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IITM BS Week 9 Mathematics 2 Questions | Prasnya
Mathematics 2 > Week 9
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58Q
01
Consider the function
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
defined as follows:
f
(
x
,
y
)
=
{
sin
(
x
2
−
y
2
)
x
2
+
y
2
if
(
x
,
y
)
≠
(
0
,
0
)
0
else
f(x, y) = \begin{cases} \frac{\sin(x^2 - y^2)}{\sqrt{x^2 + y^2}} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{else} \end{cases}
f
(
x
,
y
)
=
⎩
⎨
⎧
x
2
+
y
2
s
i
n
(
x
2
−
y
2
)
0
if
(
x
,
y
)
=
(
0
,
0
)
else
Choose the correct option.
Single correct
MEDIUM
5 marks
10 May 2026
02
Consider the following statements about the function f. S1: The range of f is the subset {z in R | z >= 0} of R. S2: The gradient of f is defined at all points in
R
2
\mathbb{R}^2
R
2
. S3: The directional derivative of f in the direction of v = (1/sqrt(2), -1/sqrt(2)) at
(
1
,
0
)
(1, 0)
(
1
,
0
)
is 0. What is the number of correct statements? Write 0 if none of the statements is correct.
Comprehension
MEDIUM
7 marks
10 May 2026
03
Which of the following options depicts the correct geometric shape of the boundary of D?
Comprehension
EASY
3 marks
10 May 2026
04
Which of the following options depicts the correct geometric shape of the boundary of D?
Comprehension
EASY
3 marks
10 May 2026
05
Consider the following statements about the function f. S1: The range of f contains 0. S2: The domain of the gradient is
R
3
\mathbb{R}^3
R
3
. S3: The directional derivative of f in the direction of v =
(
−
1
,
3
,
−
1
)
(-1, 3, -1)
(
−
1
,
3
,
−
1
)
at
(
0
,
0
,
0
)
(0, 0, 0)
(
0
,
0
,
0
)
is 2. What is the number of correct statements? Write 0 if none of the statements is correct.
Comprehension
MEDIUM
7 marks
10 May 2026
06
Suppose the function f describes the pressure in a room at point
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
in
R
3
\mathbb{R}^3
R
3
. If the rate of change of pressure at
(
−
1
,
−
1
,
1
)
(-1, -1, 1)
(
−
1
,
−
1
,
1
)
in the direction of the unit vector 1/sqrt(3)
(
1
,
1
,
1
)
(1, 1, 1)
(
1
,
1
,
1
)
is k * e^3. Find k up to two decimal points.
Comprehension
HARD
7 marks
10 May 2026
07
Let f, g be scalar-valued functions defined on
R
2
\mathbb{R}^2
R
2
\ {(0, 0)} for which the following is known. - f is bounded, i.e., there exist m, M in R such that m <= f(x, y) <= M, for all
(
x
,
y
)
(x, y)
(
x
,
y
)
in
R
2
\mathbb{R}^2
R
2
\ {(0, 0)}. - The function g satisfies -(x^2 + y^2) <= g(x, y) <= (x^2 + y^2) for all
(
x
,
y
)
(x, y)
(
x
,
y
)
in
R
2
\mathbb{R}^2
R
2
\ {(0, 0)}. Find the value of lim_{(x, y) -> (0, 0)} f(x, y)g(x, y).
Numerical
EASY
6 marks
21 December 2025
08
Let
f
f
f
be a scalar-valued function defined on
R
2
∖
{
(
0
,
0
)
}
\mathbb{R}^2 \setminus \{(0, 0)\}
R
2
∖
{(
0
,
0
)}
such that
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
=
3
\lim_{(x, y) \to (0, 0)} f(x, y) = 3
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
=
3
. Let
g
:
R
→
R
g: \mathbb{R} \to \mathbb{R}
g
:
R
→
R
be the one-variable function defined by
g
(
t
)
=
t
2
g(t) = t^2
g
(
t
)
=
t
2
, for all
t
∈
R
t \in \mathbb{R}
t
∈
R
. Find the value of
lim
(
x
,
y
)
→
(
0
,
0
)
(
g
∘
f
)
(
x
,
y
)
\lim_{(x, y) \to (0, 0)} (g \circ f)(x, y)
lim
(
x
,
y
)
→
(
0
,
0
)
(
g
∘
f
)
(
x
,
y
)
.
