Consider the function defined as follows. Choose the correct option. Consider the function defined as follows.

(B) is not defined for any non-zero .

Let and . Based on the above data, answer the given subquestions. Let Let Choose all the correct options from the following:

(A) and are neither equivalent nor similar.

(C) The columns of are linearly dependent and the columns of are linearly independent.

(B) has a unique solution but does not have a unique solution.

(D) Let , and be the vectors representing the columns of . The vectors , and lie on the same plane.

Let and . Based on the above data, answer the given subquestions. Let Let Choose all the correct options from the following:

(B) has a unique solution but does not have a unique solution.

(D) Let , and be the vectors representing the columns of . The vectors , and lie on the same plane.

(A) and are equivalent but not similar.

(C) The columns of are linearly dependent and the columns of are linearly independent.

January 2026 term · Mathematics for Data Science II · BSMA1003

Mathematics for Data Science II End Term: 10 May 2026 (January 2026 term)

The IIT Madras BS Mathematics for Data Science II (Maths 2) End Term paper sat on 10 May 2026, in the January 2026 term: 46 questions for 208 marks in 180 minutes. Every question is below with its answer. Take it as a timed mock test to be marked, or read it through first.

Questions: 46
Marks: 208
Duration: 180 min
MCQ: 8
MSQ: 8
Written: 30

Take as mock test Practise with answers

Updated 14 Sept 2026

Official paper: Maths2 06 May 26 · No negative marking.

Question 1

+5 marksOne correct option

Consider the function defined as follows.

Choose the correct option.

A
.
B
is not defined for any non-zero .
C
.
D
exists for any non-zero .

Show answer

Correct answer

D
exists for any non-zero .

Question 1 on its own page

Question 2

+5 marksOne correct option

Let be a linear transformation given by , and . Choose the correct option from the following:

Show answer

Correct answer

Question 2 on its own page

Question 3

+5 marksOne correct option

Let be a square matrix of order . If , which of the following sequences of row operations can be performed on so that the resulting matrix has determinant ?

A
Interchange two rows of and then multiply one row by
B
Multiply one row of by .
C
Interchange two rows of and then multiply one row by .
D
Multiply all rows of by .

Show answer

Correct answer

C
Interchange two rows of and then multiply one row by .

Question 3 on its own page

Question 4

+5 marksOne correct option

Consider the function defined as follows.

Choose the correct option.

Show answer

Correct answer

Question 4 on its own page

Question 5

+6 marksOne correct option

Let be a linear transformation. Let and Suppose that the matrix representation of with respect to ordered bases for the domain and for the co-domain is given by

. Then which of the following defines the transformation ?

A
—
B
—
C
—
D
—

Show answer

Correct answer

B
—

Question 5 on its own page

Question 6

+5 marksOne correct option

Let be a linear transformation defined by . Let and Choose the correct option for the matrix representation of with respect to ordered bases for the domain and for the co-domain.

Show answer

Correct answer

Question 6 on its own page

Question 7

+6 marksOne or more correct options

Define a function

Select all that apply.

A
is continuous at .
B
.
C
.
D
is differentiable at .

Show answer

Correct answers

A
is continuous at .
B
.
D
is differentiable at .

Question 7 on its own page

Question 8

+6 marksOne or more correct options

Define a function

Choose the correct options.

Select all that apply.

A
is continuous at .
B
.
C
.
D
is differentiable at .

Show answer

Correct answers

A
is continuous at .
B
.
D
is differentiable at .

Question 8 on its own page

Question 9

+6 marksOne or more correct options

Choose all the subspaces isomorphic to .

Select all that apply.

A
.
B
.
C
.
D
—

Show answer

Correct answers

A
.
B
.
D
—

Question 9 on its own page

Question 10

+7 marksOne or more correct options

Choose all the subspaces isomorphic to .

Select all that apply.

A
.
B
.
C
.
D
is the column space of the matrix .
E
.

