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Mathematics 2 > Week 2
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Type:
All
Difficulty:
All
Year:
All
28Q
01
Consider the system of equations in three variables
x
,
y
,
z
x,y,z
x
,
y
,
z
:
x
+
y
+
z
=
3
2
x
+
3
y
+
k
z
=
6
3
x
+
5
y
+
(
k
+
1
)
z
=
9
\begin{aligned}x+y+z&=3\\2x+3y+kz&=6\\3x+5y+(k+1)z&=9\end{aligned}
x
+
y
+
z
2
x
+
3
y
+
k
z
3
x
+
5
y
+
(
k
+
1
)
z
=
3
=
6
=
9
where
k
k
k
is a real parameter. Consider the following statements:
S
1
S_1
S
1
: The system has a unique solution if
k
≠
2
k\ne 2
k
=
2
.
S
2
S_2
S
2
: The system has no solution if
k
=
2
k=2
k
=
2
.
S
3
S_3
S
3
: The system has infinitely many solutions for some value of
k
k
k
. Choose the correct option.
Single correct
MEDIUM
4 marks
15 March 2026
02
If
A
A
A
is a
3
×
4
3\times4
3
×
4
matrix and
b
b
b
is a
3
×
1
3\times1
3
×
1
matrix, then choose the set of correct options.
Multiple correct
MEDIUM
5 marks
15 March 2026
03
A company produces two types of gadgets,
A
A
A
and
B
B
B
. Each gadget
A
A
A
requires
2
2
2
hours of assembly and
3
3
3
hours of testing. Each gadget
B
B
B
requires
3
3
3
hours of assembly and
2
2
2
hours of testing. The company has
8
8
8
hours of assembly and
7
7
7
hours of testing available per week. The profit from gadget
A
A
A
is Rs
5
5
5
, and from gadget
B
B
B
is Rs
4
4
4
. Determine the total profit if the company uses all available assembly and testing hours.
Numerical
EASY
4 marks
15 March 2026
04
Consider a shop that sells muffins, cookies, and cakes. The total sales on three consecutive days are as follows: on Monday,
1
1
1
muffin,
2
2
2
cookies, and
3
3
3
cakes were sold for a total of
140
140
140
. On Tuesday,
2
2
2
muffins,
1
1
1
cookie, and
4
4
4
cakes were sold for a total of
160
160
160
. On Wednesday,
3
3
3
muffins,
3
3
3
cookies, and
5
5
5
cakes were sold for a total of
240
240
240
. Let
m
,
c
,
k
m,c,k
m
,
c
,
k
denote the price of a muffin, a cookie, and a cake, respectively. Formulate a system of linear equations representing the above scenario and find the value of
m
+
c
+
k
m+c+k
m
+
c
+
k
.
Numerical
MEDIUM
4 marks
15 March 2026
05
Consider the system of linear equations with unknowns
x
,
y
,
z
x,y,z
x
,
y
,
z
given by
[
1
2
1
3
0
−
1
1
8
5
]
[
x
y
z
]
=
[
b
1
b
2
b
3
]
,
\begin{bmatrix}1&2&1\\3&0&-1\\1&8&5\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix},
1
3
1
2
0
8
1
−
1
5
x
y
z
=
b
1
b
2
b
3
,
where
b
1
,
b
2
,
b
3
∈
R
b_1,b_2,b_3\in\mathbb{R}
b
1
,
b
2
,
b
3
∈
R
. Choose the equation to be satisfied by
b
1
,
b
2
,
b
3
b_1,b_2,b_3
b
1
,
b
2
,
b
3
to ensure that the given system of equations has a solution.
Single correct
MEDIUM
4 marks
26 October 2025
06
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations with a square coefficient matrix
A
A
A
. Consider the following statements.
S
1
S_1
S
1
: If the RREF of
A
A
A
is the identity matrix, then the given system of equations has a solution.
S
2
S_2
S
2
: If the given system of equations has a solution, then the RREF of
A
A
A
is the identity matrix. Choose the correct option from the following.
