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IITM BS Week 10 Mathematics 2 Questions | Prasnya
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63Q
01
Consider the function
f
:
R
3
→
R
f: \mathbb{R}^3 \to \mathbb{R}
f
:
R
3
→
R
defined as f(x, y, z) = xy + yz + zx + x^3 + y^3 + z^3. If the unit direction of the steepest ascent of f at
(
1
,
0
,
1
)
(1, 0, 1)
(
1
,
0
,
1
)
is
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, then find the value of a + 2b + c.
Numerical
MEDIUM
5 marks
10 May 2026
02
Consider a function
f
:
R
3
→
R
f: \mathbb{R}^3 \to \mathbb{R}
f
:
R
3
→
R
defined as f(x, y, z) = 3xy - x^2 - y^2 + z^3. If the unit direction of the steepest descent of f at
(
1
,
−
1
,
0
)
(1, -1, 0)
(
1
,
−
1
,
0
)
is
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, then find the value of sqrt(17)(a + b + c).
Numerical
MEDIUM
5 marks
10 May 2026
03
Which of the following points lie in T'(2, 3)?
Comprehension
MEDIUM
6 marks
10 May 2026
04
What is the dimension of the subspace corresponding to the affine subspace T'(0, 0)?
Comprehension
EASY
4 marks
10 May 2026
05
What is the number of critical points inside the domain D of f? Write zero if there is no critical point.
Comprehension
MEDIUM
4 marks
10 May 2026
06
Which of the following points are in T'(1, 1)?
Comprehension
MEDIUM
6 marks
10 May 2026
07
What is the dimension of the subspace corresponding to the affine subspace T'(0, 0)?
Comprehension
EASY
4 marks
10 May 2026
08
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a differentiable function. Consider the following statements. S1: The tangent plane to the graph of f at
(
c
,
c
)
(c, c)
(
c
,
c
)
is parallel to the xy-plane. S2:
(
c
,
c
)
(c, c)
(
c
,
c
)
is a critical point of f, i.e., grad f(c, c) = 0. Which of the following is true?
Single correct
EASY
5 marks
21 December 2025
09
Let L_{(x, y)}(v_1, v_2) denote the tangent line to the graph of f at the point
(
x
,
y
)
(x, y)
(
x
,
y
)
and in the direction (v_1, v_2). Choose all the correct statements from the following.
Comprehension
MEDIUM
8 marks
21 December 2025
10
Consider the sequence of points x_n= (-2 + 1/n, 1 + 1/n) for n in {1, 2, }. The tangent hyperplane to the graph of f at x_n has a unique normal of the form (x_n, x_n, -1). Find lim_{n -> infinity} x_n+ lim_{n -> infinity} x_n.
Comprehension
MEDIUM
5 marks
21 December 2025
11
Let L_{(x, y)}(v_1, v_2) denote the tangent line to the graph of f at the point
(
x
,
y
)
(x, y)
(
x
,
y
)
and in the direction (v_1, v_2). Choose all the correct statements from the following.
Comprehension
HARD
8 marks
21 December 2025
12
Consider the sequence of points x_n= (1/n, 1/n) for n in {1, 2, }. The tangent plane to the graph of f at x_n has a unique normal x_nof the form (x_n, x_n, -1). If theta_n denotes the angle (in degrees) between x_n and the vector
(
0
,
0
,
−
1
)
(0, 0, -1)
(
0
,
0
,
−
1
)
, then find the value of lim_{n -> infinity} theta_n.
Comprehension
MEDIUM
5 marks
21 December 2025
13
Find the maximum value of R > 0 such that f, when restricted to D = {
(
x
,
y
)
(x, y)
(
x
,
y
)
in
R
2
\mathbb{R}^2
R
2
| x^2 + y^2 < R } contains a unique critical point. (Enter the answer rounded to two decimal places.)
Comprehension
MEDIUM
5 marks
21 December 2025
14
Find the value of c.
Comprehension
MEDIUM
2 marks
31 August 2025
15
Which of the following are possible candidates for v?
Comprehension
MEDIUM
2 marks
31 August 2025
16
Which of the following is a basis for U?
Comprehension
MEDIUM
2 marks
31 August 2025
17
Which of the following is the only critical point of T in Int(D)?
Comprehension
MEDIUM
1 marks
31 August 2025
18
Let v_1 and v_2 denote the one-variable functions obtained by restricting T on gamma_1 and gamma_2, respectively, i.e., v_1(x) = T(x, -x^2 + 1), v_2(x) = T(x, 0), for x in
(
−
1
,
1
)
(-1, 1)
(
−
1
,
1
)
. Which of the following statements is true?
