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IITM BS Week 7 Mathematics 2 Questions | Prasnya
Mathematics 2 > Week 7
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Difficulty:
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32Q
01
Let u =
(
2
,
−
1
,
4
,
3
)
(2, -1, 4, 3)
(
2
,
−
1
,
4
,
3
)
and v =
(
1
,
c
,
−
2
,
6
)
(1, c, -2, 6)
(
1
,
c
,
−
2
,
6
)
be two vectors in
R
4
\mathbb{R}^4
R
4
. If u and v are orthogonal to each other, find the value of c.
Numerical
EASY
5 marks
10 May 2026
02
Let u =
(
1
,
3
,
−
5
)
(1, 3, -5)
(
1
,
3
,
−
5
)
and v =
(
5
,
c
,
7
)
(5, c, 7)
(
5
,
c
,
7
)
be two vectors in
R
3
\mathbb{R}^3
R
3
such that u and v are perpendicular to each other. Find the value of c.
Numerical
EASY
5 marks
10 May 2026
03
Suppose function f describes the amount of rain at a point
(
x
,
y
)
(x, y)
(
x
,
y
)
in
R
2
\mathbb{R}^2
R
2
on a map. If along the unit direction
(
a
,
b
)
(a, b)
(
a
,
b
)
the amount of rain remains constant starting at point
(
1
,
1
)
(1, 1)
(
1
,
1
)
, then find a^2 + b^2.
Comprehension
MEDIUM
5 marks
10 May 2026
04
Choose all the correct options from the following.
Comprehension
MEDIUM
8 marks
10 May 2026
05
Let
T
:
R
3
→
R
3
T: \mathbb{R}^3 \to \mathbb{R}^3
T
:
R
3
→
R
3
be the linear transformation defined by
T
(
x
,
y
,
z
)
=
(
3
x
+
y
+
z
,
a
x
+
b
z
,
x
−
y
)
T(x, y, z) = (3x + y + z, ax + bz, x - y)
T
(
x
,
y
,
z
)
=
(
3
x
+
y
+
z
,
a
x
+
b
z
,
x
−
y
)
and
A
A
A
be the matrix representation of
T
T
T
with respect to the standard ordered basis
{
(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)
}
\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}
{(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)}
for both the domain and the co-domain. Consider the following statements. S1: If
a
=
1
,
b
=
0
a = 1, b = 0
a
=
1
,
b
=
0
, then
A
A
A
is similar to
B
=
[
2
0
0
1
−
1
0
2
1
3
]
B = \begin{bmatrix} 2 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 1 & 3 \end{bmatrix}
B
=
2
1
2
0
−
1
1
0
0
3
. S2: If
a
=
0
,
b
=
−
1
a = 0, b = -1
a
=
0
,
b
=
−
1
, then
A
A
A
is equivalent to
B
=
[
1
−
1
0
4
0
0
0
0
1
]
B = \begin{bmatrix} 1 & -1 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}
B
=
1
4
0
−
1
0
0
0
0
1
. Choose the correct option.
Single correct
HARD
5 marks
21 December 2025
06
Let A_{5 x 5} be similar to B_{5 x 5} and B_{5 x 5} be equivalent to C_{5 x 5}. If A is an orthogonal matrix then what is the nullity of C?
Numerical
MEDIUM
3 marks
21 December 2025
07
Let
T
:
R
2
→
R
2
T: \mathbb{R}^2 \to \mathbb{R}^2
T
:
R
2
→
R
2
be the linear transformation
T
(
x
,
y
)
=
(
2
x
+
y
,
3
x
+
y
)
T(x, y) = (2x + y, 3x + y)
T
(
x
,
y
)
=
(
2
x
+
y
,
3
x
+
y
)
and
A
A
A
denote the matrix representation of
T
T
T
with respect to the ordered basis
{
(
1
,
0
)
,
(
0
,
1
)
}
\{(1, 0), (0, 1)\}
{(
1
,
0
)
,
(
0
,
1
)}
of domain and
{
(
1
,
1
)
,
(
−
1
,
1
)
}
\{(1, 1), (-1, 1)\}
{(
1
,
1
)
,
(
−
1
,
1
)}
of co-domain. Consider the following statements. S1:
A
A
A
is similar to
[
5
2
1
0
]
\begin{bmatrix} 5 & 2 \\ 1 & 0 \end{bmatrix}
[
5
1
2
0
]
. S2:
A
A
A
is equivalent to
[
5
2
1
0
]
\begin{bmatrix} 5 & 2 \\ 1 & 0 \end{bmatrix}
[
5
1
2
0
]
. Choose the correct option.