Single correct
MEDIUM
6 marks
26 October 2025
07
Find the number of zero rows of
M
M
M
.
Comprehension
MEDIUM
3 marks
13 July 2025
08
Suppose Shakti chooses some values for
a
a
a
and
b
b
b
such that multiple sets of values could represent the prices of
1
1
1
pencil and
1
1
1
pen. What should the value of
a
a
a
be?
Comprehension
MEDIUM
4 marks
13 July 2025
09
Which of the following conditions is equivalent to having no solutions for the given system of equations?
Comprehension
MEDIUM
2 marks
23 February 2025
10
Which of the following conditions is equivalent to having a unique solution for the given system of equations?
Comprehension
EASY
2 marks
23 February 2025
11
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
. Choose all the options from the following which guarantee a unique solution.
Multiple correct
EASY
3 marks
27 October 2024
12
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations, where
A
=
[
1
0
0
−
2
0
0
1
3
0
0
0
1
]
A=\begin{bmatrix}1&0&0&-2\\0&0&1&3\\0&0&0&1\end{bmatrix}
A
=
1
0
0
0
0
0
0
1
0
−
2
3
1
. Find the number of independent variables.
Comprehension
MEDIUM
1 marks
27 October 2024
13
Consider the matrix
A
=
[
1
2
−
3
−
2
1
−
4
1
0
1
]
A=\begin{bmatrix}1&2&-3\\-2&1&-4\\1&0&1\end{bmatrix}
A
=
1
−
2
1
2
1
0
−
3
−
4
1
. Consider the system of linear equations
A
x
=
[
3
b
1
]
Ax=\begin{bmatrix}3\\b\\1\end{bmatrix}
A
x
=
3
b
1
. Find the value of
b
b
b
if the system has infinitely many solutions.
Comprehension
HARD
2 marks
27 October 2024
14
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
, where
A
∈
M
n
×
n
(
R
)
A\in M_{n\times n}(\mathbb{R})
A
∈
M
n
×
n
(
R
)
. Which of the following conditions guarantees that the system is consistent for any
b
∈
R
n
b\in\mathbb{R}^n
b
∈
R
n
? Select all true statements.
Multiple correct
MEDIUM
3 marks
07 July 2024
15
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations, where
A
=
[
1
0
0
−
1
0
0
1
5
0
0
0
1
]
A=\begin{bmatrix}1&0&0&-1\\0&0&1&5\\0&0&0&1\end{bmatrix}
A
=
1
0
0
0
0
0
0
1
0
−
1
5
1
,
x
=
[
x
1
x
2
x
3
x
4
]
T
x=\begin{bmatrix}x_1&x_2&x_3&x_4\end{bmatrix}^T
x
=
[
x
1
x
2
x
3
x
4
]
T
, and
b
∈
R
3
b\in\mathbb{R}^3
b
∈
R
3
. Which of the following statements are true?
Multiple correct
MEDIUM
3 marks
07 July 2024
16
Select all true statements.
Multiple correct
MEDIUM
3 marks
25 February 2024
17
Does there exist an
a
a
a
such that the system has infinitely many solutions? If yes, find the value of
a
a
a
; else write the answer as
100
100
100
.
Comprehension
MEDIUM
2 marks
25 February 2024
18
Find the values of
k
k
k
for which the system of equations has no solution.
Comprehension
EASY
1 marks
29 October 2023
19
Find the values of
k
k
k
for which the system of equations has infinitely many solutions.
Comprehension
EASY
1 marks
29 October 2023
20
If the system has a unique solution
(
a
,
b
)
(a,b)
(
a
,
b
)
, what is
a
−
b
a-b
a
−
b
?