Show answer

Correct answers

A
.
D
is the column space of the matrix .
E
.

Question 10 on its own page

Question 11

+5 marksWritten answer

If is an orthogonal matrix of order 5, then find the nullity of the matrix .

Show answer

A written answer, not marked automatically.

Question 11 on its own page

Question 12

+5 marksWritten answer

Let be a vector such that . If is the vector obtained from after the anti- clockwise rotation in the -plane by an angle about the -axis, then find the length of the vector .

Show answer

A written answer, not marked automatically.

Question 12 on its own page

Question 13

+5 marksWritten answer

Let and be two vectors in . If and are orthogonal to each other, find the value of .

Show answer

A written answer, not marked automatically.

Question 13 on its own page

Question 14

+5 marksWritten answer

Let and be two vectors in such that and are perpendicular to each other. Find the value of .

Show answer

A written answer, not marked automatically.

Question 14 on its own page

Question 15

+5 marksWritten answer

Consider the function defined as If the unit direction of the steepest ascent of at is , then find the value of ?

Show answer

A written answer, not marked automatically.

Question 15 on its own page

Question 16

+5 marksWritten answer

Consider a function defined as . If the unit direction of the steepest descent of at is , then find the value of ?

Show answer

A written answer, not marked automatically.

Question 16 on its own page

Question 17

+6 marksWritten answer

Let be a linear transformation given by , and . If , find .

Show answer

A written answer, not marked automatically.

Question 17 on its own page

Question 18

+6 marksWritten answer

is a square matrix of order 4. When the following sequence of row operations are performed on , the result is the identity matrix: Swap the third and fourth rows. Add two times the second row to the third row. Divide the third row by 6. Multiply the first row by 3. Compute the determinant of .

Show answer

A written answer, not marked automatically.

Question 18 on its own page

Question 19

+6 marksOne or more correct options

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

Which of the following points lie in ?

Select all that apply.

Show answer

Correct answers

Question 19 on its own page

Question 20

+6 marksOne or more correct options

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

Which of the following points are in ?

Select all that apply.

Show answer

Correct answers

Question 20 on its own page

Question 21

+4 marksWritten answer

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

What is the dimension of the subspace corresponding to the affine subspace ?

Show answer

A written answer, not marked automatically.

Question 21 on its own page

Question 22

+4 marksWritten answer

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

What is the dimension of the subspace corresponding to the affine subspace ?

Show answer

A written answer, not marked automatically.

Question 22 on its own page

Question 23

+7 marksWritten answer

Consider the function defined as

Based on the above data, answer the given subquestions.

Consider the following statements about the function . S1: The range of is the subset of . S2: The gradient of is defined at all points in . S3: The directional derivative of in the direction of at is . What is the number of correct statements? Write 0 if none of the statements is correct.

Show answer

A written answer, not marked automatically.

Question 23 on its own page

Question 24

+3 marksOne correct option

Allowed : No Group Comprehension Questions : No Question Pattern Type : NonMatrix Consider the following domain Suppose is a function defined by Based on the above data, answer the given subquestions.

Which of the following options depict the correct geometric shape of the boundary of ?

A
A square.
B
A rectangle.
C
A triangle.
D
A circle.

Show answer

Correct answer

B
A rectangle.

Question 24 on its own page

Question 25

+5 marksWritten answer

Consider the function defined as

Based on the above data, answer the given subquestions.

Suppose function describes the amount of rain at a point on a map. If along the unit direction the amount of rain remains constant starting at point , then find .

Show answer

A written answer, not marked automatically.

Question 25 on its own page

Question 26

+4 marksWritten answer

Suppose denotes the absolute maximum value. Find .

Show answer

A written answer, not marked automatically.

Question 26 on its own page

Question 27

+3 marksOne correct option

Consider the following domain Suppose is a function defined by Based on the above data, answer the given subquestions.

