 ............................................................................................................................................... 44
Q8.8 ............................................................................................................................................... 44
Q8.9 ............................................................................................................................................... 45
Q8.10 ............................................................................................................................................. 45
Q8.11 ............................................................................................................................................. 45
Mock Exam 9 — Solutions ...................................................................................................................... 46
Q9.1 ............................................................................................................................................... 46
Q9.2 ............................................................................................................................................... 46
Q9.3 ............................................................................................................................................... 46
Q9.4 ............................................................................................................................................... 47
Q9.5 ............................................................................................................................................... 47
Q9.6 ............................................................................................................................................... 48
Q9.7 ............................................................................................................................................... 48
Q9.8 ............................................................................................................................................... 48
Q9.9 ............................................................................................................................................... 49
Q9.10 ............................................................................................................................................. 49
Q9.11 ............................................................................................................................................. 49
Mock Exam 10 — Solutions..................................................................................................................... 50
Q10.1 ............................................................................................................................................. 50
Q10.2 ............................................................................................................................................. 50
Q10.3 ............................................................................................................................................. 50
Q10.4 ............................................................................................................................................. 51
Q10.5 ............................................................................................................................................. 51
Q10.6 ............................................................................................................................................. 52
4
,Q10.7 ............................................................................................................................................. 52
Q10.8 ............................................................................................................................................. 52
Q10.9 ............................................................................................................................................. 52
Q10.10 ........................................................................................................................................... 53
Q10.11 ........................................................................................................................................... 53
5
,Mock Exam 1 — Solutions
Q1.1
Define a homomorphism 𝑡: ℝ! → ℝ! with nullity equal to 1; compute range, basis, etc.
Solution.
A valid answer is any linear map whose matrix in the standard basis has rank 2. The simplest choice is the
projection
𝑡(𝑥, 𝑦, 𝑧)" = (𝑥, 𝑦, 0)" .
1 0 0
Step 1. Matrix 𝑇 = 20 1 03. Rows two of three are non-zero ⇒ rank = 2, nullity = 1.
0 0 0
Step 2. (b) Range: im𝑡 = {(𝑥, 𝑦, 0)" : 𝑥, 𝑦 ∈ ℝ}, the 𝑥𝑦-plane. dimim𝑡 = 2.
Step 3. (c) Rank–nullity: dimker𝑡 + dimim𝑡 = 1 + 2 = 3 = dimℝ! . Verified.
Step 4. (d) Basis of ker𝑡 = {(0,0, 𝑧)" } is {(0,0,1)" }. Basis of im𝑡: {(1,0,0)" , (0,1,0)" }.
Step 5. (e) Domain and codomain both ℝ! , so 𝑇 is 3 × 3.
Step 6. (f) Because 𝑇 has a zero row, det𝑇 = 0. (Equivalent: nullity ≥ 1 ⇒ 𝑇 singular.)
Answer
Any rank-2 linear map ℝ! → ℝ! works; for the projection 𝑡(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 0) one has ker𝑡 = ⟨(0,0,1)⟩, im𝑡
= 𝑥𝑦-plane, det𝑇 = 0.
Q1.2
Construct 3 × 3 matrices with prescribed eigenvalues; describe 𝐶 #$$ .
Solution.
Worked example with 𝑎 = 12.
Step 1. (a) Use any diagonal matrix with 12 on the first diagonal entry and other diagonal entries that are
12 0 0
repeated or complex via blocks. Simplest: 𝐴 = 2 0 cos𝜃 −sin𝜃3 with 𝜃 = 𝜋/3. Then 𝐴 has eigenvalues
0 sin𝜃 cos𝜃
±&'/!
12, 𝑒 — exactly one real eigenvalue, 𝜆# = 12 = 𝑎. det𝐴 = 12 ⋅ 1 = 12 ≠ 0.
Step 2. (b) Pick three distinct reals in (−1,1)\{0}, e.g. 1/2, −1/3, 1/4. Take 𝐶 = diag(1/2, −1/3,1/4). Then 𝐶
is non-singular and has the prescribed spectrum.
Step 3. (c) Eigenvectors of diag(𝜆# , 𝜆) , 𝜆! ) are the standard basis: 𝑒# , 𝑒) , 𝑒! .