Comprehension
MEDIUM
2 marks
29 October 2023
21
Consider the system of linear equations represented in the matrix form
A
x
=
b
Ax=b
A
x
=
b
, where
A
=
[
1
1
3
1
2
4
1
3
α
]
A=\begin{bmatrix}1&1&3\\1&2&4\\1&3&\alpha\end{bmatrix}
A
=
1
1
1
1
2
3
3
4
α
and
b
=
[
2
3
β
]
b=\begin{bmatrix}2\\3\\\beta\end{bmatrix}
b
=
2
3
β
. What is the value of
α
+
β
\alpha+\beta
α
+
β
if the above system has infinitely many solutions?
Numerical
HARD
3 marks
16 July 2023
22
How many solutions does the given system
A
x
=
b
Ax=b
A
x
=
b
have?
Comprehension
MEDIUM
2 marks
16 July 2023
23
Suppose at
t
=
2
t=2
t
=
2
the distribution vector is
X
2
=
[
1
3
1
2
2
3
]
T
X_2=\begin{bmatrix}\frac13&\frac12&\frac23\end{bmatrix}^T
X
2
=
[
3
1
2
1
3
2
]
T
. Which of the following options are true?
Comprehension
MEDIUM
2 marks
16 October 2022
24
Suppose at
t
=
1
t=1
t
=
1
the distribution vector is
X
1
=
[
1
2
1
2
0
]
T
X_1=\begin{bmatrix}\frac12&\frac12&0\end{bmatrix}^T
X
1
=
[
2
1
2
1
0
]
T
. Which of the following options is true?
Comprehension
EASY
2 marks
16 October 2022
25
Choose the set of correct option(s).
Comprehension
MEDIUM
2 marks
16 October 2022
26
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
0.5
c=0.5
c
=
0.5
and
d
=
100
d=100
d
=
100
, then:
Comprehension
MEDIUM
4 marks
05 June 2022
27
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
2
c=2
c
=
2
and
d
≠
160
d\ne160
d
=
160
, then which option is always true?
Comprehension
MEDIUM
4 marks
05 June 2022
28
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
3
c=3
c
=
3
and
d
=
100
d=100
d
=
100
, then which option is always true?
Comprehension
MEDIUM
4 marks
05 June 2022
Showing 28 questions.
Mathematics 2 > Week 2
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
28Q
01
Consider the system of equations in three variables
x
,
y
,
z
x,y,z
x
,
y
,
z
:
x
+
y
+
z
=
3
2
x
+
3
y
+
k
z
=
6
3
x
+
5
y
+
(
k
+
1
)
z
=
9
\begin{aligned}x+y+z&=3\\2x+3y+kz&=6\\3x+5y+(k+1)z&=9\end{aligned}
x
+
y
+
z
2
x
+
3
y
+
k
z
3
x
+
5
y
+
(
k
+
1
)
z
=
3
=
6
=
9
where
k
k
k
is a real parameter. Consider the following statements:
S
1
S_1
S
1
: The system has a unique solution if
k
≠
2
k\ne 2
k
=
2
.
S
2
S_2
S
2
: The system has no solution if
k
=
2
k=2
k
=
2
.
S
3
S_3
S
3
: The system has infinitely many solutions for some value of
k
k
k
. Choose the correct option.
Single correct
MEDIUM
4 marks
15 March 2026
02
If
A
A
A
is a
3
×
4
3\times4
3
×
4
matrix and
b
b
b
is a
3
×
1
3\times1
3
×
1
matrix, then choose the set of correct options.
Multiple correct
MEDIUM
5 marks
15 March 2026
03
A company produces two types of gadgets,
A
A
A
and
B
B
B
. Each gadget
A
A
A
requires
2
2
2
hours of assembly and
3
3
3
hours of testing. Each gadget
B
B
B
requires
3
3
3
hours of assembly and
2
2
2
hours of testing. The company has
8
8
8
hours of assembly and
7
7
7
hours of testing available per week. The profit from gadget
A
A
A
is Rs
5
5
5
, and from gadget
B
B
B
is Rs
4
4
4
. Determine the total profit if the company uses all available assembly and testing hours.