Which of the following options depicts the correct geometric shape of the boundary of ?

A
A triangle.
B
A rectangle.
C
A pentagon.
D
A hexagon.

Show answer

Correct answer

A
A triangle.

Question 27 on its own page

Question 28

+4 marksWritten answer

Suppose denote the absolute minimum value. Find .

Show answer

A written answer, not marked automatically.

Question 28 on its own page

Question 29

+4 marksWritten answer

Consider the following domain Suppose is a function defined by Based on the above data, answer the given subquestions.

What is the number of critical points inside the domain of ? Write zero if there is no critical point.

Show answer

A written answer, not marked automatically.

Question 29 on its own page

Question 30

+7 marksWritten answer

Consider the function defined as Based on the above data, answer the given subquestions.

Consider the following statements about the function . S1: The range of contains . S2: The domain of the gradient is . S3: The directional derivative of in the direction of at is . What is the number of correct statements? Write 0 if none of the statements is correct.

Show answer

A written answer, not marked automatically.

Question 30 on its own page

Question 31

+6 marksWritten answer

Consider the following domain Suppose is a function defined by Based on the above data, answer the given subquestions.

Find the absolute maximum of over the domain .

Show answer

A written answer, not marked automatically.

Question 31 on its own page

Question 32

+7 marksWritten answer

Consider the function defined as Based on the above data, answer the given subquestions.

Suppose the function describes the pressure in a room at point . If the rate of change of pressure at in the direction of the unit vector is . Find up to two decimal points.

Show answer

A written answer, not marked automatically.

Question 32 on its own page

Question 33

+4 marksWritten answer

Let

and

. Based on the above data, answer the given subquestions.

Find the rank of .

Show answer

A written answer, not marked automatically.

Question 33 on its own page

Question 34

+4 marksWritten answer

Let

and

. Based on the above data, answer the given subquestions.

Find the determinant of .

Show answer

A written answer, not marked automatically.

Question 34 on its own page

Question 35

+4 marksWritten answer

Let

and

. Based on the above data, answer the given subquestions.

Find the determinant of .

Show answer

A written answer, not marked automatically.

Question 35 on its own page

Question 36

+4 marksWritten answer

Let

and

. Based on the above data, answer the given subquestions.

Find the rank of .

Show answer

A written answer, not marked automatically.

Question 36 on its own page

Question 37

+8 marksOne or more correct options

Let

and

. Based on the above data, answer the given subquestions.

Choose all the correct options from the following:

Select all that apply.

A
and are neither equivalent nor similar.
B
has a unique solution but does not have a unique solution.
C
The columns of are linearly dependent and the columns of are linearly independent.
D
Let , and be the vectors representing the columns of . The vectors , and lie on the same plane.

Show answer

Correct answers

A
and are neither equivalent nor similar.
C
The columns of are linearly dependent and the columns of are linearly independent.

Question 37 on its own page

Question 38

+8 marksOne or more correct options

Let

and

. Based on the above data, answer the given subquestions.

Choose all the correct options from the following:

Select all that apply.

A
and are equivalent but not similar.
B
has a unique solution but does not have a unique solution.
C
The columns of are linearly dependent and the columns of are linearly independent.
D
Let , and be the vectors representing the columns of . The vectors , and lie on the same plane.

Show answer

Correct answers

B
has a unique solution but does not have a unique solution.
D
Let , and be the vectors representing the columns of . The vectors , and lie on the same plane.

Question 38 on its own page

Question 39

+1 markWritten answer

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 39 on its own page

Question 40

+1 markWritten answer

Consider the function defined as

Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 40 on its own page

Question 41

+1 markWritten answer

Consider the following domain Suppose is a function defined by Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 41 on its own page

Question 42

+1 markWritten answer

Let

and

. Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 42 on its own page

Question 43

+1 markWritten answer

Let be the real valued function defined by for all . Let denote the tangent plane to the graph of at the point . Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 43 on its own page