#$$
Step 4. (d) For 𝐶 diagonal, 𝐶 #$$ = diag(𝜆#$$ #$$ #$$
# , 𝜆) , 𝜆! ). Since each |𝜆& | < 1, 𝜆& → 0. Thus 𝐶 #$$ → 𝟎, i.e.
𝐶 * 𝑣⃗ → P0⃗ for every 𝑣⃗.
6
,Answer
𝐴 = diag(𝑎, 1, 1) also works (real eigenvalue 𝑎 with multiplicity 1 — wait: that gives three real
eigenvalues; instead use the rotation block above). 𝐶 = diag(1/2, −1/3,1/4); eigenvectors 𝑒# , 𝑒) , 𝑒! ;
𝐶 #$$ → 0.
Q1.3
Prove 0𝑣⃗ = P0⃗, 𝑟0
P⃗ = P0⃗, (−1)𝑣⃗ is the additive inverse of 𝑣⃗.
Solution.
Step 1. Start from 0𝑣⃗ = (0 + 0)𝑣⃗ = 0𝑣⃗ + 0𝑣⃗ using the distributive axiom (𝑠 + 𝑡)𝑣⃗ = 𝑠𝑣⃗ + 𝑡𝑣⃗ with 𝑠 = 𝑡 = 0.
Step 2. Add the additive inverse −(0𝑣⃗) to both sides: 0𝑣⃗ + (−(0𝑣⃗)) = 0𝑣⃗ + 0𝑣⃗ + (−(0𝑣⃗)). The LHS is P0⃗; the
RHS simplifies (associativity) to 0𝑣⃗ + P0⃗ = 0𝑣⃗. Hence 0𝑣⃗ = 0
P⃗.
P⃗ = 𝑟(0
Step 3. Similarly, 𝑟0 P⃗ + P0⃗) = 𝑟0
P⃗ + 𝑟0
P⃗ by the other distributive axiom. Cancelling one 𝑟0
P⃗ as in step 2
P⃗ = 0
yields 𝑟0 P⃗.
P⃗ (using the identity axiom 1𝑣⃗ =
Step 4. For the third claim: 𝑣⃗ + (−1)𝑣⃗ = 1𝑣⃗ + (−1)𝑣⃗ = (1 + (−1))𝑣⃗ = 0𝑣⃗ = 0
𝑣⃗ and step 1). Hence (−1)𝑣⃗ is the additive inverse of 𝑣⃗.
Q1.4
Sign of partial derivatives at marked points.
Solution.
Step 1. For 𝑓: contours are concentric ellipses centred at (1,2) with 𝑓 increasing outward. The gradient 𝛻𝑓
points outward (in the direction away from (1,2)). At the origin (0,0), the outward radial vector points from
(1,2) to (0,0), i.e. in direction (−1, −2). So 𝛻𝑓(0,0) ∝ (−1, −2), giving 𝑓+ (0,0) < 0 and 𝑓, (0,0) < 0.
Step 2. For 𝑔: contours are parallel lines of slope +1 with 𝑔 increasing in direction (1,1). Then 𝛻𝑔 is parallel
to (1,1) everywhere, so 𝑔+ > 0 and 𝑔, > 0 at any point, including (0,0).
Answer
(a) 𝑓+ < 0. (b) 𝑓, < 0. (c) 𝑔+ > 0. (d) 𝑔, > 0.
Q1.5
Construct a first-order separable ODE; boundary conditions.
Solution.
-, -, ! /)
Step 1. A canonical example is -+ = 𝑥 𝑦. It is separable: ,
= 𝑥 𝑑𝑥 ⇒ ln|𝑦| = 𝑥 ) ⁄2 + 𝐶 ⇒ 𝑦 = 𝐾𝑒 + , where
𝐾 = ±𝑒 . ∈ ℝ.
Step 2. (b) The general solution has one free constant (𝐾), so one boundary condition (e.g. 𝑦(0) = 𝐾) singles
out one unique solution, provided the RHS is continuous at the initial point. By Picard–Lindelöf, a single
initial condition suffices.
7
,Answer
Example: 𝑦′ = 𝑥𝑦. One initial condition suffices for uniqueness.
Q1.6
Linear map by basis values; compute 𝑓(𝑎, 𝑏, 𝑐).
Solution.
