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IITM BS Week 11 Mathematics 2 Questions | Prasnya
Mathematics 2 > Week 11
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40Q
01
Define a function f(x, y) = { (x - y)^2 sin(1/(x-y)) if x != y { 0, if x = y. Choose all correct options.
Multiple correct
HARD
6 marks
10 May 2026
02
Find the absolute maximum of f over the domain D.
Comprehension
MEDIUM
6 marks
10 May 2026
03
Suppose M denotes the absolute maximum value. Find M.
Comprehension
MEDIUM
4 marks
10 May 2026
04
Suppose m denotes the absolute minimum value. Find m.
Comprehension
MEDIUM
4 marks
10 May 2026
05
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a function. Consider the following statements. S1: The partial derivatives of
f
f
f
exist and are continuous at all points. S2: The function
f
f
f
is continuous. Which of the following is true?
Single correct
EASY
5 marks
21 December 2025
06
Which of the following is true for the given function f?
Comprehension
EASY
5 marks
21 December 2025
07
In the interior of D, the Hessian test is inconclusive for all but one critical points. Find the determinant of the Hessian of f at the critical point where the Hessian test does not fail.
Comprehension
HARD
5 marks
21 December 2025
08
Let M and m denote the maximum and minimum values attained by f, respectively, in the domain D. Find M - m. (Enter the answer rounded to two decimal places.)
Comprehension
MEDIUM
6 marks
21 December 2025
09
Consider the set of all values of determinants of the Hessian matrix at the critical points, i.e., consider the set { det(v_{f}(x, y)) |(x, y) is a critical point of f }. Find the minimum value from the above set.
Comprehension
HARD
6 marks
21 December 2025
10
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a function. Choose all the true statements from the options below if it is known that the function is differentiable at a point
(
c
,
c
)
(c, c)
(
c
,
c
)
in
R
2
\mathbb{R}^2
R
2
.
Multiple correct
EASY
2 marks
31 August 2025
11
What can be concluded by the Hessian test for T at its critical point in Int(D)?
Comprehension
EASY
1 marks
31 August 2025
12
It can be checked that
(
0
,
0
)
(0,0)
(
0
,
0
)
is a critical point of f. Find the determinant of the Hessian matrix of f at
(
0
,
0
)
(0,0)
(
0
,
0
)
.
Comprehension
EASY
1 marks
13 April 2025
13
What can be concluded about the critical point
(
0
,
0
)
(0,0)
(
0
,
0
)
from the Hessian test?
Comprehension
EASY
1 marks
13 April 2025
14
Choose the correct statement from the following.
Comprehension
MEDIUM
2 marks
13 April 2025
15
Using the function
ϕ
\phi
ϕ
, compute the distance between
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
and a point on the graph of f(x,y) = -2 - x^2 - y^2 that is closest to
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
.
Comprehension
EASY
1 marks
13 April 2025
16
Let f be a scalar-valued multivariable function defined on a domain D subset
R
2
\mathbb{R}^2
R
2
. Among the following statements, choose all the options which guarantee that the directional derivative of f exists in all directions, at a given point
(
a
,
b
)
(a, b)
(
a
,
b
)
in D.
Multiple correct
MEDIUM
3 marks
22 December 2024
17
What can we conclude from the Hessian test for the function f?
Comprehension
MEDIUM
2 marks
22 December 2024
18
Find the determinant of the Hessian at
(
a
,
b
)
(a, b)
(
a
,
b
)
.
Comprehension
EASY
1 marks
01 September 2024
19
What is the nature of the critical point?
Comprehension
EASY
1 marks
01 September 2024
20
How many saddle points are there for f?
Comprehension
HARD
1 marks
28 April 2024
21
Choose the correct option(s) from the following:
Comprehension
MEDIUM
2 marks
28 April 2024
22
Let f(x, y) = x^2 - y^2 + 4 on the disc S = {
(
x
,
y
)
(x,y)
(
x
,
y
)
: x^2 + y^2 <= 1 }. Choose the correct option(s) from the following:
Multiple correct
MEDIUM
3 marks
24 December 2023
23
Find the number of local maxima.
Comprehension
HARD
2 marks
24 December 2023
24
Find the number of local minima.
Comprehension
EASY
2 marks
24 December 2023
25
Find the number of saddle points.
Comprehension
EASY
2 marks
24 December 2023
26
If there are no local maxima, enter the answer as 100. Else, let
(
a
,
b
)
(a, b)
(
a
,
b
)
be a local maximum such that it is farthest from the origin, and if there are more than one such points, then it has the largest
x
x
x
-coordinate amongst them. Find
f
(
a
,
b
)
f(a,b)
f
(
a
,
b
)
.