Numerical
EASY
4 marks
15 March 2026
04
Consider a shop that sells muffins, cookies, and cakes. The total sales on three consecutive days are as follows: on Monday,
1
1
1
muffin,
2
2
2
cookies, and
3
3
3
cakes were sold for a total of
140
140
140
. On Tuesday,
2
2
2
muffins,
1
1
1
cookie, and
4
4
4
cakes were sold for a total of
160
160
160
. On Wednesday,
3
3
3
muffins,
3
3
3
cookies, and
5
5
5
cakes were sold for a total of
240
240
240
. Let
m
,
c
,
k
m,c,k
m
,
c
,
k
denote the price of a muffin, a cookie, and a cake, respectively. Formulate a system of linear equations representing the above scenario and find the value of
m
+
c
+
k
m+c+k
m
+
c
+
k
.
Numerical
MEDIUM
4 marks
15 March 2026
05
Consider the system of linear equations with unknowns
x
,
y
,
z
x,y,z
x
,
y
,
z
given by
[
1
2
1
3
0
−
1
1
8
5
]
[
x
y
z
]
=
[
b
1
b
2
b
3
]
,
\begin{bmatrix}1&2&1\\3&0&-1\\1&8&5\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix},
1
3
1
2
0
8
1
−
1
5
x
y
z
=
b
1
b
2
b
3
,
where
b
1
,
b
2
,
b
3
∈
R
b_1,b_2,b_3\in\mathbb{R}
b
1
,
b
2
,
b
3
∈
R
. Choose the equation to be satisfied by
b
1
,
b
2
,
b
3
b_1,b_2,b_3
b
1
,
b
2
,
b
3
to ensure that the given system of equations has a solution.
Single correct
MEDIUM
4 marks
26 October 2025
06
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations with a square coefficient matrix
A
A
A
. Consider the following statements.
S
1
S_1
S
1
: If the RREF of
A
A
A
is the identity matrix, then the given system of equations has a solution.
S
2
S_2
S
2
: If the given system of equations has a solution, then the RREF of
A
A
A
is the identity matrix. Choose the correct option from the following.
Single correct
MEDIUM
6 marks
26 October 2025
07
Find the number of zero rows of
M
M
M
.
Comprehension
MEDIUM
3 marks
13 July 2025
08
Suppose Shakti chooses some values for
a
a
a
and
b
b
b
such that multiple sets of values could represent the prices of
1
1
1
pencil and
1
1
1
pen. What should the value of
a
a
a
be?
Comprehension
MEDIUM
4 marks
13 July 2025
09
Which of the following conditions is equivalent to having no solutions for the given system of equations?
Comprehension
MEDIUM
2 marks
23 February 2025
10
Which of the following conditions is equivalent to having a unique solution for the given system of equations?
Comprehension
EASY
2 marks
23 February 2025
11
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
. Choose all the options from the following which guarantee a unique solution.
Multiple correct
EASY
3 marks
27 October 2024
12
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations, where
A
=
[
1
0
0
−
2
0
0
1
3
0
0
0
1
]
A=\begin{bmatrix}1&0&0&-2\\0&0&1&3\\0&0&0&1\end{bmatrix}
A
=
1
0
0
0
0
0
0
1
0
−
2
3
1
. Find the number of independent variables.
Comprehension
MEDIUM
1 marks
27 October 2024
13
Consider the matrix
A
=
[
1
2
−
3
−
2
1
−
4
1
0
1
]
A=\begin{bmatrix}1&2&-3\\-2&1&-4\\1&0&1\end{bmatrix}
A
=
1
−
2
1
2
1
0
−
3
−
4
1
. Consider the system of linear equations
A
x
=
[
3
b
1
]
Ax=\begin{bmatrix}3\\b\\1\end{bmatrix}
A
x
=
3
b
1
. Find the value of
b
b
b
if the system has infinitely many solutions.
Comprehension
HARD
2 marks
27 October 2024
14
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
, where
A
∈
M
n
×
n
(
R
)
A\in M_{n\times n}(\mathbb{R})
A
∈
M
n
×
n
(
R
)
. Which of the following conditions guarantees that the system is consistent for any
b
∈
R
n
b\in\mathbb{R}^n
b
∈
R
n
? Select all true statements.
