Figure from the original question paper

(A) Option figure from the original question paper

(B) Option figure from the original question paper

(D) Option figure from the original question paper

(C) Option figure from the original question paper

Figure from the original question paper

(C) Option figure from the original question paper

(D) Option figure from the original question paper

(E) Option figure from the original question paper

(A) Option figure from the original question paper

(B) Option figure from the original question paper

Figure from the original question paper

(B) Option figure from the original question paper

(D) Option figure from the original question paper

(A) Option figure from the original question paper

(C) Option figure from the original question paper

(E) Option figure from the original question paper

Figure of the shaded region D bounded by y = -x^2 + 1 and the x-axis, with the temperature function and notation for interior, boundary components and corners Based on the above data, answer the given subquestions. Which of the following is the only critical point of T in Int( D )?

(B) Option figure from the original question paper

(A) Option figure from the original question paper

(C) Option figure from the original question paper

(D) Option figure from the original question paper

Figure of the shaded region D bounded by y = -x^2 + 1 and the x-axis, with the temperature function and notation for interior, boundary components and corners Based on the above data, answer the given subquestions. What can be concluded by the Hessian test for T at its critical point in Int( D )?

(C) The critical point is a saddle point.

(A) The critical point is a local minimum.

(B) The critical point is a local maximum.

(D) The Hessian test is inconclusive.

May 2025 term · Mathematics for Data Science II · BSMA1003

Mathematics for Data Science II End Term: 31 August 2025, Set QDF3 (May 2025 term)

The IIT Madras BS Mathematics for Data Science II (Maths 2) End Term paper sat on 31 Aug 2025, in the May 2025 term, set QDF3: 26 questions for 50 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: 26
Marks: 50
Duration: 180 min
MCQ: 6
MSQ: 6
Numerical: 14

Take as mock test Practise with answers

Updated 25 Sept 2026

Official paper: IIT M FOUNDATION AN EXAM QDF3 31 Aug 2025 · No negative marking.

Question 1

+3 marksOne correct option

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined as:

$T(x,y,z) = (x + y - z, ay + 2bz, x - y + z)$

Let $A$ be the matrix representation of $T$ with respect to the standard bases of $\mathbb{R}^3$ and let $B = \begin{bmatrix} 3 & 0 & 3 \\ 1 & 0 & 1 \\ 1 & 0 & 3 \end{bmatrix}$ .

Consider the following statements:

S1: If $A$ and $B$ are similar, then the range of the linear transformation $T$ is a plane in $\mathbb{R}^3$ .

S2: If $a = 2$ and $b = -1$ , then $A$ and $B$ are equivalent matrices, but not similar.

Which of the following options is true?

A
S1 is true and S2 is false.
B
S1 is false and S2 is true.
C
Both S1 and S2 are true.
D
Both S1 and S2 are false.

Show answer

Correct answer

C
Both S1 and S2 are true.

Question 1 on its own page

Question 2

+2 marksOne or more correct options

Select all that apply.

Show answer

Correct answers

Question 2 on its own page

Question 3

+3 marksOne or more correct options

Which of the following options are correct?

Select all that apply.

Show answer

Correct answers

Question 3 on its own page

Question 4

+3 marksOne or more correct options

Select all that apply.

A
(0, 0, 1)
B
(1, 1, 2)
C
(1,−2, 1)
D
(−4,−1,−1)

Show answer

Correct answers

A
(0, 0, 1)
B
(1, 1, 2)

Question 4 on its own page

Question 5

+3 marksOne or more correct options

Select all that apply.

Show answer

Correct answers

Question 5 on its own page

Question 6

+3 marksOne or more correct options

Select all that apply.

Show answer

Correct answers

Question 6 on its own page

Question 7

+1 markNumerical answer

Show answer

Correct answer: 1.5

Question 7 on its own page

Question 8

+2 marksNumerical answer

Show answer

Correct answer: 2

Question 8 on its own page

Question 9

+2 marksNumerical answer

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function for which the following is known.

At every point, the partial derivatives of $f$ exist and are continuous.
$f(2,3) = -1$
The tangent line to the graph of $f$ at $(2,3)$ in the direction $(1,0)$ contains the point $(1,3,0)$ .
The tangent line to the graph of $f$ at $(2,3)$ in the direction $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ contains the point $(3,4,-1)$ .

Find the partial derivative of $f$ with respect to $y$ (the second variable) at $(2,3)$ .

Show answer

Correct answer: 1

Question 9 on its own page

Question 10

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

Find the value of k.

Show answer

Correct answer: 5

Question 10 on its own page

Question 11

+1 markNumerical answer

Based on the above data, answer the given subquestions.

Show answer

Correct answer: 3

Question 11 on its own page

Question 12

+1 markOne correct option

Based on the above data, answer the given subquestions.

The directional derivatives of f exist in all directions at the origin.

A
True
B
False

Show answer

Correct answer

A
True

Question 12 on its own page

Question 13

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

Show answer

Correct answer: 0.5

Question 13 on its own page

Question 14

+1 markNumerical answer

Based on the above data, answer the given subquestions.

Show answer

Correct answer: 3

Question 14 on its own page

Question 15

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

Find the value of c.

Show answer

Correct answer: -3

Question 15 on its own page

Question 16

+1 markNumerical answer

Based on the above data, answer the given subquestions.

Show answer

Correct answer: 1

Question 16 on its own page

Question 17

+1 markNumerical answer

Based on the above data, answer the given subquestions.

Show answer

Correct answer: 2

Question 17 on its own page

Question 18

+2 marksOne or more correct options

Based on the above data, answer the given subquestions.

Which of the following are possible candidates for v ?

Select all that apply.

A
(2, 1, 2)
B
(0, 7, 0)
C
(1,−2, 0)
D
(2, 0,−1)

Show answer

Correct answers

A
(2, 1, 2)
C
(1,−2, 0)

Question 18 on its own page

Question 19

+2 marksOne correct option

Based on the above data, answer the given subquestions.

Which of the following is a basis for U ?

A
{(1,−1,−1), (1, 3, 2)}
B
{(1, 2, 3), (−1,−1,−1)}
C
{(0, 1,−1), (0,−1, 1)}
D
{(1, 2, 3), (0,−1, 1)}

Show answer

Correct answer

D
{(1, 2, 3), (0,−1, 1)}

Question 19 on its own page

Question 20

+1 markOne correct option

Based on the above data, answer the given subquestions.

Which of the following is the only critical point of T in Int( D )?

Show answer

Correct answer

Question 20 on its own page

Question 21

+1 markOne correct option

Based on the above data, answer the given subquestions.

What can be concluded by the Hessian test for T at its critical point in Int( D )?

A
The critical point is a local minimum.
B
The critical point is a local maximum.
C
The critical point is a saddle point.
D
The Hessian test is inconclusive.

Show answer

Correct answer

C
The critical point is a saddle point.

Question 21 on its own page

Question 22

+1 markOne correct option

Based on the above data, answer the given subquestions.

Let $f_1$ and $f_2$ denote the one-variable functions obtained by restricting $T$ on $\gamma_1$ and $\gamma_2$ , respectively, i.e.,

$f_1(x) = T(x, -x^2 + 1), \qquad f_2(x) = T(x, 0),$

for $x \in (-1,1)$ . Which of the following statements is true?

Show answer

Correct answer

Question 22 on its own page

Question 23

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

Comparing the critical points on the interior, the boundary components, and the corners, find the temperature at the hottest point on the metallic plate.

Show answer

Correct answer: 10.66 (accepted within ±0.01)

Question 23 on its own page