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IITM BS Week 8 Mathematics 2 Questions | Prasnya
Mathematics 2 > Week 8
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Difficulty:
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33Q
01
Let v in
R
3
\mathbb{R}^3
R
3
be a vector such that ||v|| = 5. If u is the vector obtained from v after the anti-clockwise rotation in the XY-plane by an angle 70 degrees about the Z-axis, then find the length of the vector u.
Numerical
EASY
5 marks
10 May 2026
02
Choose the set of correct options.
Multiple correct
MEDIUM
8 marks
21 December 2025
03
Consider the subspace
W
=
span
{
(
0
,
1
,
0
)
,
(
−
1
,
0
,
1
)
}
W = \text{span}\{(0, 1, 0), (-1, 0, 1)\}
W
=
span
{(
0
,
1
,
0
)
,
(
−
1
,
0
,
1
)}
of
R
3
\mathbb{R}^3
R
3
with the usual inner product. Consider the projection map
P
W
:
R
3
→
R
3
P_W: \mathbb{R}^3 \to \mathbb{R}^3
P
W
:
R
3
→
R
3
and assume that
P
W
(
1
,
3
,
1
)
=
(
v
1
,
v
2
,
v
3
)
P_W(1, 3, 1) = (v_1, v_2, v_3)
P
W
(
1
,
3
,
1
)
=
(
v
1
,
v
2
,
v
3
)
. Calculate
v
1
+
v
2
+
v
3
v_1 + v_2 + v_3
v
1
+
v
2
+
v
3
.
Numerical
MEDIUM
5 marks
21 December 2025
04
Consider the subspace
W
=
span
{
(
1
,
2
,
3
,
0
)
,
(
0
,
1
,
0
,
0
)
}
W = \text{span}\{(1, 2, 3, 0), (0, 1, 0, 0)\}
W
=
span
{(
1
,
2
,
3
,
0
)
,
(
0
,
1
,
0
,
0
)}
of
R
4
\mathbb{R}^4
R
4
with the usual inner product. Define
W
⊥
=
{
v
∈
R
4
∣
w
⋅
v
=
0
for all
w
∈
W
}
W^\perp = \{v \in \mathbb{R}^4 \mid w \cdot v = 0 \text{ for all } w \in W\}
W
⊥
=
{
v
∈
R
4
∣
w
⋅
v
=
0
for all
w
∈
W
}
. What is the dimension of
W
⊥
W^\perp
W
⊥
?
Numerical
EASY
6 marks
21 December 2025
05
Consider the subspace
W
=
span
{
(
1
,
0
,
1
)
,
(
1
,
4
,
−
1
)
}
W = \text{span}\{(1, 0, 1), (1, 4, -1)\}
W
=
span
{(
1
,
0
,
1
)
,
(
1
,
4
,
−
1
)}
in
R
3
\mathbb{R}^3
R
3
with the usual inner product and a point
P
P
P
with coordinates
(
0
,
3
,
0
)
(0, 3, 0)
(
0
,
3
,
0
)
in
R
3
\mathbb{R}^3
R
3
. What is the minimum distance between the point
P
P
P
and the subspace
W
W
W
?
Numerical
MEDIUM
5 marks
21 December 2025
06
Define
S
=
{
A
=
[
a
b
c
3
2
]
|
A
is orthogonal, and
a
,
b
,
c
∈
R
}
S = \left\{ A = \begin{bmatrix} a & b \\ c & \frac{\sqrt{3}}{2} \end{bmatrix} \;\middle|\; A \text{ is orthogonal, and } a, b, c \in \mathbb{R} \right\}
S
=
{
A
=
[
a
c
b
2
3
]
A
is orthogonal, and
a
,
b
,
c
∈
R
}
. What is the cardinality of
S
S
S
?
Numerical
MEDIUM
6 marks
21 December 2025
07
Consider
R
4
\mathbb{R}^4
R
4
with the usual inner product and a subspace
W
=
span
{
(
2
,
0
,
2
,
0
)
,
(
0
,
−
3
,
−
3
,
0
)
,
(
1
,
1
,
2
,
1
)
,
(
4
,
0
,
4
,
−
3
)
}
W = \text{span}\{(2, 0, 2, 0), (0, -3, -3, 0), (1, 1, 2, 1), (4, 0, 4, -3)\}
W
=
span
{(
2
,
0
,
2
,
0
)
,
(
0
,
−
3
,
−
3
,
0
)
,
(
1
,
1
,
2
,
1
)
,
(
4
,
0
,
4
,
−
3
)}
of
R
4
\mathbb{R}^4
R
4
. Define
W
⊥
=
{
v
∈
R
4
∣
w
⋅
v
=
0
for all
w
∈
W
}
W^\perp = \{v \in \mathbb{R}^4 \mid w \cdot v = 0 \text{ for all } w \in W\}
W
⊥
=
{
v
∈
R
4
∣
w
⋅
v
=
0
for all
w
∈
W
}
. What is the dimension of
W
⊥
W^\perp
W
⊥
?
Numerical
MEDIUM
6 marks
21 December 2025
08
Consider the following sets. v_1(2) = { A in M_{2x2}(R) | A^T A = v_2 } and v_2(2) = { A in M_{2x2}(R) | A A^T = v_2 } Choose all the correct options from the following.
Multiple correct
MEDIUM
3 marks
31 August 2025
09
Let S denote the set of all unit directions along which there is no change in f, and which are perpendicular to (1/sqrt(10), 0, -3/sqrt(10)). Find dim(Span{S}).
