Indian Institute of Technology Madras

BS Degree Programme · Examination & Study Paper

Mathematics for Data Science II

All 2025 papers

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.

Quiz 1 · 26 Oct 2025 (AN)Quiz 1

Q1MCQ4 Marks·Choose one option
Key ↓
1. A
2. B
3. C
4. D
Q2MCQ6 Marks·Choose one option
Key ↓
1. ABoth S1 and S2 are true.
2. BBoth S1 and S2 are false.
3. CS1 is true and S2 is false.
4. DS1 is false and S2 is true.
Q3MSQ6 Marks·Choose all correct options
Key ↓
Choose all the correct statements from the following.
1. A
2. B
3. C
4. D
5. E
Q4MSQ6 Marks·Choose all correct options
Key ↓
Let A be a square matrix of order n > 1. Select all the statements which imply that det(A) = 0.
1. AAll the entries of one of the columns of A are zero.
2. BOne of the columns of A is a scalar multiple of another column.
3. CThe diagonal entries of A are all zero.
4. DThe determinant of the transpose of A is zero.
5. EA^2 = 0.
Q5NAT3 Marks·Write the number
Key ↓
Your answer:
Q6NAT3 Marks·Write the number
Key ↓
Your answer:
Q7NAT4 Marks·Write the number
Key ↓
Your answer:
Q8NAT4 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Q9NAT4 Marks·Write the number
Key ↓
Let A be a 5 × 7 matrix. If n_1 and n_2 are the minimum and maximum possible values for the rank of A, respectively, then find the value of n_2 − n_1.
Your answer:
Q10NAT4 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q11–Q12
Based on the above data, answer the given subquestions.
Q11MCQ3 Marks·Choose one option
Key ↓
1. ATRUE
2. BFALSE
Q12NAT3 Marks·Write the number
Key ↓
Your answer:

End Term · 31 Aug 2025 (FN)End Term

Q13MCQ3 Marks·Choose one option
Asked 2 times Key ↓
1. AS1 is false and S2 is true.
2. BS1 is true and S2 is false.
3. CBoth S1 and S2 are false.
4. DBoth S1 and S2 are true.
Q14MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q15MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following options are correct? Let A and B be square matrices of the same order.
1. AThe rows of AB are linear combinations of the rows of A.
2. BThe rows of AB are linear combinations of the rows of B.
3. CThe columns of AB are linear combinations of the columns of A.
4. DThe columns of AB are linear combinations of the columns of B.
Q16MSQ3 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)
Q17MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
5. E
Q18MSQ3 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
5. E
Q19NAT1 Mark·Write the number
Key ↓
Your answer:
Q20NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Q21NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q22–Q23
Based on the above data, answer the given subquestions.
Q22NAT2 Marks·Write the number
Key ↓
Find the value of k.
Your answer:
Q23NAT1 Mark·Write the number
Key ↓
Find the dimension of the affine subspace S.
Your answer:
Read the passage below, then answer Q24–Q26
Based on the above data, answer the given subquestions.
Q24MCQ1 Mark·Choose one option
Key ↓
1. ATrue
2. BFalse
Q25NAT2 Marks·Write the number
Key ↓
Your answer:
Q26NAT1 Mark·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q27–Q29
Based on the above data, answer the given subquestions.
Q27NAT2 Marks·Write the number
Asked 3 times Key ↓
If (a, 3,−2) is a vector which lies along a direction in which there is no change in f, then find the value of a.
Your answer:
Q28NAT1 Mark·Write the number
Asked 2 times Key ↓
Let S denote the set of all unit directions along which there is no change in f. Find dim(Span(S)).
Your answer:
Q29NAT1 Mark·Write the number
Key ↓
If the gradient of f at (1,−1, 1) is given by (12, 4, c), then find the value of c.
Your answer:
Read the passage below, then answer Q30–Q31
Based on the above data, answer the given subquestions.
Q30MSQ2 Marks·Choose all correct options
Key ↓
Which of the following points lie on T(1,−1)?
1. A(0, 0, 1)
2. B(0, 1, 0)
3. C(1, 0, 0)
4. D(2, 0, 1)
5. E(1, 1, 1)
Q31MCQ2 Marks·Choose one option
Key ↓
Let W denote the subspace corresponding to the affine subspace T(1,−1). Which of the following is a basis for the orthogonal complement of W?
1. A{(1,−1,−1)}
2. B{(−1,−1,−1)}
3. C{(1, 1,−1)}
4. D{(1,−1, 1)}
Read the passage below, then answer Q32–Q35
Q32MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q33MCQ1 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)}
Q34MCQ1 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.
Q35NAT2 Marks·Write the number
Asked 4 times Key ↓
Your answer:
Read the passage below, then answer Q36–Q39
Q36NAT2 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Q37NAT2 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Q38NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Q39NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:

End Term · 31 Aug 2025 (AN)End Term

Q40MCQ3 Marks·Choose one option
Asked 2 times Key ↓
1. AS1 is true and S2 is false.
2. BS1 is false and S2 is true.
3. CBoth S1 and S2 are true.
4. DBoth S1 and S2 are false.
Q41MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q42MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following options are correct?
1. A
2. B
3. C
4. D
Q43MSQ3 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)
Q44MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
5. E
Q45MSQ3 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
5. E
Q46NAT1 Mark·Write the number
Key ↓
Your answer:
Q47NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Q48NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q49–Q50
Based on the above data, answer the given subquestions.
Q49NAT2 Marks·Write the number
Key ↓
Find the value of k.
Your answer:
Q50NAT1 Mark·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q51–Q53
Based on the above data, answer the given subquestions.
Q51MCQ1 Mark·Choose one option
Key ↓
The directional derivatives of f exist in all directions at the origin.
1. ATrue
2. BFalse
Q52NAT2 Marks·Write the number
Key ↓
Your answer:
Q53NAT1 Mark·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q54–Q56
Based on the above data, answer the given subquestions.
Q54NAT2 Marks·Write the number
Key ↓
Find the value of c.
Your answer:
Q55NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q56NAT1 Mark·Write the number
Asked 3 times Key ↓
Your answer:
Read the passage below, then answer Q57–Q58
Based on the above data, answer the given subquestions.
Q57MSQ2 Marks·Choose all correct options
Key ↓
Which of the following are possible candidates for v ?
1. A(2, 1, 2)
2. B(0, 7, 0)
3. C(1,−2, 0)
4. D(2, 0,−1)
Q58MCQ2 Marks·Choose one option
Key ↓
Which of the following is a basis for U ?
1. A{(1,−1,−1), (1, 3, 2)}
2. B{(1, 2, 3), (−1,−1,−1)}
3. C{(0, 1,−1), (0,−1, 1)}
4. D{(1, 2, 3), (0,−1, 1)}
Read the passage below, then answer Q59–Q62
Based on the above data, answer the given subquestions.
Q59MCQ1 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
Q60MCQ1 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.
Q61MCQ1 Mark·Choose one option
Key ↓
1. A
2. B
3. C
4. D
Q62NAT2 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:
Read the passage below, then answer Q63–Q65
Q63NAT3 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Q64NAT3 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Q65NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:

Quiz 2 · 3 Aug 2025 (AN)Quiz 2

Q66MCQ2 Marks·Choose one option
Key ↓
1. AThe nullity of A is independent of the value of k.
2. BThe nullity of A depends on the value of k, and the set of all possible values for the nullity of A is {1, 2}.
3. CThe nullity of A depends on the value of k, and the set of all possible values for the nullity of A is {0, 1, 2, 3, 4, 5}.
4. DThe nullity of A depends on the value of k, and the set of all possible values for the nullity of A is {2, 3}.
Q67MCQ4 Marks·Choose one option
Key ↓
1. Aa) → iii), b) → i), c) → ii)
2. Ba) → ii), b) → i), c) → iii)
3. Ca) → iii), b) → ii), c) → i)
4. Da) → ii), b) → iii), c) → i)
Q68MSQ6 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
Q69NAT3 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Q70NAT4 Marks·Write the number
Key ↓
Your answer:
Q71NAT5 Marks·Write the number
Key ↓
Your answer:
Q72NAT5 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q73–Q74
Q73MSQ4 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.
Q74NAT3 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q75–Q77
Based on the above data, answer the given subquestions.
Q75NAT2 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:
Q76MCQ2 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.
Q77NAT3 Marks·Write the number
Key ↓
Your answer:
Read the passage below, then answer Q78–Q79
Based on the above data, answer the given subquestions.
Q78MCQ2 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
Q79MSQ5 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