Comprehension
MEDIUM
1 marks
31 August 2025
19
Comparing the critical points on the interior, the boundary components, and the corners, find the temperature at the hottest point on the metallic plate.
Comprehension
MEDIUM
2 marks
31 August 2025
20
Find the value of
2
a
\sqrt{2} a
2
a
.
Comprehension
MEDIUM
1 marks
13 April 2025
21
Find the value of
2
\sqrt{2}
2
b.
Comprehension
EASY
1 marks
13 April 2025
22
If
(
c
,
d
)
(c, d)
(
c
,
d
)
denotes a direction vector along which there is no change in f at
(
−
1
,
1
)
(-1,1)
(
−
1
,
1
)
, find the value of c + d.
Comprehension
EASY
1 marks
13 April 2025
23
Suppose
(
1
,
b
,
c
)
(1, b, c)
(
1
,
b
,
c
)
is a point on the tangent plane at
(
0
,
π
2
)
\left(0, \frac{\pi}{2}\right)
(
0
,
2
π
)
to the surface
z
=
f
(
x
,
y
)
z = f(x, y)
z
=
f
(
x
,
y
)
. Find the value of
b
−
c
b - c
b
−
c
.
Comprehension
MEDIUM
2 marks
13 April 2025
24
Which of the following are critical points of f?
Comprehension
MEDIUM
2 marks
13 April 2025
25
Find the number of critical points of the function
ϕ
\phi
ϕ
obtained in the previous subquestion that lie in
{
(
x
,
y
)
∈
R
2
∣
x
2
+
y
2
<
1
}
\{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}
{(
x
,
y
)
∈
R
2
∣
x
2
+
y
2
<
1
}
, the interior of
D
D
D
.
Comprehension
MEDIUM
1 marks
13 April 2025
26
Express the volume of a cuboid with perimeter 12, as a function
ϕ
\phi
ϕ
of two variables x and y (length and breadth).
Comprehension
EASY
1 marks
22 December 2024
27
Using the function given, calculate the maximum possible volume of a cuboid whose perimeter is 12.
Comprehension
MEDIUM
2 marks
22 December 2024
28
Find the value of a.
Comprehension
MEDIUM
1.5 marks
22 December 2024
29
Find the value of b.
Comprehension
MEDIUM
1.5 marks
22 December 2024
30
How many critical points does the function f have?
Comprehension
HARD
2 marks
22 December 2024
31
The equation of the tangent plane to the function f(x, y) = e^y * (x + y) at the point
(
1
,
1
)
(1, 1)
(
1
,
1
)
is given by A*x + B*y + C = z. Find the value of (A + B + C) / e.
Numerical
MEDIUM
3 marks
01 September 2024
32
If f has a critical point at
(
a
,
b
)
(a, b)
(
a
,
b
)
, find a + b.
Comprehension
EASY
1 marks
01 September 2024
33
Enter the value of x.
Comprehension
MEDIUM
1 marks
01 September 2024
34
Enter the value of y.
Comprehension
MEDIUM
1 marks
01 September 2024
35
Enter the value of z.
Comprehension
MEDIUM
1 marks
01 September 2024
36
Suppose the function T(x,y,z) = -xz/(x^2+y^2) represents temperature at the point
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in a room and (u1, u2, u3) is the unit vector in the direction in which the temperature increases most rapidly at the point
(
1
,
0
,
1
)
(1,0,1)
(
1
,
0
,
1
)
. Find u1 + u2 + u3.
Comprehension
HARD
2 marks
28 April 2024
37
Suppose a is the rate of change of the function f(x,y,z) = x^3*y*z^2 at the point
(
−
1
,
2
,
1
)
(-1,2,1)
(
−
1
,
2
,
1
)
in the direction where f decreases most rapidly. Find a^2.
Comprehension
HARD
2 marks
28 April 2024
38
How many critical points are there for f?
Comprehension
MEDIUM
1 marks
28 April 2024
39
If D_u(x, y) is the linear approximation of the function u(x, y) at point
(
1
,
2
)
(1, 2)
(
1
,
2
)
and D_u(2, 3) = a + b*e, where a, b are integers, find a + b.
Comprehension
MEDIUM
2 marks
28 April 2024
40
If 1_{v}(x, y) is the linear approximation of the function v(x, y) at point
(
2
,
1
)
(2, 1)
(
2
,
1
)
and 1_{v}(2, 3) = c + d*e^2, where c, d are integers, find c + d.