Single correct
HARD
5 marks
21 December 2025
08
Let A be a non-zero 3 x 3 matrix such that A^2 = 0. If A is similar to the matrix B and B is equivalent to a matrix C, then what is the rank of C?
Numerical
MEDIUM
3 marks
21 December 2025
09
Let
T
:
R
3
→
R
3
T: \mathbb{R}^3 \to \mathbb{R}^3
T
:
R
3
→
R
3
be a linear transformation defined as:
T
(
x
,
y
,
z
)
=
(
x
+
y
−
z
,
a
y
+
2
b
z
,
x
−
y
+
z
)
T(x, y, z) = (x + y - z, ay + 2bz, x - y + z)
T
(
x
,
y
,
z
)
=
(
x
+
y
−
z
,
a
y
+
2
b
z
,
x
−
y
+
z
)
. Let
A
A
A
be the matrix representation of
T
T
T
with respect to the standard bases of
R
3
\mathbb{R}^3
R
3
and let
B
=
[
3
0
3
1
0
1
1
0
3
]
B = \begin{bmatrix} 3 & 0 & 3 \\ 1 & 0 & 1 \\ 1 & 0 & 3 \end{bmatrix}
B
=
3
1
1
0
0
0
3
1
3
. Consider the following statements: S1: If
A
A
A
and
B
B
B
are similar, then the range of the linear transformation
T
T
T
is a plane in
R
3
\mathbb{R}^3
R
3
. S2: If
a
=
2
a = 2
a
=
2
and
b
=
−
1
b = -1
b
=
−
1
, then
A
A
A
and
B
B
B
are equivalent matrices, but not similar. Which of the following options is true?
Single correct
MEDIUM
3 marks
31 August 2025
10
Find the value of k.
Comprehension
MEDIUM
2 marks
31 August 2025
11
If
b
=
(
α
,
α
+
1
,
10
)
T
b = (\alpha, \alpha + 1, 10)^T
b
=
(
α
,
α
+
1
,
10
)
T
, find
α
\alpha
α
.
Comprehension
MEDIUM
1 marks
31 August 2025
12
Suppose there is another bridge B2 along the line ax + by = 0 which is orthogonal to the bridge B1. Find the value of b/a.
Comprehension
MEDIUM
3 marks
31 August 2025
13
Let
θ
\theta
θ
be the acute angle between bridge B1 and bridge B3 given by x + y = 0 and then what will be the value of 1/cos^2(
θ
\theta
θ
)?
Comprehension
MEDIUM
2 marks
31 August 2025
14
Find the dimension of the affine subspace corresponding to the set of all solutions to the following system:
[
1
2
0
1
0
1
1
2
1
3
1
3
]
\begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 1 & 3 & 1 & 3 \end{bmatrix}
1
0
1
2
1
3
0
1
1
1
2
3
* [x1, x2, x3, x4]^T = [1, 2, 3]^T
Numerical
MEDIUM
3 marks
01 September 2024
15
Let
U
=
{
x
∈
R
4
:
⟨
x
,
u
⟩
=
0
where
u
=
(
1
,
−
1
,
2
,
−
1
)
}
U = \{x \in \mathbb{R}^4 : \langle x, u \rangle = 0 \text{ where } u = (1, -1, 2, -1)\}
U
=
{
x
∈
R
4
:
⟨
x
,
u
⟩
=
0
where
u
=
(
1
,
−
1
,
2
,
−
1
)}
be a subspace of
R
4
\mathbb{R}^4
R
4
with the dot product as the inner product. Find the dimension of
U
U
U
.
