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Complex Analysis & Modern Abstract Algebra (Pure & Applied Mathematics) Solved Questions & Notes (2026) - Apex Rankers

Natural & Physical Sciences > Pure & Applied Mathematics > Complex Analysis & Modern Abstract Algebra

100 Total Questions
~150 mins Read Time
1 Subject Areas
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Q. 1 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
The Cauchy-Riemann equations for an analytic complex function f(z) = u(x,y) + i v(x,y) are:
A
∂v/∂x = ∂v/∂y
B
∂u/∂x = ∂u/∂y
C
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
✓ Correct
D
∂u/∂x = -∂v/∂y
💡 Step-by-Step Explanation & Concept Rationale
Satisfying CR equations with continuous partial derivatives guarantees complex differentiability.
Q. 2 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Cauchy's Integral Theorem states that for an analytic function f(z) inside a closed contour C:
A
∮_C f(z) dz = ∞
B
∮_C f(z) dz = 1
C
∮_C f(z) dz = 2πi
D
∮_C f(z) dz = 0
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Analytic functions have conservative complex path integrals vanishing on closed loops.
Q. 3 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
The divergence of the curl of any twice continuously differentiable vector field F (∇ · (∇ × F)) is identically:
A
∇² F
B
0
✓ Correct
C
1
D
F
💡 Step-by-Step Explanation & Concept Rationale
The vector identity div(curl F) = 0 holds universally in vector calculus.
Q. 4 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Green's Theorem in the plane relates a line integral around a closed curve C to a double integral over region D as:
A
∮_C (L dx + M dy) = ∬_D (∂M/∂x - ∂L/∂y) dA
✓ Correct
B
∮ M dy = 0
C
∬ (L + M) dA = 1
D
∮ L dx = ∬ L dA
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem connecting 2D circulation with macroscopic curl.
Q. 5 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
The residue of f(z) = 1 / (z - 3) at its simple pole z = 3 is:
A
3
B
2πi
C
0
D
1
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Res(f, 3) = lim (z->3) (z - 3) * (1 / (z - 3)) = 1.
Q. 6 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
In professional practice: The Cauchy-Riemann equations for an analytic complex function f(z) = u(x,y) + i v(x,y) are:
A
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
✓ Correct
B
∂v/∂x = ∂v/∂y
C
∂u/∂x = ∂u/∂y
D
∂u/∂x = -∂v/∂y
💡 Step-by-Step Explanation & Concept Rationale
Satisfying CR equations with continuous partial derivatives guarantees complex differentiability.
Q. 7 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: Cauchy's Integral Theorem states that for an analytic function f(z) inside a closed contour C:
A
∮_C f(z) dz = 2πi
B
∮_C f(z) dz = ∞
C
∮_C f(z) dz = 0
✓ Correct
D
∮_C f(z) dz = 1
💡 Step-by-Step Explanation & Concept Rationale
Analytic functions have conservative complex path integrals vanishing on closed loops.
Q. 8 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: The divergence of the curl of any twice continuously differentiable vector field F (∇ · (∇ × F)) is identically:
A
F
B
0
✓ Correct
C
1
D
∇² F
💡 Step-by-Step Explanation & Concept Rationale
The vector identity div(curl F) = 0 holds universally in vector calculus.
Q. 9 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
In professional practice: Green's Theorem in the plane relates a line integral around a closed curve C to a double integral over region D as:
A
∮_C (L dx + M dy) = ∬_D (∂M/∂x - ∂L/∂y) dA
✓ Correct
B
∬ (L + M) dA = 1
C
∮ L dx = ∬ L dA
D
∮ M dy = 0
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem connecting 2D circulation with macroscopic curl.
Q. 10 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: The residue of f(z) = 1 / (z - 3) at its simple pole z = 3 is:
A
0
B
2πi
C
1
✓ Correct
D
3
💡 Step-by-Step Explanation & Concept Rationale
Res(f, 3) = lim (z->3) (z - 3) * (1 / (z - 3)) = 1.
Q. 11 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: The Cauchy-Riemann equations for an analytic complex function f(z) = u(x,y) + i v(x,y) are:
A
∂u/∂x = ∂u/∂y
B
∂u/∂x = -∂v/∂y
C
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
✓ Correct
D
∂v/∂x = ∂v/∂y
💡 Step-by-Step Explanation & Concept Rationale
Satisfying CR equations with continuous partial derivatives guarantees complex differentiability.
Q. 12 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: Cauchy's Integral Theorem states that for an analytic function f(z) inside a closed contour C:
A
∮_C f(z) dz = 1
B
∮_C f(z) dz = 0
✓ Correct
C
∮_C f(z) dz = ∞
D
∮_C f(z) dz = 2πi
💡 Step-by-Step Explanation & Concept Rationale
Analytic functions have conservative complex path integrals vanishing on closed loops.