Numerical
EASY
6 marks
21 December 2025
09
For any function
g
:
R
2
→
R
g: \mathbb{R}^2 \to \mathbb{R}
g
:
R
2
→
R
, the level set of
g
g
g
at
c
∈
R
c \in \mathbb{R}
c
∈
R
, denoted by
L
c
(
g
)
L_c(g)
L
c
(
g
)
, is defined as the set of points in
R
2
\mathbb{R}^2
R
2
that are mapped to
c
c
c
, i.e.,
L
c
(
g
)
=
{
(
x
,
y
)
∈
R
2
∣
g
(
x
,
y
)
=
c
}
L_c(g) = \{(x, y) \in \mathbb{R}^2 \mid g(x, y) = c\}
L
c
(
g
)
=
{(
x
,
y
)
∈
R
2
∣
g
(
x
,
y
)
=
c
}
. Choose all the correct statements from the following.
Comprehension
MEDIUM
7 marks
21 December 2025
10
Let M_band M_b denote the maximum and minimum value of f, respectively, when restricted to the boundary of the domain. Find the value of M_b- M_b.
Comprehension
EASY
3 marks
21 December 2025
11
Select all options containing functions
f
f
f
and
c
∈
R
c \in \mathbb{R}
c
∈
R
such that
L
c
(
f
)
L_c(f)
L
c
(
f
)
is a straight line in
R
2
\mathbb{R}^2
R
2
.
Comprehension
MEDIUM
7 marks
21 December 2025
12
Consider the following statements. S1: The function f: D -> R is injective (one-to-one). S2: The set L_{c}(f) contains exactly one element, for all c in Image(f). Which of the following is true?
Comprehension
EASY
5 marks
21 December 2025
13
Let M_band M_b denote the maximum and minimum values of f, respectively. Find M_b- M_b. (Note that this does not require the Hessian test.)
Comprehension
EASY
3 marks
21 December 2025
14
Consider the function
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
defined by
f
(
x
,
y
)
=
{
y
2
−
x
2
x
4
+
(
y
−
sin
(
x
2
)
)
2
if
(
x
,
y
)
≠
(
0
,
0
)
0
if
(
x
,
y
)
=
(
0
,
0
)
f(x, y) = \begin{cases} \frac{y^2 - x^2}{x^4 + (y - \sin(x^2))^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}
f
(
x
,
y
)
=
{
x
4
+
(
y
−
s
i
n
(
x
2
)
)
2
y
2
−
x
2
0
if
(
x
,
y
)
=
(
0
,
0
)
if
(
x
,
y
)
=
(
0
,
0
)
Choose all curves along which the limit of
f
f
f
exists at
(
0
,
0
)
(0, 0)
(
0
,
0
)
.
Multiple correct
HARD
3 marks
31 August 2025
15
Let D(r) denote the open disc of radius r in
R
2
\mathbb{R}^2
R
2
, i.e., D(r) = {
(
x
,
y
)
(x, y)
(
x
,
y
)
in
R
2
\mathbb{R}^2
R
2
| x^2 + y^2 < r^2 }. Let f: D(r) -> [-1, 1] be the scalar-valued function defined by f(x, y) = sin(sqrt(x^2 + y^2)). If cis the largest possible value of r such that f is not surjective on the domain D(r), then find c/
π
\pi
π
.
Numerical
MEDIUM
1 marks
31 August 2025
16
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a function for which the following is known: At every point, the partial derivatives of
f
f
f
exist and are continuous;
f
(
2
,
3
)
=
−
1
f(2, 3) = -1
f
(
2
,
3
)
=
−
1
; The tangent line to the graph of
f
f
f
at
(
2
,
3
)
(2, 3)
(
2
,
3
)
in the direction
(
1
,
0
)
(1, 0)
(
1
,
0
)
contains the point
(
1
,
3
,
0
)
(1, 3, 0)
(
1
,
3
,
0
)
; The tangent line to the graph of
f
f
f
at
(
2
,
3
)
(2, 3)
(
2
,
3
)
in the direction
(
1
/
2
,
1
/
2
)
(1/\sqrt{2}, 1/\sqrt{2})
(
1/
2
,
1/
2
)
contains the point
(
3
,
4
,
−
1
)
(3, 4, -1)
(
3
,
4
,
−
1
)
. Find the partial derivative of
f
f
f
with respect to
y
y
y
(the second variable) at
(
2
,
3
)
(2, 3)
(
2
,
3
)
.
Numerical
HARD
2 marks
31 August 2025
17
The directional derivatives of f exist in all directions at the origin.
Comprehension
MEDIUM
1 marks
31 August 2025
18
Find the limit of f at
(
0
,
0
)
(0, 0)
(
0
,
0
)
along the curve {(y^3, y) | y in R}.
Comprehension
EASY
2 marks
31 August 2025
19
Find the directional derivative of f at
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the direction (1/2, -sqrt(3)/2).
Comprehension
EASY
1 marks
31 August 2025
20
Find the value of b if the directional derivative at
(
−
1
,
−
1
,
−
1
)
(-1, -1, -1)
(
−
1
,
−
1
,
−
1
)
in the direction (4/5, 3/5, 0) is equal to -6.