Comprehension
EASY
1 marks
28 April 2024
16
Choose the correct option(s) from the following:
Comprehension
MEDIUM
2 marks
24 December 2023
17
Let W = {(x, y, z) in
R
3
\mathbb{R}^3
R
3
: 2x - y - z = 3}. Then which of the following affine subspace(s) of
R
3
\mathbb{R}^3
R
3
is/are same as the subspace W?
Multiple correct
MEDIUM
2 marks
03 September 2023
18
Which of the following statement(s) about similar matrices is/are true?
Multiple correct
EASY
3 marks
03 September 2023
19
Consider the following matrices. v_1 =
[
2
2
0
2
]
\begin{bmatrix} 2 & 2 \\ 0 & 2 \end{bmatrix}
[
2
0
2
2
]
, v_2 =
[
−
1
2
3
1
]
\begin{bmatrix} -1 & 2 \\ 3 & 1 \end{bmatrix}
[
−
1
3
2
1
]
, v_3 =
[
2
0
0
2
]
\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}
[
2
0
0
2
]
, and 1_{4} =
[
−
5
−
3
6
5
]
\begin{bmatrix} -5 & -3 \\ 6 & 5 \end{bmatrix}
[
−
5
6
−
3
5
]
Choose the correct option(s) from the following:
Multiple correct
HARD
3 marks
03 September 2023
20
Let u, v be vectors in
R
3
\mathbb{R}^3
R
3
such that u + v and u - v are orthogonal. If u =
(
1
,
−
2
,
2
)
(1, -2, 2)
(
1
,
−
2
,
2
)
, then ||v|| =
Numerical
EASY
2 marks
03 September 2023
21
Choose the correct option to match the affine subspace of Row 1 with the associated subspaces and dimension.
Comprehension
MEDIUM
1 marks
03 September 2023
22
Choose the correct option to match the affine subspace of Row 2 with the associated subspaces and dimension.
Comprehension
MEDIUM
1 marks
03 September 2023
23
Choose the correct option to match the affine subspace of Row 3 with the associated subspaces and dimension.
Comprehension
MEDIUM
1 marks
03 September 2023
24
Let B be the matrix corresponding to the linear transformation T with respect to the basis {(1, 0, 0),
(
0
,
1
,
0
)
(0, 1, 0)
(
0
,
1
,
0
)
, (0, 0, 1)} for both domain and codomain. The trace of B is
Comprehension
EASY
1 marks
03 September 2023
25
Let B be the matrix corresponding to the linear transformation T with respect to the basis {(1, 0, 0),
(
0
,
1
,
0
)
(0, 1, 0)
(
0
,
1
,
0
)
, (0, 0, 1)} for both domain and codomain. The determinant of B is
Comprehension
EASY
1 marks
03 September 2023
26
Let V1, V2 and V3 represent the subspaces {
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
| 2x + 3y + 4z = 0 }, {
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
| x + 2y + z = 0 } and {
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
| x + y + 2z = 0 }, respectively of
R
3
\mathbb{R}^3
R
3
. Let Ai (i = 1, 2, 3) be the affine spaces corresponding to Vi. Suppose A1 contains
(
1
,
1
,
−
2
)
(1, 1, -2)
(
1
,
1
,
−
2
)
, A2 contains
(
−
2
,
1
,
1
)
(-2, 1, 1)
(
−
2
,
1
,
1
)
and A3 contains
(
1
,
−
2
,
1
)
(1, -2, 1)
(
1
,
−
2
,
1
)
. If (x1, x2, x3) is the point of intersection of A1, A2 and A3, find x1 + 2x2 + x3.
Numerical
HARD
2 marks
30 April 2023
27
Let B be the matrix representation of T with respect to the ordered basis
β
\beta
β
for the domain and the ordered basis {(1, -1), (1, 1)} for the co-domain
R
2
\mathbb{R}^2
R
2
. Which of the following is/are true?