Q. 13 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: The divergence of the curl of any twice continuously differentiable vector field F (∇ · (∇ × F)) is identically:
A
∇² F
B
0
✓ Correct
C
F
D
1
💡 Step-by-Step Explanation & Concept Rationale
The vector identity div(curl F) = 0 holds universally in vector calculus.
Q. 14 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: Green's Theorem in the plane relates a line integral around a closed curve C to a double integral over region D as:
A
∮_C (L dx + M dy) = ∬_D (∂M/∂x - ∂L/∂y) dA
✓ Correct
B
∮ M dy = 0
C
∬ (L + M) dA = 1
D
∮ L dx = ∬ L dA
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem connecting 2D circulation with macroscopic curl.
Q. 15 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: The residue of f(z) = 1 / (z - 3) at its simple pole z = 3 is:
A
1
✓ Correct
B
0
C
3
D
2πi
💡 Step-by-Step Explanation & Concept Rationale
Res(f, 3) = lim (z->3) (z - 3) * (1 / (z - 3)) = 1.
Q. 16 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: The Cauchy-Riemann equations for an analytic complex function f(z) = u(x,y) + i v(x,y) are:
A
∂u/∂x = -∂v/∂y
B
∂v/∂x = ∂v/∂y
C
∂u/∂x = ∂u/∂y
D
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Satisfying CR equations with continuous partial derivatives guarantees complex differentiability.
Q. 17 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: Cauchy's Integral Theorem states that for an analytic function f(z) inside a closed contour C:
A
∮_C f(z) dz = 1
B
∮_C f(z) dz = ∞
C
∮_C f(z) dz = 0
✓ Correct
D
∮_C f(z) dz = 2πi
💡 Step-by-Step Explanation & Concept Rationale
Analytic functions have conservative complex path integrals vanishing on closed loops.
Q. 18 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: The divergence of the curl of any twice continuously differentiable vector field F (∇ · (∇ × F)) is identically:
A
F
B
1
C
0
✓ Correct
D
∇² F
💡 Step-by-Step Explanation & Concept Rationale
The vector identity div(curl F) = 0 holds universally in vector calculus.
Q. 19 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: Green's Theorem in the plane relates a line integral around a closed curve C to a double integral over region D as:
A
∮ L dx = ∬ L dA
B
∮_C (L dx + M dy) = ∬_D (∂M/∂x - ∂L/∂y) dA
✓ Correct
C
∮ M dy = 0
D
∬ (L + M) dA = 1
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem connecting 2D circulation with macroscopic curl.
Q. 20 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: The residue of f(z) = 1 / (z - 3) at its simple pole z = 3 is:
A
1
✓ Correct
B
3
C
0
D
2πi
💡 Step-by-Step Explanation & Concept Rationale
Res(f, 3) = lim (z->3) (z - 3) * (1 / (z - 3)) = 1.
Q. 21 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: The Cauchy-Riemann equations for an analytic complex function f(z) = u(x,y) + i v(x,y) are:
A
∂u/∂x = -∂v/∂y
B
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
✓ Correct
C
∂v/∂x = ∂v/∂y
D
∂u/∂x = ∂u/∂y
💡 Step-by-Step Explanation & Concept Rationale
Satisfying CR equations with continuous partial derivatives guarantees complex differentiability.
Q. 22 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: Cauchy's Integral Theorem states that for an analytic function f(z) inside a closed contour C:
A
∮_C f(z) dz = ∞
B
∮_C f(z) dz = 0
✓ Correct
C
∮_C f(z) dz = 1
D
∮_C f(z) dz = 2πi
💡 Step-by-Step Explanation & Concept Rationale
Analytic functions have conservative complex path integrals vanishing on closed loops.
Q. 23 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: The divergence of the curl of any twice continuously differentiable vector field F (∇ · (∇ × F)) is identically:
A
1
B
0
✓ Correct
C
∇² F
D
F
💡 Step-by-Step Explanation & Concept Rationale
The vector identity div(curl F) = 0 holds universally in vector calculus.
Q. 24 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: Green's Theorem in the plane relates a line integral around a closed curve C to a double integral over region D as:
A
∮ M dy = 0
B
∮_C (L dx + M dy) = ∬_D (∂M/∂x - ∂L/∂y) dA
✓ Correct
C
∬ (L + M) dA = 1
D
∮ L dx = ∬ L dA
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem connecting 2D circulation with macroscopic curl.
Q. 25 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: The residue of f(z) = 1 / (z - 3) at its simple pole z = 3 is:
A
0
B
2πi
C
1
✓ Correct
D
3
💡 Step-by-Step Explanation & Concept Rationale
Res(f, 3) = lim (z->3) (z - 3) * (1 / (z - 3)) = 1.
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