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Maths II · Most repeated questions

148 questions · 240 marks · 8 papers · 65 question types

Indian Institute of Technology Madras

BS Degree Programme · Examination & Study Paper

Mathematics for Data Science II

Most repeated questions

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.

Type 1 · asked 6 times · Critical points, Hessian & optimisation

Use the Hessian test to classify a critical point

Read the passage below, then answer Q1
Based on the above data, answer the given subquestions.
Q1MCQ2 Marks·Choose one option
End Term · 22 Dec 2024 (AN)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 Q2
Based on the above data answer the given subquestions.
Q2MCQ2 Marks·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D
5. E
Read the passage below, then answer Q3
Based on the above data, answer the given subquestions.
Q3MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)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 Q4
Based on the above data, answer the given subquestions.
Q4MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)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 Q5
Based on the above data, answer the given subquestions.
Q5MCQ1 Mark·Choose one option
End Term · 31 Aug 2025 (AN)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.
Read the passage below, then answer Q6
Q6MCQ1 Mark·Choose one option
End Term · 31 Aug 2025 (FN)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.

Type 2 · asked 5 times · Multivariable functions: domain & range

Find the range of f on a given domain D

Read the passage below, then answer Q7
Based on the above data, answer the given subquestions.
Q7MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q8–Q9
Based on the above data answer the given subquestions.
Q8MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D
Q9MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q10
Based on the above data, answer the given subquestions.
Q10MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q11
Based on the above data, answer the given subquestions.
Q11MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 3 · asked 4 times · Critical points, Hessian & optimisation

Write phi, the squared distance from a point to the graph of f

Read the passage below, then answer Q12
Based on the above data, answer the given subquestions.
Q12MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q13
Based on the above data, answer the given subquestions.
Q13MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q14
Based on the above data, answer the given subquestions.
Q14MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q15
Q15MCQ1 Mark·Choose one option
End Term · 31 Aug 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 4 · asked 4 times · Critical points, Hessian & optimisation

Find the minimum distance from a point to a graph or subspace

Read the passage below, then answer Q16
Based on the above data, answer the given subquestions.
Q16NAT2 Marks·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q17
Based on the above data, answer the given subquestions.
Q17NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q18
Based on the above data, answer the given subquestions.
Q18NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:
Read the passage below, then answer Q19
Q19NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 5 · asked 3 times · Basis, dimension & rank

Find the dimension of a subspace given by linear conditions

Q20MCQ2 Marks·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Q21MCQ2 Marks·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. Adim(W) = 1
2. Bdim(W) = 2
3. Cdim(W) = 3
4. Ddim(W) = 4
Q22NAT4 Marks·Write the number
Quiz 1 · 26 Oct 2025 (AN)Key ↓
Your answer:

Type 6 · asked 3 times · Linear maps & their matrices

Write the matrix of T with respect to standard bases

Read the passage below, then answer Q23
Based on the above data, answer the given subquestions.
Q23MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)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 Q24
Based on the above data, answer the given subquestions.
Q24MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)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 Q25
Based on the above data, answer the given subquestions.
Q25MCQ2 Marks·Choose one option
Quiz 2 · 3 Aug 2025 (AN)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

Type 7 · asked 3 times · Inner products & projections

Find the projection of a vector onto a line through the origin

Q26NAT2 Marks·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Q27NAT2 Marks·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:
Q28NAT3 Marks·Write the number
Quiz 2 · 3 Aug 2025 (AN)Key ↓
Your answer:

Type 8 · asked 3 times · Inner products & projections

Find the foot of the perpendicular from a point to a line

Read the passage below, then answer Q29
Q29NAT3 Marks·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q30–Q31
Q30NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:
Q31NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 9 · asked 3 times · Multivariable functions: domain & range

Decide if f is injective on a given domain

Read the passage below, then answer Q32
Based on the above data, answer the given subquestions.
Q32MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q33
Based on the above data, answer the given subquestions.
Q33MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q34
Based on the above data, answer the given subquestions.
Q34MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. ATRUE
2. BFALSE

Type 10 · asked 3 times · Partial & directional derivatives

Given a directional derivative value, find the unknown in f or point

Read the passage below, then answer Q35
Based on the above data answer the given subquestions.
Q35NAT1 Mark·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:
Read the passage below, then answer Q36
Based on the above data, answer the given subquestions.
Q36NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:
Read the passage below, then answer Q37
Based on the above data, answer the given subquestions.
Q37NAT1 Mark·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:

Type 11 · asked 3 times · Partial & directional derivatives

Find a direction along which there is no change in f

Read the passage below, then answer Q38
Based on the above data, answer the given subquestions.
Q38NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q39
Based on the above data, answer the given subquestions.
Q39NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:
Read the passage below, then answer Q40
Based on the above data, answer the given subquestions.
Q40NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)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:

Type 12 · asked 2 times · Determinants & their properties

Choose row operations that change det(A) to a given new value

Q41MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (AN)Key ↓
1. AAdd second row to fourth row and multiply second row with −5.
2. BMultiply matrix A with −5.
3. CInterchange first and third row and multiply fourth row with 5.
4. DMultiply second row of A with 5 and multiply third row with -1.
Q42MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (FN)Key ↓
1. AMultiply first row of A with 2 and interchange second and third row.
2. BMultiply matrix A with −2.
3. CAdd third row to first row and multiply second row with −2.
4. DInterchange first and second row and multiply third row with −2.

Type 13 · asked 2 times · Determinants & their properties

Find det(B) when B is A with scaled rows and columns

Q43NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Q44NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 14 · asked 2 times · Determinants & their properties

Find det(A^n - A^(n-1)) for a given 2x2 matrix

Q45NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Q46NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 15 · asked 2 times · Determinants & their properties

Find det(AB) when A^2 = 2I, B^2 = 3I and the signs of det are given

Q47NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Q48NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 16 · asked 2 times · Systems of linear equations

Find k for which Ax = 0 has infinitely many solutions

Read the passage below, then answer Q49
Q49NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q50
Q50NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 17 · asked 2 times · Systems of linear equations

For given k and b with unknown alpha, decide if Ax = b has solutions

Read the passage below, then answer Q51
Q51MCQ1.5 Marks·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q52
Q52MCQ1.5 Marks·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 18 · asked 2 times · Span & linear independence

True/false: rows or columns of A span R^2 for values of k

Read the passage below, then answer Q53
Q53MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q54
Q54MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. ATRUE
2. BFALSE

Type 19 · asked 2 times · Span & linear independence

True/false: rank of A is 1 for some k

Read the passage below, then answer Q55
Q55MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q56
Q56MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. ATRUE
2. BFALSE

Type 20 · asked 2 times · Span & linear independence

True/false: for given k the row/column vectors span a given plane

Read the passage below, then answer Q57
Q57MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. ATRUE
2. BFALSE
Read the passage below, then answer Q58
Q58MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. ATRUE
2. BFALSE

Type 21 · asked 2 times · Systems of linear equations

True/false statements on solutions of Ax = b and Ax = 0

Q59MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (AN)Key ↓
Which of the following options is/are true?
1. A
2. B
3. C
4. D
Q60MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)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.

Type 22 · asked 2 times · Vector spaces & subspaces

Which sets with usual operations are vector spaces

Q61MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Q62MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (FN)Key ↓
Assume the usual vector addition and scalar multiplication for all the options below. Choose the correct option(s) from the following statements.
1. A
2. B
3. C
4. D

Type 23 · asked 2 times · Span & linear independence

Find all alpha that make a set of three vectors in R^3 dependent

Q63MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Q64MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 24 · asked 2 times · Span & linear independence

Find the parameter so that v3 lies in span{v1, v2}

Read the passage below, then answer Q65
Based on the above data, answer the given subquestions.
Q65NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q66
Based on the above data, answer the given subquestions.
Q66NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 25 · asked 2 times · Span & linear independence

Write v3 = a v1 + b v2 and find a combination of a and b

Read the passage below, then answer Q67
Based on the above data, answer the given subquestions.
Q67NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q68
Based on the above data, answer the given subquestions.
Q68NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 26 · asked 2 times · Basis, dimension & rank

Find the maximum possible rank of a nonzero A with A^2 = 0

Q69NAT2 Marks·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Q70NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 27 · asked 2 times · Basis, dimension & rank

Row space and column space of AB compared with those of A and B

Q71MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (AN)Key ↓
Which of the following options are correct?
1. A
2. B
3. C
4. D
Q72MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (FN)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.

