BTCS-110 Applied Mathematics Final Exam | Complete
Practice QUESTIONs, Verified Answers & Detailed
Solutions (2026/2027)
QUESTION 1
What is the value of (1 + 𝑖)4 expressed in standard form?
• A. −4
• B. 4𝑖
• C. −4𝑖
• D. 4
Correct Answer: A. −4
Detailed Rationale: Using polar form or binomial expansion: (1 + 𝑖)2 =
1 + 2𝑖 + 𝑖 2 = 2𝑖. Therefore, (1 + 𝑖)4 = (2𝑖)2 = 4𝑖 2 = −4.
Alternatively, using De Moivre's theorem: 1 + 𝑖 = √2(cos(𝜋/4) +
4
𝑖 sin(𝜋/4)), so (1 + 𝑖)4 = (√2) (cos 𝜋 + 𝑖 sin 𝜋) = 4(−1) = −4.
QUESTION 2
Which of the following convergence tests is most suitable for testing the
𝑛𝑛
convergence of the series ∑∞
𝑛=1 ?
𝑛!
• A. Comparison test
• B. d'Alembert's ratio test
• C. Integral test
, • D. Leibniz alternating series test
Correct Answer: B. d'Alembert's ratio test
Detailed Rationale: The presence of factorial terms (𝑛!) along with
𝑢𝑛+1
powers involving 𝑛 (𝑛𝑛 ) makes d'Alembert's ratio test ( lim ) highly
𝑛→∞ 𝑢𝑛
effective, as the factorial and exponential components simplify readily.
QUESTION 3
What is the 𝑛-th derivative of the function 𝑦 = log(𝑎𝑥 + 𝑏)?
(−1)𝑛−1 (𝑛−1)!𝑎𝑛
• A. (𝑎𝑥+𝑏)𝑛
(−1)𝑛 𝑛!𝑎𝑛
• B. (𝑎𝑥+𝑏)𝑛
𝑛!𝑎𝑛
• C. (𝑎𝑥+𝑏)𝑛
(−1)𝑛−1 𝑛!𝑎𝑛
• D. (𝑎𝑥+𝑏)𝑛+1
(−1)𝑛−1 (𝑛−1)!𝑎𝑛
Correct Answer: A. (𝑎𝑥+𝑏)𝑛
𝑎
Detailed Rationale: Differentiating successively: 𝑦′ = , 𝑦′′ =
𝑎𝑥+𝑏
𝑎2 2!𝑎3
− (𝑎𝑥+𝑏)2, 𝑦′′′ = (𝑎𝑥+𝑏)3
. Extending this pattern gives the general
(𝑛) (−1)𝑛−1 (𝑛−1)!𝑎𝑛
formula 𝑦 = (𝑎𝑥+𝑏)𝑛
.
QUESTION 4
What is the Maclaurin series expansion for the function 𝑓(𝑥 ) = cos 𝑥?
𝑥2 𝑥4 𝑥6
• A. 1 − + − +⋯
2! 4! 6!
, 𝑥3 𝑥5
• B. 𝑥 − + −⋯
3! 5!
𝑥2 𝑥3
• C. 1 + 𝑥 + + +⋯
2! 3!
𝑥2 𝑥3
• D. 1 − 𝑥 + − +⋯
2! 3!
𝑥2 𝑥4 𝑥6
Correct Answer: A. 1 − + − +⋯
2! 4! 6!
Detailed Rationale: Maclaurin's series expansion is given by 𝑓(𝑥 ) =
𝑥2
𝑓(0) + 𝑥𝑓′(0) + 𝑓′′(0) + ⋯. For cos 𝑥, 𝑓 (0) = 1, 𝑓′(0) = 0,
2!
𝑓′′(0) = −1, 𝑓′′′(0) = 0, 𝑓 (𝑖𝑣) (0) = 1, yielding the alternating series of
even powers with factorial denominators.
QUESTION 5
What is the radius of curvature 𝜌 at any point on a curve where the
curvature 𝜅 is given as 0.05?
• A. 0.05
• B. 5
• C. 20
• D. 100
Correct Answer: C. 20
Detailed Rationale: The radius of curvature 𝜌 is defined as the
1 1
reciprocal of the curvature 𝜅, i.e., 𝜌 = . Substituting 𝜅 = 0.05 (or ),
𝜅 20
1
we get 𝜌 = = 20.
0.05
QUESTION 6
, What is the rank of a 3 × 3 identity matrix 𝐼3 ?
• A. 0
• B. 1
• C. 2
• D. 3
Correct Answer: D. 3
Detailed Rationale: The rank of a matrix is the maximum number of
linearly independent row or column vectors. Since an identity matrix of
order 𝑛 has 𝑛 non-zero pivot elements in row-echelon form, its rank is
equal to its order, which is 3.
QUESTION 7
3 5
What are the eigenvalues of the upper triangular matrix 𝐴 = [ ]?
0 −2
• A. 3 and −2
• B. 3 and 5
• C. −3 and 2
• D. 0 and 1
Correct Answer: A. 3 and −2
Detailed Rationale: The eigenvalues of a triangular (upper or lower) or
diagonal matrix are simply the elements located along its main
diagonal. For matrix 𝐴, the diagonal elements are 3 and −2.