Multiple correct
MEDIUM
3 marks
07 July 2024
15
Let
A
x
=
b
Ax=b
A
x
=
b
be a system of linear equations, where
A
=
[
1
0
0
−
1
0
0
1
5
0
0
0
1
]
A=\begin{bmatrix}1&0&0&-1\\0&0&1&5\\0&0&0&1\end{bmatrix}
A
=
1
0
0
0
0
0
0
1
0
−
1
5
1
,
x
=
[
x
1
x
2
x
3
x
4
]
T
x=\begin{bmatrix}x_1&x_2&x_3&x_4\end{bmatrix}^T
x
=
[
x
1
x
2
x
3
x
4
]
T
, and
b
∈
R
3
b\in\mathbb{R}^3
b
∈
R
3
. Which of the following statements are true?
Multiple correct
MEDIUM
3 marks
07 July 2024
16
Select all true statements.
Multiple correct
MEDIUM
3 marks
25 February 2024
17
Does there exist an
a
a
a
such that the system has infinitely many solutions? If yes, find the value of
a
a
a
; else write the answer as
100
100
100
.
Comprehension
MEDIUM
2 marks
25 February 2024
18
Find the values of
k
k
k
for which the system of equations has no solution.
Comprehension
EASY
1 marks
29 October 2023
19
Find the values of
k
k
k
for which the system of equations has infinitely many solutions.
Comprehension
EASY
1 marks
29 October 2023
20
If the system has a unique solution
(
a
,
b
)
(a,b)
(
a
,
b
)
, what is
a
−
b
a-b
a
−
b
?
Comprehension
MEDIUM
2 marks
29 October 2023
21
Consider the system of linear equations represented in the matrix form
A
x
=
b
Ax=b
A
x
=
b
, where
A
=
[
1
1
3
1
2
4
1
3
α
]
A=\begin{bmatrix}1&1&3\\1&2&4\\1&3&\alpha\end{bmatrix}
A
=
1
1
1
1
2
3
3
4
α
and
b
=
[
2
3
β
]
b=\begin{bmatrix}2\\3\\\beta\end{bmatrix}
b
=
2
3
β
. What is the value of
α
+
β
\alpha+\beta
α
+
β
if the above system has infinitely many solutions?
Numerical
HARD
3 marks
16 July 2023
22
How many solutions does the given system
A
x
=
b
Ax=b
A
x
=
b
have?
Comprehension
MEDIUM
2 marks
16 July 2023
23
Suppose at
t
=
2
t=2
t
=
2
the distribution vector is
X
2
=
[
1
3
1
2
2
3
]
T
X_2=\begin{bmatrix}\frac13&\frac12&\frac23\end{bmatrix}^T
X
2
=
[
3
1
2
1
3
2
]
T
. Which of the following options are true?
Comprehension
MEDIUM
2 marks
16 October 2022
24
Suppose at
t
=
1
t=1
t
=
1
the distribution vector is
X
1
=
[
1
2
1
2
0
]
T
X_1=\begin{bmatrix}\frac12&\frac12&0\end{bmatrix}^T
X
1
=
[
2
1
2
1
0
]
T
. Which of the following options is true?
Comprehension
EASY
2 marks
16 October 2022
25
Choose the set of correct option(s).
Comprehension
MEDIUM
2 marks
16 October 2022
26
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
0.5
c=0.5
c
=
0.5
and
d
=
100
d=100
d
=
100
, then:
Comprehension
MEDIUM
4 marks
05 June 2022
27
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
2
c=2
c
=
2
and
d
≠
160
d\ne160
d
=
160
, then which option is always true?
Comprehension
MEDIUM
4 marks
05 June 2022
28
If we try to find
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
using the appropriate matrix representation by taking
c
=
3
c=3
c
=
3
and
d
=
100
d=100
d
=
100
, then which option is always true?
Comprehension
MEDIUM
4 marks
05 June 2022
Showing 28 questions.