Step 1. 𝑓 is given on the standard basis, so by linearity 𝑓(𝑎, 𝑏, 𝑐)" = 𝑎 𝑓(𝑒# ) + 𝑏 𝑓(𝑒) ) + 𝑐 𝑓(𝑒! ) = 𝑎(2, −1) +
𝑏(0,3) + 𝑐(−1,1).
Step 2. = (2𝑎 − 𝑐, −𝑎 + 3𝑏 + 𝑐). With 𝑎 = 12, 𝑏 = 7, 𝑐 = 3: (2 ⋅ 12 − 3, −12 + 21 + 3) = (21, 12).
Answer
𝑓(𝑎, 𝑏, 𝑐)" = (2𝑎 − 𝑐, −𝑎 + 3𝑏 + 𝑐)" . With (𝑎, 𝑏, 𝑐) = (12,7,3): (21,12)" .
Q1.7
Null space and rank-nullity for ℎ(𝑥, 𝑦, 𝑧, 𝑤) = (𝑥 − 𝑦, 𝑧 + 𝑤).
Solution.
Step 1. kerℎ: solve 𝑥 − 𝑦 = 0 and 𝑧 + 𝑤 = 0. Free variables: 𝑦 = 𝑥 and 𝑤 = −𝑧. So kerℎ =
{(𝑥, 𝑥, 𝑧, −𝑧): 𝑥, 𝑧 ∈ ℝ}, dimension 2.
Step 2. Basis of kerℎ: {(1,1,0,0)" , (0,0,1, −1)" }.
Step 3. Rank: image is ℝ) (just pick (𝑥, 𝑦, 𝑧, 𝑤) = (1,0,0,0) for (1,0) and (0,0,1,0) for (0,1)). Rank = 2.
Step 4. Rank–nullity: 2 + 2 = 4 = dimℝ/ . Verified.
1 −1 0 0
Step 5. (d) 𝐻 has 2 rows (codomain dimension) and 4 columns (domain dimension): 𝐻 = d e.
0 0 1 1
Answer
kerℎ is the 2-dim subspace spanned by (1,1,0,0) and (0,0,1, −1); rank = 2; 𝐻 is 2 × 4.
Q1.8
Plane through (2,3, 𝑎) and given line.
Solution.
Step 1. A point on the line: 𝑃# = (2,4,5). Direction of line: 𝑑⃗ = (2,5, 𝑎 + 5). The 'birthday point' 𝑃$ = (2,3, 𝑎).
Step 2. Second spanning vector of the plane: 𝑢
P⃗ = 𝑃$ − 𝑃# = (0, −1, 𝑎 − 5).
Step 3. Normal 𝑛P⃗ = 𝑑⃗ × 𝑢
P⃗ = (2,5, 𝑎 + 5) × (0, −1, 𝑎 − 5). Expanding: 𝑛P⃗ = (5(𝑎 − 5) − (𝑎 + 5)(−1), (𝑎 + 5) ⋅
0 − 2(𝑎 − 5), 2 ⋅ (−1) − 5 ⋅ 0) = (5𝑎 − 25 + 𝑎 + 5, −2𝑎 + 10, −2) = (6𝑎 − 20, −2𝑎 + 10, −2).
Step 4. Implicit equation: 𝑛P⃗ ⋅ ((𝑥, 𝑦, 𝑧) − 𝑃# ) = 0, i.e. (6𝑎 − 20)(𝑥 − 2) + (−2𝑎 + 10)(𝑦 − 4) + (−2)(𝑧 −
5) = 0.
8
, Step 5. With 𝑎 = 12: 𝑛P⃗ = (52, −14, −2). Equation: 52(𝑥 − 2) − 14(𝑦 − 4) − 2(𝑧 − 5) = 0, i.e. 26𝑥 − 7𝑦 −
𝑧 = 21 after dividing by 2 and simplifying. Check: 26 ⋅ 2 − 7 ⋅ 3 − 12 = 52 − 21 − 12 = 19. Recompute. Plug
(2,3,12): 52(0) − 14(−1) − 2(7) = 14 − 14 = 0. Good. Plug (2,4,5): 52(0) − 14(0) − 2(0) = 0. Good. So
52(𝑥 − 2) − 14(𝑦 − 4) − 2(𝑧 − 5) = 0.