Comprehension
MEDIUM
1 marks
24 December 2023
27
If there are no local minima, enter the answer as 100. Else, let
(
a
,
b
)
(a,b)
(
a
,
b
)
be a local minimum such that it is farthest from the origin, and if there are more than one such points, then it has the largest
x
x
x
-coordinate amongst them. Find
f
(
a
,
b
)
f(a,b)
f
(
a
,
b
)
.
Comprehension
MEDIUM
1 marks
24 December 2023
28
The number of saddle points of f(x, y) is
Comprehension
MEDIUM
1 marks
30 April 2023
29
The number of local maxima of f(x, y) is
Comprehension
MEDIUM
1 marks
30 April 2023
30
The number of local minima of f(x, y) is
Comprehension
MEDIUM
1 marks
30 April 2023
31
If the tangent plane exists for a function at a point then at that point . (Enter 5 best possible options. Enter only the serial numbers of those options in increasing order without adding any comma or space in between them.)
Comprehension
MEDIUM
2 marks
30 April 2023
32
Which of the following matrices represent the transpose of the Hessian matrix of the function at any critical point?
Comprehension
EASY
2 marks
11 December 2022
33
Which of the following options is/are true?
Comprehension
HARD
2 marks
11 December 2022
34
Which of the following options is/are true?
Comprehension
MEDIUM
2 marks
11 December 2022
35
Let v_1 =
(
1
,
−
2
)
(1, -2)
(
1
,
−
2
)
, v_2 = (0, sqrt(2)), v_3 = (sqrt(5), 0), 1_{4} =
(
−
1
,
2
)
(-1, 2)
(
−
1
,
2
)
, 1_{5} = (-sqrt(5), 0). Answer the following questions with respect to these points. Write down the values of i in increasing order for which v_i, i = 1, 2, 3, 4, 5, is a point of local minima according to the Hessian test. e.g. if v_1 and v_2 are points of local minima, your answer should be 12. If there is no such i, enter 0 as your answer.
Comprehension
HARD
3 marks
11 December 2022
36
Let v_1 =
(
1
,
−
2
)
(1, -2)
(
1
,
−
2
)
, v_2 = (0, sqrt(2)), v_3 = (sqrt(5), 0), 1_{4} =
(
−
1
,
2
)
(-1, 2)
(
−
1
,
2
)
, 1_{5} = (-sqrt(5), 0). Answer the following questions with respect to these points. Write down the values of i in increasing order for which v_i, i = 1, 2, 3, 4, 5, is a saddle point according to the Hessian test. e.g. if v_1 and v_2 are saddle points, your answer should be 12. If there is no such i, enter 0 as your answer.
Comprehension
HARD
3 marks
11 December 2022
37
Find the number of local maxima using the Hessian test.
Comprehension
HARD
2 marks
07 August 2022
38
Find the number of local minima using the Hessian test.
Comprehension
HARD
2 marks
07 August 2022
39
Find the number of saddle points using the Hessian test.
Comprehension
HARD
2 marks
07 August 2022
40
Find the number of points at which the Hessian test is indeterminate.
Comprehension
EASY
1 marks
07 August 2022
Showing 40 questions.
Complete question index for this section:
Question 1 (multiple):
Question 2 (comprehension):
Question 3 (comprehension):
Question 4 (comprehension):
Question 5 (single):
Question 6 (comprehension):
Question 7 (comprehension):
Question 8 (comprehension):
Question 9 (comprehension):
Question 10 (multiple):
Question 11 (comprehension):
Question 12 (comprehension):
Question 13 (comprehension):
Question 14 (comprehension):
Question 15 (comprehension):
Question 16 (multiple):
Question 17 (comprehension):
Question 18 (comprehension):
Question 19 (comprehension):
Question 20 (comprehension):
Question 21 (comprehension):
Question 22 (multiple):
Question 23 (comprehension):
Question 24 (comprehension):
Question 25 (comprehension):
Question 26 (comprehension):
Question 27 (comprehension):
Question 28 (comprehension):
Question 29 (comprehension):
Question 30 (comprehension):
Question 31 (comprehension):
Question 32 (comprehension):
Question 33 (comprehension):
Question 34 (comprehension):
Question 35 (comprehension):
Question 36 (comprehension):
Question 37 (comprehension):
Question 38 (comprehension):
Question 39 (comprehension):
Question 40 (comprehension):