Comprehension
EASY
1 marks
31 August 2025
21
For which of the following functions defined on
R
2
\mathbb{R}^2
R
2
\ {(0,0)} does the limit of the function exist at (0,0)?
Single correct
MEDIUM
2 marks
13 April 2025
22
Which of the following sequences in
R
2
\mathbb{R}^2
R
2
are convergent?
Single correct
EASY
2 marks
13 April 2025
23
Let
f
:
R
n
→
R
f: \mathbb{R}^n \to \mathbb{R}
f
:
R
n
→
R
be a scalar-valued multivariable function (
n
≥
2
n \ge 2
n
≥
2
) such that the partial derivatives exist at all points and are continuous. Choose the correct statements from the following.
Multiple correct
MEDIUM
3 marks
13 April 2025
24
What is the minimum number of points required to be removed from
R
2
\mathbb{R}^2
R
2
to obtain a valid domain for f?
Comprehension
EASY
1 marks
13 April 2025
25
What is the range of the function
f
f
f
when defined on the domain
D
=
{
(
x
,
y
)
∈
R
2
∣
x
2
+
y
2
≥
2
}
D = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \ge 2\}
D
=
{(
x
,
y
)
∈
R
2
∣
x
2
+
y
2
≥
2
}
?
Comprehension
EASY
1 marks
13 April 2025
26
The function f defined on the domain specified in the previous subquestion is injective.
Comprehension
EASY
1 marks
13 April 2025
27
If the directional derivative of f at the point (0,
π
\pi
π
/2) in the direction (t,
1
−
t
2
\sqrt{1-t^2}
1
−
t
2
) is equal to 0, then find the value of t^2.
Comprehension
MEDIUM
1 marks
13 April 2025
28
Using the procedure described in the main data, choose the expression for the function
ϕ
\phi
ϕ
that measures the square of the distance between
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
and a generic point on the graph of f.
Comprehension
EASY
1 marks
13 April 2025
29
Consider the function
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
defined as follows:
f
(
x
,
y
)
=
{
(
x
−
y
)
3
sin
(
1
x
−
y
)
for
x
≠
y
0
for
x
=
y
f(x, y) = \begin{cases} (x - y)^3 \sin\left(\frac{1}{x - y}\right) & \text{for } x \neq y \\ 0 & \text{for } x = y \end{cases}
f
(
x
,
y
)
=
{
(
x
−
y
)
3
sin
(
x
−
y
1
)
0
for
x
=
y
for
x
=
y
Choose all the correct statements from the following.
Multiple correct
MEDIUM
3 marks
22 December 2024
30
Which of the following is the largest possible set that would be a valid domain D for the given function f?
Comprehension
EASY
1 marks
22 December 2024
31
For D = {(x, y) in
R
2
\mathbb{R}^2
R
2
| x^2 + y^2 < 1}, what is the range of f?
Comprehension
MEDIUM
1 marks
22 December 2024
32
For D = {(x, y) in
R
2
\mathbb{R}^2
R
2
| x^2 + y^2 > 2}, what is the range of f?
Comprehension
MEDIUM
1 marks
22 December 2024
33
If the gradient of f at the point
(
3
,
0
)
(3, 0)
(
3
,
0
)
is given by
(
a
,
b
)
(a, b)
(
a
,
b
)
, then what is the value of b - sqrt(3)a?
Comprehension
MEDIUM
2 marks
22 December 2024
34
Find the directional derivative of f at the point
(
3
,
0
)
(3, 0)
(
3
,
0
)
in the direction (sqrt(3)/2, -1/2).
Comprehension
MEDIUM
2 marks
22 December 2024
35
If the directional derivative of f at the point
(
2
,
−
1
)
(2, -1)
(
2
,
−
1
)
in the direction (-1/sqrt(2), 1/sqrt(2)) is -17/sqrt(2), then find the value of k.
Comprehension
MEDIUM
1 marks
22 December 2024
36
Find the value of k for which the limit of f exists at
(
0
,
0
)
(0, 0)
(
0
,
0
)
.
Comprehension
EASY
1 marks
22 December 2024
37
For the given value of k, find the value of f(0, 0) for which f is a continuous function on
R
2
\mathbb{R}^2
R
2
.
Comprehension
EASY
1 marks
22 December 2024
38
Consider the function f: f(x, y) = y^4 / (x^2 + y^2) for
(
x
,
y
)
(x, y)
(
x
,
y
)
!=
(
0
,
0
)
(0, 0)
(
0
,
0
)
and f(0, 0) = 0. If f_x, f_y denote the partial derivatives, select all true statements from the options given below.
Multiple correct
MEDIUM
3 marks
01 September 2024
39
Evaluate lim_{(x, y) -> (0, 0)} sin(4*x^2 + 4*y^2) / (x^2 + y^2).