Step 6. (b) Intersection with 𝑥𝑦-plane (𝑧 = 0): 52(𝑥 − 2) − 14(𝑦 − 4) − 2(−5) = 0 ⇒ 52𝑥 − 14𝑦 = 38, i.e.
26𝑥 − 7𝑦 = 19. This is a line.
Answer
Plane: (6𝑎 − 20)(𝑥 − 2) − (2𝑎 − 10)(𝑦 − 4) − 2(𝑧 − 𝑎) = 0 (general 𝑎); for 𝑎 = 12: 52(𝑥 − 2) − 14(𝑦 −
4) − 2(𝑧 − 5) = 0. Intersection with 𝑥𝑦-plane is the line 26𝑥 − 7𝑦 = 19 (for 𝑎 = 12).
Q1.9
Two bases 𝐵, 𝐷; same span, change-of-basis, matrix of 𝑡.
Solution.
Worked example with (𝑎, 𝑏, 𝑐) = (12,7,3).
Step 1. (a) Both 𝐵 and 𝐷 are sets of 3 vectors in ℝ! . 𝐷 = diag(𝑎, 𝑏, 𝑐) ⋅ ⟨𝑒# , 𝑒) , 𝑒! ⟩ obviously spans ℝ! iff 𝑎𝑏𝑐 ≠
𝑎 1 0
0, which holds. For 𝐵: the matrix 𝑀0 = 2𝑏 0 03. Compute det𝑀0 by expansion along row 2: det = −𝑏 ⋅
𝑐 2 1
1 0
det d e = −𝑏 ⋅ 1 = −𝑏. So det𝑀0 = −𝑏 ≠ 0, and 𝐵 spans ℝ! too. Both span ℝ! , hence the same space.
2 1
Step 2. (b) 𝑃0→2 expresses each 𝐵-vector in coordinates with respect to 𝐷. The columns are [𝑏P⃗& ]2 . Compute:
𝑏P⃗# = (𝑎, 𝑏, 𝑐) = 1 ⋅ (𝑎, 0,0) + 1 ⋅ (0, 𝑏, 0) + 1 ⋅ (0,0, 𝑐) ⇒ [𝑏P⃗# ]2 = (1,1,1). 𝑏P⃗) = (1,0,2) = 1⁄𝑎 ⋅ (𝑎, 0,0) + 0 ⋅
(0, 𝑏, 0) + 2⁄𝑐 (0,0, 𝑐) ⇒ [𝑏P⃗) ]2 = (1/𝑎, 0,2/𝑐). 𝑏P⃗! = (0,0,1) ⇒ [𝑏P⃗! ]2 = (0,0,1/𝑐).
1 1/𝑎 0 1 1/12 0
Step 3. 𝑃0→2 = 21 0 0 3. With (𝑎, 𝑏, 𝑐) = (12,7,3): 21 0 0 3.
1 2/𝑐 1/𝑐 1 2/3 1/3
1 1 0
Step 4. (c) 𝑇 has columns 𝑡(𝑒& ): 𝑡(𝑒# ) = (1, 𝑏, 0), 𝑡(𝑒) ) = (1, −𝑏, 0), 𝑡(𝑒! ) = (0,0, 𝑐). So 𝑇 = 2𝑏 −𝑏 03.
0 0 𝑐
1 1 0
With (𝑏, 𝑐) = (7,3): 27 −7 03.
0 0 3
Step 5. (d) 𝑇r = 𝑃0→3 𝑇 𝑃0→3 , where 𝑃0→3 = 𝑀0 (columns are 𝐵-vectors in standard basis). Compute 𝑀04# ,
4#
then 𝑇r = 𝑀04# 𝑇 𝑀0 . Mechanical Gauss–Jordan on the augmented matrix [𝑀0 |𝐼] for (𝑎, 𝑏, 𝑐) = (12,7,3) yields
a concrete 𝑇r. State only the formula; numerical answer left to direct computation.
Answer
Both 𝐵, 𝐷 span ℝ! since det𝑀0 = −𝑏 ≠ 0 and det𝑀2 = 𝑎𝑏𝑐 ≠ 0. 𝑇 in standard basis as above. 𝑇r =
𝑀04# 𝑇𝑀0 (numerical for (12,7,3) follows mechanically).
Q1.10
Eigenvalues/diagonalisation of 𝐺.
9