Comprehension
HARD
1 marks
30 April 2023
28
Consider the affine subspaces
W
1
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
}
W_1 = \{(x, y, z) \in \mathbb{R}^3 \mid x + y + z = 1\}
W
1
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
}
and
W
2
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
=
3
}
W_2 = \{(x, y, z) \in \mathbb{R}^3 \mid x = 3\}
W
2
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
=
3
}
. Let
V
1
V_1
V
1
and
V
2
V_2
V
2
be the subspaces corresponding to
W
1
W_1
W
1
and
W
2
W_2
W
2
respectively. Choose the correct option(s).
Multiple correct
MEDIUM
3 marks
11 December 2022
29
Let A be the matrix representation of T with respect to a basis {(1, 1), (1, 2)} for the domain and {(1, 1), (1, -1)} for the co-domain (obtained in the previous question) and B be the matrix representation of T with respect to the standard ordered basis for
R
2
\mathbb{R}^2
R
2
for both the domain and co-domain. Choose the correct option(s).
Multiple correct
MEDIUM
3 marks
11 December 2022
30
An inner product on a vector space V is a function <·, ·> : V x V -> R satisfying the following conditions: Condition 1:
⟨
v
,
v
⟩
\langle v, v \rangle
⟨
v
,
v
⟩
> 0 for all v in V \ {0};
⟨
v
,
v
⟩
\langle v, v \rangle
⟨
v
,
v
⟩
= 0 if and only if v = 0. Condition 2:
⟨
v
1
+
v
2
,
w
⟩
\langle v1 + v2, w \rangle
⟨
v
1
+
v
2
,
w
⟩
=
⟨
v
1
,
w
⟩
\langle v1, w \rangle
⟨
v
1
,
w
⟩
+
⟨
v
2
,
w
⟩
\langle v2, w \rangle
⟨
v
2
,
w
⟩
. Condition 3:
⟨
v
1
,
v
2
⟩
\langle v1, v2 \rangle
⟨
v
1
,
v
2
⟩
=
⟨
v
2
,
v
1
⟩
\langle v2, v1 \rangle
⟨
v
2
,
v
1
⟩
. Condition 4:
⟨
c
v
1
,
v
2
⟩
\langle cv1, v2 \rangle
⟨
c
v
1
,
v
2
⟩
= c
⟨
v
1
,
v
2
⟩
\langle v1, v2 \rangle
⟨
v
1
,
v
2
⟩
. Let V =
R
2
\mathbb{R}^2
R
2
and consider the function defined as: <·, ·> : V x V -> R, <(x1, x2), (y1, y2)> = x1 y1 - 4x2 y2. Which of the following are satisfied by the above function?
Multiple correct
MEDIUM
2 marks
07 August 2022
31
Let A =
[
1
2
3
2
1
2
−
1
1
1
]
\begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 2 \\ -1 & 1 & 1 \end{bmatrix}
1
2
−
1
2
1
1
3
2
1
and B =
[
−
1
3
4
0
2
1
−
1
1
3
]
\begin{bmatrix} -1 & 3 & 4 \\ 0 & 2 & 1 \\ -1 & 1 & 3 \end{bmatrix}
−
1
0
−
1
3
2
1
4
1
3
. Choose the correct option(s).
Multiple correct
MEDIUM
2 marks
07 August 2022
32
Which of the following option is/are true?
Single correct
MEDIUM
2 marks
07 August 2022
Showing 32 questions.
Complete question index for this section:
Question 1 (integer):
Question 2 (integer):
Question 3 (comprehension):
Question 4 (comprehension):
Question 5 (single):
Question 6 (integer):
Question 7 (single):
Question 8 (integer):
Question 9 (single):
Question 10 (comprehension):
Question 11 (comprehension):
Question 12 (comprehension):
Question 13 (comprehension):
Question 14 (integer):
Question 15 (comprehension):
Question 16 (comprehension):
Question 17 (multiple):
Question 18 (multiple):
Question 19 (multiple):
Question 20 (integer):
Question 21 (comprehension):
Question 22 (comprehension):
Question 23 (comprehension):
Question 24 (comprehension):
Question 25 (comprehension):
Question 26 (integer):
Question 27 (comprehension):
Question 28 (multiple):
Question 29 (multiple):
Question 30 (multiple):
Question 31 (multiple):
Question 32 (single):
