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Maths II · Critical points, Hessian & optimisation

31 questions · 42 marks · 6 papers

Answer key

Indian Institute of Technology Madras

BS Degree Programme · Examination & Study Paper

Mathematics for Data Science II

Critical points, Hessian & optimisation

Official Paper

Examination Instructions & Guidelines

Solve on notebook/paper first before checking solutions.
MCQ: Exactly 1 option is correct. Negative marking applies.
MSQ: One or more options correct. Select all to score.
NAT: Numerical response. No negative marking.
Scientific calculator allowed for all calculations.
Official answer key is sealed at the end of this sheet.

End Term · 31 Aug 2025 (FN)End Term

Read the passage below, then answer Q1–Q4
Q1MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q2MCQ1 Mark·Choose one option
Key ↓
Which of the following is the only critical point for the function defined in the previous part?
1. A{(2,−2)}
2. B{(−2, 2)}
3. C{(−3, 0)}
4. D{(3, 0)}
Q3MCQ1 Mark·Choose one option
Asked 6 times Key ↓
What can be concluded from the Hessian test at the only critical point for ϕ?
1. AThe critical point is a local minimum.
2. BThe critical point is a local maximum.
3. CThe critical point is a saddle point.
4. DThe Hessian test is inconclusive.
Q4NAT2 Marks·Write the number
Asked 4 times Key ↓
Your answer:

End Term · 31 Aug 2025 (AN)End Term

Read the passage below, then answer Q5–Q8
Based on the above data, answer the given subquestions.
Q5MCQ1 Mark·Choose one option
Key ↓
Which of the following is the only critical point of T in Int( D )?
1. A
2. B
3. C
4. D
Q6MCQ1 Mark·Choose one option
Asked 6 times Key ↓
What can be concluded by the Hessian test for T at its critical point in Int( D )?
1. AThe critical point is a local minimum.
2. BThe critical point is a local maximum.
3. CThe critical point is a saddle point.
4. DThe Hessian test is inconclusive.
Q7MCQ1 Mark·Choose one option
Key ↓
1. A
2. B
3. C
4. D
Q8NAT2 Marks·Write the number
Key ↓
Comparing the critical points on the interior, the boundary components, and the corners, find the temperature at the hottest point on the metallic plate.
Your answer:

End Term · 13 Apr 2025 (FN)End Term

Read the passage below, then answer Q9–Q11
Based on the above data, answer the given subquestions.
Q9MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q10NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q11MCQ1 Mark·Choose one option
Asked 6 times Key ↓
What can be concluded about the critical point (0, 0) from the Hessian test?
1. AThe Hessian test is inconclusive.
2. B(0, 0) is a local minimum.
3. C(0, 0) is a local maximum.
4. D(0, 0) is a saddle point.
Read the passage below, then answer Q12–Q15
Based on the above data, answer the given subquestions.
Q12MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q13MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q14MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
Choose the correct statements from the following.
1. A
2. B
3. C
4. D
5. E
Q15NAT1 Mark·Write the number
Asked 4 times Key ↓
Your answer:

End Term · 13 Apr 2025 (AN)End Term

Read the passage below, then answer Q16–Q18
Based on the above data, answer the given subquestions.
Q16MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A(1, 0)
2. B(0, 1)
3. C(−1, 1)
4. D(1,−1)
Q17NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q18MCQ1 Mark·Choose one option
Asked 6 times Key ↓
What can be concluded about the critical point (0, 0) from the Hessian test?
1. AThe Hessian test is inconclusive.
2. B(0, 0) is a local minimum.
3. C(0, 0) is a local maximum.
4. D(0, 0) is a saddle point.
Read the passage below, then answer Q19–Q22
Based on the above data, answer the given subquestions.
Q19MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q20MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A0
2. B1
3. C2
4. DInfinitely many
Q21MCQ2 Marks·Choose one option
Asked 2 times Key ↓
Choose the correct statement from the following.
1. A
2. B
3. C
4. D
Q22NAT1 Mark·Write the number
Asked 4 times Key ↓
Your answer:

End Term · 22 Dec 2024 (FN)End Term

Read the passage below, then answer Q23–Q24
Based on the above data answer the given subquestions.
Q23MCQ1 Mark·Choose one option
Key ↓
1. A
2. B
3. C
4. D
Q24NAT2 Marks·Write the number
Key ↓
Using the function given, calculate the maximum possible volume of a cuboid whose perimeter is 12.
Your answer:
Read the passage below, then answer Q25–Q26
Based on the above data answer the given subquestions.
Q25MCQ2 Marks·Choose one option
Asked 2 times Key ↓
1. A0
2. B1
3. C2
4. DInfinitely many
Q26MCQ2 Marks·Choose one option
Asked 6 times Key ↓
1. A
2. B
3. C
4. D
5. E

End Term · 22 Dec 2024 (AN)End Term

Read the passage below, then answer Q27–Q28
Based on the above data, answer the given subquestions.
Q27MCQ1 Mark·Choose one option
Asked 2 times Key ↓
How many critical points does the function f have?
1. A0
2. B1
3. C2
4. DInfinitely many
Q28MCQ2 Marks·Choose one option
Asked 6 times Key ↓
What can we conclude from the Hessian test for the function f at (0, 0)?
1. A(0, 0) is not a critical point.
2. BThe Hessian test is inconclusive.
3. CThe Hessian test can be used to conclude that (0, 0) is a local maximum.
4. DThe Hessian test can be used to conclude that (0, 0) is a local minimum.
5. EThe Hessian test can be used to conclude that (0, 0) is a saddle point.
Read the passage below, then answer Q29–Q31
Based on the above data, answer the given subquestions.
Q29MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q30NAT2 Marks·Write the number
Asked 4 times Key ↓
Your answer:
Q31MCQ1 Mark·Choose one option
Key ↓
1. A
2. B
3. C
4. D

Official Key

Master Answer Key & Explanations

Official answer keys from IIT Madras BS past papers. Click any question to review.

Reveal Key

Q1C

Q2B

Q3A

Q46

Q5B

Q6C

Q7A

Q810.65 to 10.67

Q9A, D

Q10-1

Q11D

Q12D

Q13B

Q14A, B, E

Q151

Q16C, D

Q174

Q18B

Q19D

Q20B

Q21B

Q222

Q23C

Q241

Q25C

Q26E

Q27D

Q28B

Q29B

Q301

Q31D

Questions and answer keys are reproduced from the official IIT Madras BS papers. Topics and repeated question types are BTechi’s own grouping. BTechi is not affiliated with IIT Madras.