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Calculus & Multivariable Analysis (Pure & Applied Mathematics) Solved Questions & Notes (2026) - Apex Rankers

Natural & Physical Sciences > Pure & Applied Mathematics > Calculus & Multivariable Analysis

100 Total Questions
~150 mins Read Time
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Q. 1 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
What is the derivative of f(x) = ln(sin(x)) with respect to x?
A
sec(x)
B
cot(x)
✓ Correct
C
cos(x)
D
tan(x)
💡 Step-by-Step Explanation & Concept Rationale
f'(x) = (1/sin(x)) * cos(x) = cot(x) via the chain rule.
Q. 2 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
The value of the definite integral ∫ from 0 to π/2 of sin²(x) dx is:
A
0
B
1
C
π / 2
D
π / 4
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Using Wallis formula or symmetry ∫ sin²(x) = (1/2) * (π/2) = π/4.
Q. 3 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
L'Hôpital's Rule is applicable for evaluating limits yielding which indeterminate forms?
A
0/0 and ∞/∞
✓ Correct
B
1/0 and 0/1
C
0^1
D
∞ - 0
💡 Step-by-Step Explanation & Concept Rationale
L'Hôpital's differentiates numerator and denominator for 0/0 or ∞/∞ limits.
Q. 4 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
A real function f(x) is continuous at x = a if and only if:
A
f(a) > 0
B
f''(a) exists
C
lim (x->a) f(x) exists, f(a) is defined, and lim (x->a) f(x) = f(a)
✓ Correct
D
f'(a) = 0
💡 Step-by-Step Explanation & Concept Rationale
Continuity requires identical left/right limits matching the defined function value.
Q. 5 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
The Taylor series expansion of e^x about x = 0 (Maclaurin series) is:
A
∑ x^(2n) / (2n)!
B
∑ (x^n / n!) from n=0 to ∞
✓ Correct
C
∑ (x^n / n)
D
∑ (-1)^n x^n
💡 Step-by-Step Explanation & Concept Rationale
e^x = 1 + x + x²/2! + x³/3! + ... with infinite convergence radius.
Q. 6 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: What is the derivative of f(x) = ln(sin(x)) with respect to x?
A
tan(x)
B
sec(x)
C
cot(x)
✓ Correct
D
cos(x)
💡 Step-by-Step Explanation & Concept Rationale
f'(x) = (1/sin(x)) * cos(x) = cot(x) via the chain rule.
Q. 7 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: The value of the definite integral ∫ from 0 to π/2 of sin²(x) dx is:
A
π / 2
B
π / 4
✓ Correct
C
0
D
1
💡 Step-by-Step Explanation & Concept Rationale
Using Wallis formula or symmetry ∫ sin²(x) = (1/2) * (π/2) = π/4.
Q. 8 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: L'Hôpital's Rule is applicable for evaluating limits yielding which indeterminate forms?
A
0^1
B
0/0 and ∞/∞
✓ Correct
C
∞ - 0
D
1/0 and 0/1
💡 Step-by-Step Explanation & Concept Rationale
L'Hôpital's differentiates numerator and denominator for 0/0 or ∞/∞ limits.
Q. 9 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: A real function f(x) is continuous at x = a if and only if:
A
f'(a) = 0
B
f''(a) exists
C
f(a) > 0
D
lim (x->a) f(x) exists, f(a) is defined, and lim (x->a) f(x) = f(a)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Continuity requires identical left/right limits matching the defined function value.
Q. 10 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: The Taylor series expansion of e^x about x = 0 (Maclaurin series) is:
A
∑ (-1)^n x^n
B
∑ x^(2n) / (2n)!
C
∑ (x^n / n!) from n=0 to ∞
✓ Correct
D
∑ (x^n / n)
💡 Step-by-Step Explanation & Concept Rationale
e^x = 1 + x + x²/2! + x³/3! + ... with infinite convergence radius.
Q. 11 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: What is the derivative of f(x) = ln(sin(x)) with respect to x?
A
cot(x)
✓ Correct
B
sec(x)
C
tan(x)
D
cos(x)
💡 Step-by-Step Explanation & Concept Rationale
f'(x) = (1/sin(x)) * cos(x) = cot(x) via the chain rule.
Q. 12 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: The value of the definite integral ∫ from 0 to π/2 of sin²(x) dx is:
A
0
B
π / 2
C
π / 4
✓ Correct
D
1
💡 Step-by-Step Explanation & Concept Rationale
Using Wallis formula or symmetry ∫ sin²(x) = (1/2) * (π/2) = π/4.