Q80MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q81MCQ2 Marks·Choose one option
Key ↓
1. A
2. B
3. C
4. D
Q82MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q83MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following options is/are true?
1. AIf a system of linear equations Ax = b has a solution, then b is a linear combination of the columns of the coefficient matrix A.
2. BConsider a system of linear equations Ax = b. If b is a linear combination of the columns of the coefficient matrix A, then the system Ax = b has a unique solution.
3. CIf the number of variables is more than the number of equations in a system of linear equations Ax = 0, then it has a unique solution.
4. DIf the columns of a matrix A are linearly independent, then the system Ax = 0 will have a unique solution.
Q84MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q85MSQ3 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
Q86NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q87NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q88NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q89NAT2 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Read the passage below, then answer Q90–Q91
Based on the above data, answer the given subquestions.
Q90NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q91NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q92–Q93
Based on the above data, answer the given subquestions.
Q92MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q93MCQ1 Mark·Choose one option
Asked 3 times Key ↓
Suppose A is the matrix representation of T with respect to the standard basis for both domain and co-domain. Then A is given by
1. A
2. B
3. C
4. D
Read the passage below, then answer Q94–Q96
Based on the above data, answer the given subquestions.
Q94NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the maximum rank of A?
Your answer:
Q95NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the minimum value for nullity of A?
Your answer:
Q96MCQ1 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.
Read the passage below, then answer Q97–Q99
Based on the above data, answer the given subquestions.
Q97MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A0
2. B1
3. CInfinitely many
Q98MCQ1 Mark·Choose one option
Asked 5 times Key ↓
1. A
2. B
3. C
4. D
Q99MCQ1 Mark·Choose one option
Asked 3 times Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q100–Q102
Based on the above data, answer the given subquestions.
Q100NAT1 Mark·Write the number
Key ↓
Your answer:
Q101NAT1 Mark·Write the number
Key ↓
Your answer:
Q102NAT1 Mark·Write the number
Asked 3 times Key ↓
Your answer:
Read the passage below, then answer Q103–Q104
Based on the above data, answer the given subquestions.
Q103NAT1 Mark·Write the number
Asked 3 times Key ↓
Your answer:
Q104NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q105–Q107
Based on the above data, answer the given subquestions.
Q105MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q106NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q107MCQ1 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 Q108–Q111
Based on the above data, answer the given subquestions.
Q108MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q109MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q110MSQ2 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
Q111NAT1 Mark·Write the number
Asked 4 times Key ↓
Your answer:
Read the passage below, then answer Q112–Q115
Based on the above data, answer the given subquestions.
Q112NAT1 Mark·Write the number
Asked 2 times Key ↓
Find the dimension of W.
Your answer:
Q113MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q114MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q115MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following vector spaces is/are isomorphic to W?
1. A
2. B
3. C
4. D

End Term · 13 Apr 2025 (AN)End Term

Q116MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q117MSQ2 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q118MSQ2 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
Q119MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following options is/are true?
1. A
2. B
3. C
4. D
Q120MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q121MSQ3 Marks·Choose all correct options
Key ↓
1. A
2. B
3. C
4. D
Q122NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q123NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q124NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q125NAT2 Marks·Write the number
Asked 3 times Key ↓
Your answer:
Read the passage below, then answer Q126–Q127
Based on the above data, answer the given subquestions.
Q126NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q127NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q128–Q129
Based on the above data, answer the given subquestions.
Q128MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q129MCQ1 Mark·Choose one option
Asked 3 times Key ↓
Suppose A is the matrix representation of T with respect to the standard basis for both domain and co-domain. Then A is given by
1. A
2. B
3. C
4. D
Read the passage below, then answer Q130–Q132
Q130NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the maximum rank of A?
Your answer:
Q131NAT1 Mark·Write the number
Asked 2 times Key ↓
What is the minimum value for nullity of A?
Your answer:
Q132MCQ1 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.
Read the passage below, then answer Q133–Q135
Based on the above data, answer the given subquestions.
Q133MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A0
2. B1
3. CInfinitely many
Q134MCQ1 Mark·Choose one option
Asked 5 times Key ↓
1. A
2. B
3. C
4. D
Q135MCQ1 Mark·Choose one option
Asked 3 times Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q136–Q138
Based on the above data, answer the given subquestions.
Q136NAT1 Mark·Write the number
Key ↓
Your answer:
Q137NAT1 Mark·Write the number
Key ↓
Your answer:
Q138NAT1 Mark·Write the number
Asked 3 times Key ↓
Your answer:
Read the passage below, then answer Q139–Q140
Based on the above data, answer the given subquestions.
Q139NAT1 Mark·Write the number
Key ↓
Your answer:
Q140NAT2 Marks·Write the number
Asked 2 times Key ↓
Your answer:
Read the passage below, then answer Q141–Q143
Based on the above data, answer the given subquestions.
Q141MSQ2 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)
Q142NAT1 Mark·Write the number
Asked 2 times Key ↓
Your answer:
Q143MCQ1 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 Q144–Q147
Based on the above data, answer the given subquestions.
Q144MCQ1 Mark·Choose one option
Asked 4 times Key ↓
1. A
2. B
3. C
4. D
Q145MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A0
2. B1
3. C2
4. DInfinitely many
Q146MCQ2 Marks·Choose one option
Asked 2 times Key ↓
Choose the correct statement from the following.
1. A
2. B
3. C
4. D
Q147NAT1 Mark·Write the number
Asked 4 times Key ↓
Your answer:
Read the passage below, then answer Q148–Q151
Based on the above data, answer the given subquestions.
Q148NAT1 Mark·Write the number
Asked 2 times Key ↓
Find the dimension of W.
Your answer:
Q149MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q150MCQ1 Mark·Choose one option
Asked 2 times Key ↓
1. A
2. B
3. C
4. D
Q151MSQ3 Marks·Choose all correct options
Asked 2 times Key ↓
Which of the following vector spaces is/are isomorphic to W?
1. A
2. B
3. C
4. D