Numerical
EASY
3 marks
01 September 2024
40
Consider the following function: f(x, y) = (x^3 - y^3) / (sqrt(x) - sqrt(y)) if x != y and x >= 0 and y >= 0; and f(x, y) = c otherwise. For what value of c is the function f continuous at (0, 0)?
Numerical
MEDIUM
3 marks
01 September 2024
41
If the directional derivative of f(x, y) = x^3*y - x*y^2 + 8 at the point
(
1
,
2
)
(1, 2)
(
1
,
2
)
in the direction (1/sqrt(2), -1/sqrt(2)) is p/sqrt(2), find the value of p.
Numerical
MEDIUM
3 marks
01 September 2024
42
Choose the largest possible D that would be a valid domain from the options given below.
Comprehension
EASY
1 marks
01 September 2024
43
Which of the following is the range of f using the domain chosen from the previous question? Recall that [a, b] = {x : a <= x <= b, x in R}.
Comprehension
EASY
1 marks
01 September 2024
44
Let f(x, y) = 2x^4 - 8*sqrt(y) - 7. Choose the correct options from the following:
Multiple correct
MEDIUM
2 marks
28 April 2024
45
Choose the option(s) for which the limit exists.
Multiple correct
HARD
4 marks
28 April 2024
46
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a function given by:
f
(
x
,
y
)
=
{
(
x
+
y
)
2
x
2
+
y
2
if
(
x
,
y
)
≠
(
0
,
0
)
0
if
(
x
,
y
)
=
(
0
,
0
)
f(x, y) = \begin{cases} \frac{(x+y)^2}{x^2+y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}
f
(
x
,
y
)
=
{
x
2
+
y
2
(
x
+
y
)
2
0
if
(
x
,
y
)
=
(
0
,
0
)
if
(
x
,
y
)
=
(
0
,
0
)
Choose the correct option(s) from the following:
Multiple correct
MEDIUM
3 marks
24 December 2023
47
Find
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
\lim_{(x,y)\to(0,0)} f(x,y)
lim
(
x
,
y
)
→
(
0
,
0
)
f
(
x
,
y
)
.
Comprehension
EASY
1 marks
24 December 2023
48
Is the function continuous at
(
0
,
0
)
(0,0)
(
0
,
0
)
? [If the answer is yes, enter 1 and if the answer is no, enter 0.]
Comprehension
EASY
1 marks
24 December 2023
49
At how many points does
f
f
f
have a discontinuity?
Comprehension
EASY
1 marks
24 December 2023
50
Let f(x, y) = { xy/(x^2 + y^2) if
(
x
,
y
)
(x, y)
(
x
,
y
)
!=
(
0
,
0
)
(0, 0)
(
0
,
0
)
, and 0 otherwise. Then which of the following statement(s) is/are true?
Multiple correct
HARD
2 marks
03 September 2023
51
Let f(x, y, z) = x^2 y^3 - 3xz and u =
(
1
,
2
,
2
)
(1, 2, 2)
(
1
,
2
,
2
)
. Find the directional derivative of f in the direction of the vector u at the point
(
0
,
1
,
−
1
)
(0, 1, -1)
(
0
,
1
,
−
1
)
.
Numerical
MEDIUM
2 marks
03 September 2023
52
Choose the correct statements for the function: f(x, y) = x^2 * y^2 / (x^4 + y^4) for
(
x
,
y
)
(x,y)
(
x
,
y
)
!=
(
0
,
0
)
(0,0)
(
0
,
0
)
, and f(0,0) = 0
Multiple correct
HARD
3 marks
30 April 2023
53
Consider a function g(x, y) such that g(x, y) = { u(x, y) if x = y, v(x, y) if x != y } where u(x, y) = x^3 + e^y and v(x, y) = y^3 + y e^x. Which of the following options is/are true?
Multiple correct
HARD
3 marks
11 December 2022
54
Which of the following options is/are true? f(x, y) = xy^3 / (x^2 + y^6)
Multiple correct
MEDIUM
2 marks
07 August 2022
55
Which of the following is true for the function: f(x, y) = (x^3 + y^3)/(x + y)
Single correct
MEDIUM
2 marks
07 August 2022
56
What will be the ratio of the price of the raw materials and the price of the transportation (x : y) when y > x, if the rate of change of the price of the product with respect to the price of the raw materials is 0? (In this context x and y both are always positive).
Comprehension
EASY
2 marks
07 August 2022
57
Which of the following statements are true?
Comprehension
MEDIUM
3 marks
07 August 2022
58
If the rate of change of the price of the product along the direction of the vector
(
1
,
1
)
(1, 1)
(
1
,
1
)
is 1/sqrt(2) [ka^2 + lab + mb^2], when the price of raw material is a and the price of transportation of the product to the market is b (where a != b), then find the value of k - l + m.
Comprehension
MEDIUM
2 marks
07 August 2022
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