Type 28 · asked 2 times · Linear maps & their matrices

Choose the correct formula of T from given data

Read the passage below, then answer Q73
Q73MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
Read the passage below, then answer Q74
Q74MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
Choose the correct definition for T.
1. A
2. B
3. C

Type 29 · asked 2 times · Linear maps & their matrices

Find the rank of the matrix of T in a non-standard ordered basis

Read the passage below, then answer Q75
Q75NAT1 Mark·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q76
Q76NAT1 Mark·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 30 · asked 2 times · Linear maps & their matrices

Find the determinant of the matrix of T in a non-standard basis

Read the passage below, then answer Q77
Q77NAT1 Mark·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Find the determinant of B.
Your answer:
Read the passage below, then answer Q78
Q78NAT1 Mark·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Find the determinant of B.
Your answer:

Type 31 · asked 2 times · Linear maps & their matrices

Statements about the matrix of T in some ordered basis

Read the passage below, then answer Q79
Q79MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q80
Q80MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (FN)Key ↓
Let A be the matrix representation of T with respect to some ordered basis for both the domain and the codomain. Which of the following options is correct:
1. A
2. B
3. C
4. D

Type 32 · asked 2 times · Linear maps & their matrices

Find the expression for T(x, y) from given images of vectors

Read the passage below, then answer Q81
Based on the above data, answer the given subquestions.
Q81MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q82
Based on the above data, answer the given subquestions.
Q82MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 33 · asked 2 times · Kernel, range & isomorphisms

Decide if a linear map T is injective, surjective or bijective

Read the passage below, then answer Q83
Q83MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
The transformation T is
1. Abijective
2. Binjective but not surjective
3. Csurjective but not injective
4. Dneither injective nor surjective
Read the passage below, then answer Q84
Q84MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
The transformation T is
1. Abijective
2. Binjective but not surjective
3. Csurjective but not injective
4. Dneither injective nor surjective

Type 34 · asked 2 times · Kernel, range & isomorphisms

Find the maximum rank of A for a map between given spaces

Read the passage below, then answer Q85
Q85NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
What is the maximum rank of A?
Your answer:
Read the passage below, then answer Q86
Based on the above data, answer the given subquestions.
Q86NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
What is the maximum rank of A?
Your answer:

Type 35 · asked 2 times · Kernel, range & isomorphisms

Find the minimum nullity of A for a map between given spaces

Read the passage below, then answer Q87
Q87NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
What is the minimum value for nullity of A?
Your answer:
Read the passage below, then answer Q88
Based on the above data, answer the given subquestions.
Q88NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
What is the minimum value for nullity of A?
Your answer:

Type 36 · asked 2 times · Kernel, range & isomorphisms

Decide if T may be one-one or onto from the dimensions

Read the passage below, then answer Q89
Q89MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)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 Q90
Based on the above data, answer the given subquestions.
Q90MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)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.

Type 37 · asked 2 times · Kernel, range & isomorphisms

Choose T(0,0,1) so that T has the required rank or range

Q91MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (AN)Key ↓
1. A(0, 0, 1)
2. B(1, 1, 2)
3. C(1,−2, 1)
4. D(−4,−1,−1)
Q92MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (FN)Key ↓
1. A(0, 0, 0)
2. B(−1, 1, 7)
3. C(−1, 5, 5)
4. D(1,−1,−3)

Type 38 · asked 2 times · Linear maps & their matrices

Statements about two linear maps having similar matrices

Q93MCQ3 Marks·Choose one option
End Term · 31 Aug 2025 (AN)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.
Q94MCQ3 Marks·Choose one option
End Term · 31 Aug 2025 (FN)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.

Type 39 · asked 2 times · Basis, dimension & rank

Find the dimension of subspace W from given data

Read the passage below, then answer Q95
Based on the above data, answer the given subquestions.
Q95NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Find the dimension of W.
Your answer:
Read the passage below, then answer Q96
Based on the above data, answer the given subquestions.
Q96NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Find the dimension of W.
Your answer:

Type 40 · asked 2 times · Basis, dimension & rank

Find the set W intersect U where U is the plane x + y + z = 0

Read the passage below, then answer Q97
Based on the above data, answer the given subquestions.
Q97MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q98
Based on the above data, answer the given subquestions.
Q98MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 41 · asked 2 times · Basis, dimension & rank

Choose the correct option for a 3x3 matrix with given first columns

Read the passage below, then answer Q99
Based on the above data, answer the given subquestions.
Q99MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q100
Based on the above data, answer the given subquestions.
Q100MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 42 · asked 2 times · Basis, dimension & rank