Comprehension
MEDIUM
1 marks
31 August 2025
10
Suppose the ship sails along the shortest path to the bridge and meets the bridge B1 at (
α
\alpha
α
,
β
\beta
β
). Find the value of 34(
α
\alpha
α
+
β
\beta
β
).
Comprehension
HARD
3 marks
31 August 2025
11
Let V be an inner product space and W be a subspace of V. If P is the projection of V on W, choose the correct option(s).
Multiple correct
MEDIUM
3 marks
13 April 2025
12
If
(
a
,
b
)
(a, b)
(
a
,
b
)
is the projection of the vector
(
3
,
1
)
(3, 1)
(
3
,
1
)
on the line y = 2x using the standard inner product, find a + b.
Numerical
EASY
2 marks
13 April 2025
13
If P_W(0, 1, 2) =
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, find a + b + c.
Comprehension
MEDIUM
1.5 marks
22 December 2024
14
If (
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
) is a vector in the kernel of P_W, find
α
\alpha
α
+
β
\beta
β
+
γ
\gamma
γ
.
Comprehension
MEDIUM
1.5 marks
22 December 2024
15
What is the dimension of the orthogonal complement W^perp?
Comprehension
EASY
1 marks
22 December 2024
16
Which of the following is an orthonormal basis for
span
{
(
1
,
1
,
−
1
)
,
(
1
,
2
,
0
)
}
\text{span}\{(1, 1, -1), (1, 2, 0)\}
span
{(
1
,
1
,
−
1
)
,
(
1
,
2
,
0
)}
?
Single correct
MEDIUM
3 marks
01 September 2024
17
Let v = (sqrt(2), -sqrt(2), 6) in
R
3
\mathbb{R}^3
R
3
.
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is the vector obtained from v after rotating the XY plane anti-clockwise by 45 degrees about the Z axis. Find the value of a + b + c. Enter the nearest integer as your answer.
Numerical
MEDIUM
3 marks
01 September 2024
18
Choose the correct option(s) from the following:
Multiple correct
HARD
4 marks
28 April 2024
19
find a.
Comprehension
MEDIUM
2 marks
28 April 2024
20
find b.
Comprehension
MEDIUM
2 marks
28 April 2024
21
Find
dim
(
W
⊥
)
\dim(W^\perp)
dim
(
W
⊥
)
.
Comprehension
EASY
1 marks
24 December 2023
22
If
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is a vector in
W
⊥
W^\perp
W
⊥
, then
2
a
+
b
2a+b
2
a
+
b
equals
Comprehension
EASY
1 marks
24 December 2023
23
Let
W
=
span
{
(
1
,
−
1
,
1
)
,
(
0
,
2
,
1
)
}
W = \text{span}\{(1, -1, 1), (0, 2, 1)\}
W
=
span
{(
1
,
−
1
,
1
)
,
(
0
,
2
,
1
)}
in
R
3
\mathbb{R}^3
R
3
. Find a basis for
W
⊥
W^\perp
W
⊥
.
Comprehension
MEDIUM
2 marks
24 December 2023
24
Let U = {(x, y, z) in
R
3
\mathbb{R}^3
R
3
: x = y = z} and V = {(x, y, z) in
R
3
\mathbb{R}^3
R
3
: x + y + z = 0}. Let 1_{U} and 1_{V} be the projections on the spaces U and V respectively. Which of the following statement(s) is/are true?
Multiple correct
HARD
2 marks
03 September 2023
25
Let A be an orthogonal matrix. Then the sum of squares of the elements of every row is :
Numerical
EASY
1 marks
03 September 2023
26
Let A be a 3 x 3 orthogonal matrix with positive determinant. Which of the following option(s) is/are true?
Multiple correct
MEDIUM
1 marks
30 April 2023
27
If
γ
\gamma
γ
is the orthonormal basis of W obtained from
β
\beta
β
(from the previous question) by using the Gram-Schmidt process with respect to the usual inner product, and
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is the projection of
(
1
,
0
,
1
)
(1, 0, 1)
(
1
,
0
,
1
)
onto W, then what is 6(a + b + c)?
Comprehension
HARD
4 marks
30 April 2023
28
Let A be a 3x3 orthogonal matrix. Which of the following options is/are true?
Multiple correct
EASY
3 marks
11 December 2022
29
Consider V =
R
3
\mathbb{R}^3
R
3
with inner product as the dot product and W = {
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
| x = y } is a subspace of V. If
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is the projection of
(
1
,
2
,
3
)
(1, 2, 3)
(
1
,
2
,
3
)
onto W, then what is a + b + 2c?
Numerical
MEDIUM
3 marks
11 December 2022
30
Find the dimension of the image space PW.
Comprehension
EASY
1 marks
07 August 2022
31
Find the dimension of the null space of PW.
Comprehension
EASY
2 marks
07 August 2022
32
If v in W is such that ||v|| = 2, then find ||P_W(v)||.
Comprehension
EASY
1 marks
07 August 2022
33
Choose the correct options about the matrix B.
Comprehension
MEDIUM
2 marks
07 August 2022
Showing 33 questions.
Complete question index for this section:
Question 1 (integer):
Question 2 (multiple):
Question 3 (integer):
Question 4 (integer):
Question 5 (integer):
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Question 7 (integer):
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Question 10 (comprehension):
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Question 12 (integer):
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Question 17 (integer):
Question 18 (multiple):
Question 19 (comprehension):
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Question 21 (comprehension):
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Question 23 (comprehension):
Question 24 (multiple):
Question 25 (integer):
Question 26 (multiple):
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Question 28 (multiple):
Question 29 (integer):
Question 30 (comprehension):
Question 31 (comprehension):
Question 32 (comprehension):
Question 33 (comprehension):