Comprehension
MEDIUM
2 marks
28 April 2024
41
Choose the correct option(s) from the following:
Comprehension
HARD
4 marks
28 April 2024
42
Let f(x, y) = x^3 + 2x^2 + y^3 + y^2 - 2x + y - 2. Choose the correct option(s) from the following:
Multiple correct
HARD
3 marks
24 December 2023
43
If
f
f
f
increases most rapidly at the point
(
1
,
1
)
(1,1)
(
1
,
1
)
, find
5
(
a
+
b
)
5(a+b)
5
(
a
+
b
)
.
Comprehension
MEDIUM
1 marks
24 December 2023
44
If
f
f
f
decreases most rapidly at the point
(
1
,
1
)
(1,1)
(
1
,
1
)
, find
5
(
a
+
b
)
5(a+b)
5
(
a
+
b
)
.
Comprehension
EASY
1 marks
24 December 2023
45
If there is no change in
f
f
f
at the point
(
1
,
1
)
(1,1)
(
1
,
1
)
, find
∣
a
+
b
∣
|a+b|
∣
a
+
b
∣
.
Comprehension
MEDIUM
1 marks
24 December 2023
46
A skier is on a mountain with equation z = 20 - 0.4x^2 - 0.3y^2 where z denotes the height. The skier is located at the point with xy-coordinates
(
1
,
−
1
)
(1, -1)
(
1
,
−
1
)
, and wants to ski downhill along the steepest possible path. In which direction should the skier begin skiing?
Single correct
MEDIUM
2 marks
03 September 2023
47
Choose the correct option for the parametric equations of the line tangent to f at the point (
π
\pi
π
/2,
π
\pi
π
/2) in the direction of x axis:
Comprehension
MEDIUM
2 marks
03 September 2023
48
Choose the correct option for the parametric equations of the line tangent to f at the point (
π
\pi
π
/2,
π
\pi
π
/2) in the direction of y axis:
Comprehension
MEDIUM
2 marks
03 September 2023
49
Choose the correct option for the parametric equations of the line tangent to f at the point (
π
\pi
π
/2,
π
\pi
π
/2) in the direction of (-1, 1):
Comprehension
HARD
2 marks
03 September 2023
50
What is the minimum sum of three non-negative numbers whose product is 27?
Numerical
EASY
2 marks
30 April 2023
51
Consider the function f(x, y) = x^2 + xy. Which of the following lines represents the tangent line at the point
(
1
,
1
)
(1, 1)
(
1
,
1
)
in the direction of the vector (1, 1)?
Single correct
HARD
2 marks
30 April 2023
52
Match the equation of the surface in Column A with the tangent plane at the point
(
1
,
1
,
2
)
(1, 1, 2)
(
1
,
1
,
2
)
in Column B and the vector subspace corresponding to the affine subspace (of
R
3
\mathbb{R}^3
R
3
) formed by the tangent plane, in Column C. Select the correct option from the following:
Single correct
HARD
4 marks
30 April 2023
53
The number of critical points of f(x, y) is
Comprehension
MEDIUM
3 marks
30 April 2023
54
A critical point can be . (Enter 3 best possible options. Enter only the serial numbers of those options in increasing order without adding any comma or space in between them.)
Comprehension
EASY
2 marks
30 April 2023
55
Which of the following options is/are true?
Multiple correct
HARD
3 marks
11 December 2022
56
Find the value of 15(a + b).
Comprehension
EASY
2 marks
11 December 2022
57
If d is the minimum distance from the origin, then find the value of 9d^2.
Comprehension
EASY
2 marks
11 December 2022
58
Find the number of critical points.
Comprehension
MEDIUM
3 marks
11 December 2022
59
If (v_1, v_2) is the unit vector along the direction in which the directional derivative of the function u(x, y) at the point
(
1
,
1
)
(1, 1)
(
1
,
1
)
is the maximum and (v_1, v_2) is the unit vector along the direction in which the directional derivative of the function v(x, y) at the point
(
1
,
1
)
(1, 1)
(
1
,
1
)
is the minimum, then which of the following options is/are true?
Comprehension
HARD
2 marks
11 December 2022
60
If D_u(x, y) is the linear approximation of the function u(x, y) at point
(
2
,
3
)
(2, 3)
(
2
,
3
)
and D_u(3, 4) = (a + b e^3), and if 1_{v}(x, y) is the linear approximation of the function v(x, y) at point
(
1
,
2
)
(1, 2)
(
1
,
2
)
, and 1_{v}(3, 4) = (c + d e), then find the value of (a + c) - (b + d), where a, b, c, d are integers.
Comprehension
HARD
2 marks
11 December 2022
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Question 46 (single):
Question 47 (comprehension):
Question 48 (comprehension):
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Question 50 (integer):
Question 51 (single):
Question 52 (single):
Question 53 (comprehension):
Question 54 (comprehension):
Question 55 (multiple):
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Question 61 (integer):
Question 62 (single):
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