Which vector spaces are isomorphic to the given subspace W

Read the passage below, then answer Q101
Based on the above data, answer the given subquestions.
Q101MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (AN)Key ↓
Which of the following vector spaces is/are isomorphic to W?
1. A
2. B
3. C
4. D
Read the passage below, then answer Q102
Based on the above data, answer the given subquestions.
Q102MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)Key ↓
Which of the following vector spaces is/are isomorphic to W?
1. A
2. B
3. C
4. D

Type 43 · asked 2 times · Inner products & projections

Find a + b + c for the projection P_W of a given vector

Read the passage below, then answer Q103
Q103NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q104
Q104NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 44 · asked 2 times · Inner products & projections

Find a vector in the kernel of the projection P_W

Read the passage below, then answer Q105
Q105NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q106
Q106NAT1.5 Marks·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 45 · asked 2 times · Inner products & projections

Find the dimension of the orthogonal complement of W

Read the passage below, then answer Q107
Q107NAT1 Mark·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q108
Q108NAT1 Mark·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 46 · asked 2 times · Inner products & projections

True/false statements about the projection onto a subspace

Q109MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Q110MSQ3 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 47 · asked 2 times · Inner products & projections

Find a line through the origin perpendicular to a given line

Read the passage below, then answer Q111
Q111NAT3 Marks·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q112
Q112NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 48 · asked 2 times · Inner products & projections

Find the angle between two lines (cos or 1/cos^2)

Read the passage below, then answer Q113
Q113NAT2 Marks·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q114
Q114NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 49 · asked 2 times · Inner products & projections

Statements about 2x2 orthogonal matrices O(2)

Q115MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
5. E
Q116MSQ3 Marks·Choose all correct options
End Term · 31 Aug 2025 (FN)Key ↓
1. A
2. B
3. C
4. D
5. E

Type 50 · asked 2 times · Multivariable functions: domain & range

Find the largest valid domain for a given function f

Read the passage below, then answer Q117
Based on the above data, answer the given subquestions.
Q117MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Read the passage below, then answer Q118
Based on the above data answer the given subquestions.
Q118MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 51 · asked 2 times · Multivariable functions: domain & range

Find the minimum number of points to remove from R^2 for a domain

Read the passage below, then answer Q119
Based on the above data, answer the given subquestions.
Q119MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A0
2. B1
3. CInfinitely many
Read the passage below, then answer Q120
Based on the above data, answer the given subquestions.
Q120MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A0
2. B1
3. CInfinitely many

Type 52 · asked 2 times · Limits & continuity

Find k so that the limit of f at (0, 0) exists or f is continuous

Read the passage below, then answer Q121
Based on the above data, answer the given subquestions.
Q121NAT1 Mark·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q122
Based on the above data answer the given subquestions.
Q122NAT1 Mark·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 53 · asked 2 times · Limits & continuity

Choose functions whose limit at (0, 0) exists

Q123MSQ2 Marks·Choose all correct options
End Term · 13 Apr 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Q124MSQ2 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 54 · asked 2 times · Partial & directional derivatives

Statements about the directional derivative of f at a point

Q125MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Q126MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 55 · asked 2 times · Partial & directional derivatives

Statements about a piecewise function with a sin(1/r) factor

Q127MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (AN)Key ↓
1. A
2. B
3. C
4. D
Q128MSQ3 Marks·Choose all correct options
End Term · 22 Dec 2024 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 56 · asked 2 times · Partial & directional derivatives

Find the gradient of f at a point and combine its parts

Read the passage below, then answer Q129
Based on the above data, answer the given subquestions.
Q129NAT1 Mark·Write the number
End Term · 22 Dec 2024 (AN)Key ↓
Your answer:
Read the passage below, then answer Q130
Based on the above data answer the given subquestions.
Q130NAT2 Marks·Write the number
End Term · 22 Dec 2024 (FN)Key ↓
Your answer:

Type 57 · asked 2 times · Partial & directional derivatives

Find dim(Span(S)) for unit directions with no change in f

Read the passage below, then answer Q131
Based on the above data, answer the given subquestions.
Q131NAT1 Mark·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q132
Based on the above data, answer the given subquestions.
Q132NAT1 Mark·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Let S denote the set of all unit directions along which there is no change in f. Find dim(Span(S)).
Your answer:

Type 58 · asked 2 times · Partial & directional derivatives

Find a point on the tangent line to the graph in a given direction

Read the passage below, then answer Q133
Based on the above data, answer the given subquestions.
Q133NAT2 Marks·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q134
Based on the above data, answer the given subquestions.
Q134NAT2 Marks·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 59 · asked 2 times · Partial & directional derivatives

Use tangent lines in given directions to find a derivative of f

Q135NAT2 Marks·Write the number
End Term · 31 Aug 2025 (AN)Key ↓
Your answer:
Q136NAT2 Marks·Write the number
End Term · 31 Aug 2025 (FN)Key ↓
Your answer:

Type 60 · asked 2 times · Partial & directional derivatives

Statements implied by or needed for differentiability of f

Q137MSQ2 Marks·Choose all correct options
End Term · 31 Aug 2025 (AN)Key ↓
1. A
2. B
3. C
4. D
Q138MSQ2 Marks·Choose all correct options
End Term · 31 Aug 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 61 · asked 2 times · Critical points, Hessian & optimisation

Count the critical points of a function

Read the passage below, then answer Q139
Based on the above data, answer the given subquestions.
Q139MCQ1 Mark·Choose one option
End Term · 22 Dec 2024 (AN)Key ↓
How many critical points does the function f have?
1. A0
2. B1
3. C2
4. DInfinitely many
Read the passage below, then answer Q140
Based on the above data answer the given subquestions.
Q140MCQ2 Marks·Choose one option
End Term · 22 Dec 2024 (FN)Key ↓
1. A0
2. B1
3. C2
4. DInfinitely many

Type 62 · asked 2 times · Critical points, Hessian & optimisation

Choose which given points are critical points of f

Read the passage below, then answer Q141
Based on the above data, answer the given subquestions.
Q141MSQ2 Marks·Choose all correct options
End Term · 13 Apr 2025 (AN)Key ↓
1. A(1, 0)
2. B(0, 1)
3. C(−1, 1)
4. D(1,−1)
Read the passage below, then answer Q142
Based on the above data, answer the given subquestions.
Q142MSQ2 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 63 · asked 2 times · Critical points, Hessian & optimisation

Find the determinant of the Hessian of f at (0, 0)

Read the passage below, then answer Q143
Based on the above data, answer the given subquestions.
Q143NAT1 Mark·Write the number
End Term · 13 Apr 2025 (AN)Key ↓
Your answer:
Read the passage below, then answer Q144
Based on the above data, answer the given subquestions.
Q144NAT1 Mark·Write the number
End Term · 13 Apr 2025 (FN)Key ↓
Your answer:

Type 64 · asked 2 times · Critical points, Hessian & optimisation

Find the critical points of the squared distance function phi

Read the passage below, then answer Q145
Based on the above data, answer the given subquestions.
Q145MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
1. A0
2. B1
3. C2
4. DInfinitely many
Read the passage below, then answer Q146
Based on the above data, answer the given subquestions.
Q146MCQ1 Mark·Choose one option
End Term · 13 Apr 2025 (FN)Key ↓
1. A
2. B
3. C
4. D

Type 65 · asked 2 times · Critical points, Hessian & optimisation

Statements about global minima and maxima of phi on D

Read the passage below, then answer Q147
Based on the above data, answer the given subquestions.
Q147MCQ2 Marks·Choose one option
End Term · 13 Apr 2025 (AN)Key ↓
Choose the correct statement from the following.
1. A
2. B
3. C
4. D
Read the passage below, then answer Q148
Based on the above data, answer the given subquestions.
Q148MSQ2 Marks·Choose all correct options
End Term · 13 Apr 2025 (FN)Key ↓
Choose the correct statements from the following.
1. A
2. B
3. C
4. D
5. E

Official Key

Master Answer Key & Explanations

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

Reveal Key

Q39-0.5

Q41A, C, D

Q42A, C

Q43-30

Q47-36

Q48-36

Q59A, D

Q60A, D

Q61A, B

Q62B, C

Q71B, C

Q72B, C

Q79A, C, D

Q80A, C, D

Q91A, B

Q92A, C

Q101A, D

Q102A, D

Q109A, C, D

Q110A, C, D

Q115C, D, E

Q116B, D, E

Q123B, C

Q124A, D

Q125B, D

Q126B, C

Q127A, B

Q128A, C

Q137A, B, D

Q138A, C

Q141C, D

Q142A, D

Q148A, B, E

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.