QUESTION 8
Practice QUESTIONs, Verified Answers & Detailed
Solutions (2026/2027)
QUESTION 1
What is the value of (1 + 𝑖)4 expressed in standard form?
• A. −4
• B. 4𝑖
• C. −4𝑖
• D. 4
Correct Answer: A. −4
Detailed Rationale: Using polar form or binomial expansion: (1 + 𝑖)2 =
1 + 2𝑖 + 𝑖 2 = 2𝑖. Therefore, (1 + 𝑖)4 = (2𝑖)2 = 4𝑖 2 = −4.
Alternatively, using De Moivre's theorem: 1 + 𝑖 = √2(cos(𝜋/4) +
4
𝑖 sin(𝜋/4)), so (1 + 𝑖)4 = (√2) (cos 𝜋 + 𝑖 sin 𝜋) = 4(−1) = −4.
QUESTION 2
Which of the following convergence tests is most suitable for testing the
𝑛𝑛
convergence of the series ∑∞
𝑛=1 ?
𝑛!
• A. Comparison test
• B. d'Alembert's ratio test
• C. Integral test
, • D. Leibniz alternating series test
Correct Answer: B. d'Alembert's ratio test
Detailed Rationale: The presence of factorial terms (𝑛!) along with
𝑢𝑛+1
powers involving 𝑛 (𝑛𝑛 ) makes d'Alembert's ratio test ( lim ) highly
𝑛→∞ 𝑢𝑛
effective, as the factorial and exponential components simplify readily.
QUESTION 3
What is the 𝑛-th derivative of the function 𝑦 = log(𝑎𝑥 + 𝑏)?
(−1)𝑛−1 (𝑛−1)!𝑎𝑛
• A. (𝑎𝑥+𝑏)𝑛
(−1)𝑛 𝑛!𝑎𝑛
• B. (𝑎𝑥+𝑏)𝑛
𝑛!𝑎𝑛
• C. (𝑎𝑥+𝑏)𝑛
(−1)𝑛−1 𝑛!𝑎𝑛
• D. (𝑎𝑥+𝑏)𝑛+1
(−1)𝑛−1 (𝑛−1)!𝑎𝑛
Correct Answer: A. (𝑎𝑥+𝑏)𝑛
𝑎
Detailed Rationale: Differentiating successively: 𝑦′ = , 𝑦′′ =
𝑎𝑥+𝑏
𝑎2 2!𝑎3
− (𝑎𝑥+𝑏)2, 𝑦′′′ = (𝑎𝑥+𝑏)3
. Extending this pattern gives the general
(𝑛) (−1)𝑛−1 (𝑛−1)!𝑎𝑛
formula 𝑦 = (𝑎𝑥+𝑏)𝑛
.
QUESTION 4
What is the Maclaurin series expansion for the function 𝑓(𝑥 ) = cos 𝑥?
𝑥2 𝑥4 𝑥6
• A. 1 − + − +⋯
2! 4! 6!
, 𝑥3 𝑥5
• B. 𝑥 − + −⋯
3! 5!
𝑥2 𝑥3
• C. 1 + 𝑥 + + +⋯
2! 3!
𝑥2 𝑥3
• D. 1 − 𝑥 + − +⋯
2! 3!
𝑥2 𝑥4 𝑥6
Correct Answer: A. 1 − + − +⋯
2! 4! 6!
Detailed Rationale: Maclaurin's series expansion is given by 𝑓(𝑥 ) =
𝑥2
𝑓(0) + 𝑥𝑓′(0) + 𝑓′′(0) + ⋯. For cos 𝑥, 𝑓 (0) = 1, 𝑓′(0) = 0,
2!
𝑓′′(0) = −1, 𝑓′′′(0) = 0, 𝑓 (𝑖𝑣) (0) = 1, yielding the alternating series of
even powers with factorial denominators.
QUESTION 5
What is the radius of curvature 𝜌 at any point on a curve where the
curvature 𝜅 is given as 0.05?
• A. 0.05
• B. 5
• C. 20
• D. 100
Correct Answer: C. 20
Detailed Rationale: The radius of curvature 𝜌 is defined as the
1 1
reciprocal of the curvature 𝜅, i.e., 𝜌 = . Substituting 𝜅 = 0.05 (or ),
𝜅 20
1
we get 𝜌 = = 20.
0.05
QUESTION 6
, What is the rank of a 3 × 3 identity matrix 𝐼3 ?
• A. 0
• B. 1
• C. 2
• D. 3
Correct Answer: D. 3
Detailed Rationale: The rank of a matrix is the maximum number of
linearly independent row or column vectors. Since an identity matrix of
order 𝑛 has 𝑛 non-zero pivot elements in row-echelon form, its rank is
equal to its order, which is 3.
QUESTION 7
3 5
What are the eigenvalues of the upper triangular matrix 𝐴 = [ ]?
0 −2
• A. 3 and −2
• B. 3 and 5
• C. −3 and 2
• D. 0 and 1
Correct Answer: A. 3 and −2
Detailed Rationale: The eigenvalues of a triangular (upper or lower) or
diagonal matrix are simply the elements located along its main
diagonal. For matrix 𝐴, the diagonal elements are 3 and −2.
QUESTION 8