Q. 13 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: L'Hôpital's Rule is applicable for evaluating limits yielding which indeterminate forms?
A
1/0 and 0/1
B
0/0 and ∞/∞
✓ Correct
C
0^1
D
∞ - 0
💡 Step-by-Step Explanation & Concept Rationale
L'Hôpital's differentiates numerator and denominator for 0/0 or ∞/∞ limits.
Q. 14 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: A real function f(x) is continuous at x = a if and only if:
A
f(a) > 0
B
f'(a) = 0
C
f''(a) exists
D
lim (x->a) f(x) exists, f(a) is defined, and lim (x->a) f(x) = f(a)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Continuity requires identical left/right limits matching the defined function value.
Q. 15 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: The Taylor series expansion of e^x about x = 0 (Maclaurin series) is:
A
∑ (x^n / n!) from n=0 to ∞
✓ Correct
B
∑ (-1)^n x^n
C
∑ x^(2n) / (2n)!
D
∑ (x^n / n)
💡 Step-by-Step Explanation & Concept Rationale
e^x = 1 + x + x²/2! + x³/3! + ... with infinite convergence radius.
Q. 16 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: What is the derivative of f(x) = ln(sin(x)) with respect to x?
A
cos(x)
B
cot(x)
✓ Correct
C
sec(x)
D
tan(x)
💡 Step-by-Step Explanation & Concept Rationale
f'(x) = (1/sin(x)) * cos(x) = cot(x) via the chain rule.
Q. 17 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: The value of the definite integral ∫ from 0 to π/2 of sin²(x) dx is:
A
π / 4
✓ Correct
B
0
C
1
D
π / 2
💡 Step-by-Step Explanation & Concept Rationale
Using Wallis formula or symmetry ∫ sin²(x) = (1/2) * (π/2) = π/4.
Q. 18 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: L'Hôpital's Rule is applicable for evaluating limits yielding which indeterminate forms?
A
1/0 and 0/1
B
0/0 and ∞/∞
✓ Correct
C
0^1
D
∞ - 0
💡 Step-by-Step Explanation & Concept Rationale
L'Hôpital's differentiates numerator and denominator for 0/0 or ∞/∞ limits.
Q. 19 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: A real function f(x) is continuous at x = a if and only if:
A
f(a) > 0
B
lim (x->a) f(x) exists, f(a) is defined, and lim (x->a) f(x) = f(a)
✓ Correct
C
f'(a) = 0
D
f''(a) exists
💡 Step-by-Step Explanation & Concept Rationale
Continuity requires identical left/right limits matching the defined function value.
Q. 20 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: The Taylor series expansion of e^x about x = 0 (Maclaurin series) is:
A
∑ (-1)^n x^n
B
∑ (x^n / n)
C
∑ x^(2n) / (2n)!
D
∑ (x^n / n!) from n=0 to ∞
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
e^x = 1 + x + x²/2! + x³/3! + ... with infinite convergence radius.
Q. 21 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: What is the derivative of f(x) = ln(sin(x)) with respect to x?
A
tan(x)
B
sec(x)
C
cot(x)
✓ Correct
D
cos(x)
💡 Step-by-Step Explanation & Concept Rationale
f'(x) = (1/sin(x)) * cos(x) = cot(x) via the chain rule.
Q. 22 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: The value of the definite integral ∫ from 0 to π/2 of sin²(x) dx is:
A
1
B
π / 4
✓ Correct
C
π / 2
D
0
💡 Step-by-Step Explanation & Concept Rationale
Using Wallis formula or symmetry ∫ sin²(x) = (1/2) * (π/2) = π/4.
Q. 23 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: L'Hôpital's Rule is applicable for evaluating limits yielding which indeterminate forms?
A
0/0 and ∞/∞
✓ Correct
B
∞ - 0
C
0^1
D
1/0 and 0/1
💡 Step-by-Step Explanation & Concept Rationale
L'Hôpital's differentiates numerator and denominator for 0/0 or ∞/∞ limits.
Q. 24 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: A real function f(x) is continuous at x = a if and only if:
A
f''(a) exists
B
f'(a) = 0
C
f(a) > 0
D
lim (x->a) f(x) exists, f(a) is defined, and lim (x->a) f(x) = f(a)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Continuity requires identical left/right limits matching the defined function value.
Q. 25 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: The Taylor series expansion of e^x about x = 0 (Maclaurin series) is:
A
∑ x^(2n) / (2n)!
B
∑ (x^n / n)
C
∑ (-1)^n x^n
D
∑ (x^n / n!) from n=0 to ∞
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
e^x = 1 + x + x²/2! + x³/3! + ... with infinite convergence radius.
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