BTechibtechi.in
Loading Loading
Free IIT Madras BS notes, PYQs and mock tests at btechi.in

Maths II · Kernel, range & isomorphisms

18 questions · 40 marks · 7 papers

Answer key

Indian Institute of Technology Madras

BS Degree Programme · Examination & Study Paper

Mathematics for Data Science II

Kernel, range & isomorphisms

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

Q1MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A(0, 0, 0)
2. B(−1, 1, 7)
3. C(−1, 5, 5)
4. D(1,−1,−3)

End Term · 31 Aug 2025 (AN)End Term

Q2MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A(0, 0, 1)
2. B(1, 1, 2)
3. C(1,−2, 1)
4. D(−4,−1,−1)

Quiz 2 · 3 Aug 2025 (AN)Quiz 2

Q3NAT5 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q4–Q5
Q4MSQ4 Marks·Choose all correct options
Key ↓
Choose all the correct statements from the following.
1. AThe rank of the linear transformation T is 2.
2. BThe nullity of the linear transformation T is 2.
3. CThe transformation T is invertible.
4. DT cannot be an isomorphism.
Q5NAT3 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q6–Q8
Based on the above data, answer the given subquestions.
Q6NAT2 Marks·Write the number
Key ↓
Find the number of columns in the matrix (with respect to any bases) of the linear transformation T.
Your answer:
Q7MCQ2 Marks·Choose one option
Key ↓
Let A denote the matrix of the linear transformation T with respect to standard bases for both the domain and the codomain. Choose the correct statement from the following. (In the options below, REF is used for row echelon form, and RREF is used for reduced row echelon form.)
1. AA is in REF and RREF.
2. BA is neither in RREF nor in REF.
3. CA is in RREF, but not in REF.
4. DA is in REF, but not in RREF.
Q8NAT3 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q9–Q10
Based on the above data, answer the given subquestions.
Q9MCQ2 Marks·Choose one option
Asked 3 times Key ↓
Which of the following is the matrix of the linear transformation T with respect to standard bases for both the domain and the codomain.
1. A
2. B
3. C
4. D
Q10MSQ5 Marks·Choose all correct options
Key ↓
Which of the following spaces are isomorphic to the kernel of T?
1. A
2. B
3. C
4. D
5. E
6. F

End Term · 13 Apr 2025 (FN)End Term

Read the passage below, then answer Q11–Q13
Based on the above data, answer the given subquestions.
Q11NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the maximum rank of A?
Your answer:
Q12NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the minimum value for nullity of A?
Your answer:
Q13MCQ1 Mark·Choose one option
Asked 2 times Key ↓
Choose the correct option for T.
1. AT may be one-one but not onto.
2. BT may be onto but not one-one.
3. CT must be both one-one and onto.
4. DT can neither be one-one nor onto.

End Term · 13 Apr 2025 (AN)End Term

Read the passage below, then answer Q14–Q16
Q14NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the maximum rank of A?
Your answer:
Q15NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the minimum value for nullity of A?
Your answer:
Q16MCQ1 Mark·Choose one option
Asked 2 times Key ↓
Choose the correct option for T.
1. AT may be one-one but not onto.
2. BT may be onto but not one-one.
3. CT must be both one-one and onto.
4. DT can neither be one-one nor onto.

End Term · 22 Dec 2024 (FN)End Term

Read the passage below, then answer Q17
Q17MCQ1 Mark·Choose one option
Asked 2 times Key ↓
The transformation T is
1. Abijective
2. Binjective but not surjective
3. Csurjective but not injective
4. Dneither injective nor surjective

End Term · 22 Dec 2024 (AN)End Term

Read the passage below, then answer Q18
Q18MCQ1 Mark·Choose one option
Asked 2 times Key ↓
The transformation T is
1. Abijective
2. Binjective but not surjective
3. Csurjective but not injective
4. Dneither injective nor surjective

Official Key

Master Answer Key & Explanations

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

Reveal Key

Q1A, C

Q2A, B

Q35

Q4A, D

Q5-2

Q64

Q7D

Q81

Q9B

Q10D, F

Q113

Q121

Q13B

Q142

Q151

Q16B

Q17B

Q18B

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.