JEE Main Previous Year Questions (2025): Linear Inequalities
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Page 1
JEE Main Previous Year Questions
(2025): Linear Inequalities
Q1: If the equation ?? ( ?? - ?? ) ?? ?? + ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = ?? has equal roots, where
?? + ?? = ???? and ?? =
????
?? , then ?? ?? + ?? ?? is equal to _ _ _ _
JEE Main 2025 (Online) 23rd January Morning Shift
Ans: 117
Solution:
To solve the given problem, we start with the quadratic equation:
?? ( ?? - ?? ) ?? 2
+ ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = 0
Given that the roots are equal (let's assume both roots are 1 ), we know the sum of the roots,
?? + ?? , is twice the value of one root, which leads us to:
?? + ?? = 2
Using the formula for the sum of roots for a quadratic equation, ?? + ?? = -
?? ( ?? - ?? )
?? ( ?? - ?? )
, we set this
equal to 2 :
-
?? ( ?? - ?? )
?? ( ?? - ?? )
= 2
Solving for this:
- ???? + ???? = 2 ???? - 2 ???? 2 ???? = ???? + ???? 2 ???? = ?? ( ?? + ?? )
Given that ?? + ?? = 15 and ?? =
36
5
, substitute these into the equation:
2 ???? = 15 ?? 2 ???? = 15 ×
36
5
= 108 ???? = 54
Now, using the equation ?? + ?? = 15 and ???? = 54, find ?? 2
+ ?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 15
2
- 2 × 54 ?? 2
+ ?? 2
= 225 - 108 = 117
Therefore, ?? 2
+ ?? 2
is equal to 117 .
Q2: If the set of all ?? ? ?? - { ?? }, for which the roots of the equation ( ?? - ?? ) ?? ?? +
?? ( ?? - ?? ) ?? + ?? = ?? are positive is ( - 8 , - ?? ] ? [ ?? , ?? ) , then ?? ?? + ?? + ?? is equal to
JEE Main 2025 (Online) 2nd April Evening Shift
Ans: 7
Solution:
?? ( ?? ) = ( 1 - ?? ) ?? 2
+ 2 ( ?? - 3 ) ?? + 9 , ?? ( 0 ) = 9 > 0
?? = 0 ? 4 ( ?? - 3 )
2
= 4 ( 1 - ?? ) · 9
? ?? ? ( - 8 , - 3 ] ? [ 0 , 8 ) ( ?? )
Page 2
JEE Main Previous Year Questions
(2025): Linear Inequalities
Q1: If the equation ?? ( ?? - ?? ) ?? ?? + ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = ?? has equal roots, where
?? + ?? = ???? and ?? =
????
?? , then ?? ?? + ?? ?? is equal to _ _ _ _
JEE Main 2025 (Online) 23rd January Morning Shift
Ans: 117
Solution:
To solve the given problem, we start with the quadratic equation:
?? ( ?? - ?? ) ?? 2
+ ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = 0
Given that the roots are equal (let's assume both roots are 1 ), we know the sum of the roots,
?? + ?? , is twice the value of one root, which leads us to:
?? + ?? = 2
Using the formula for the sum of roots for a quadratic equation, ?? + ?? = -
?? ( ?? - ?? )
?? ( ?? - ?? )
, we set this
equal to 2 :
-
?? ( ?? - ?? )
?? ( ?? - ?? )
= 2
Solving for this:
- ???? + ???? = 2 ???? - 2 ???? 2 ???? = ???? + ???? 2 ???? = ?? ( ?? + ?? )
Given that ?? + ?? = 15 and ?? =
36
5
, substitute these into the equation:
2 ???? = 15 ?? 2 ???? = 15 ×
36
5
= 108 ???? = 54
Now, using the equation ?? + ?? = 15 and ???? = 54, find ?? 2
+ ?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 15
2
- 2 × 54 ?? 2
+ ?? 2
= 225 - 108 = 117
Therefore, ?? 2
+ ?? 2
is equal to 117 .
Q2: If the set of all ?? ? ?? - { ?? }, for which the roots of the equation ( ?? - ?? ) ?? ?? +
?? ( ?? - ?? ) ?? + ?? = ?? are positive is ( - 8 , - ?? ] ? [ ?? , ?? ) , then ?? ?? + ?? + ?? is equal to
JEE Main 2025 (Online) 2nd April Evening Shift
Ans: 7
Solution:
?? ( ?? ) = ( 1 - ?? ) ?? 2
+ 2 ( ?? - 3 ) ?? + 9 , ?? ( 0 ) = 9 > 0
?? = 0 ? 4 ( ?? - 3 )
2
= 4 ( 1 - ?? ) · 9
? ?? ? ( - 8 , - 3 ] ? [ 0 , 8 ) ( ?? )
?? 1
+ ?? 2
=
- 2 ( ?? - 3 )
1 - ?? , ?? 1
?? 2
=
9
1 - ??
?? 1
+ ?? 2
> 0 ?
?? - 3
?? - 1
> 0 ? ?? ? ( - 8 , 1 ) ? ( 3 , 8 ) … ( ???? )
?? 1
?? 2
> 0 ? 1 - ?? > 0 ? ?? ? ( - 8 , 1 ) ( ?????? )
? Interaction of (i), (ii) and (iii)
?? ? ( - 8 , - 3 ] ? [ 0 , 1 )
? ?? = 3 , ?? = 0 , ?? = 1 ? 2 ?? + ?? + ?? = 7
Q3: Let ?? ?? and ?? ?? be the distinct roots of ?? ?? ?? + ( ?? ?? ?? ? ?? ) ?? - ?? = ?? , ?? ? ( ?? , ?? ?? ) . If ??
and ?? are the minimum and the maximum values of ?? ?? ?? + ?? ?? ?? , then ???? ( ?? + ?? )
equals :
JEE Main 2025 (Online) 22nd January Evening Shift
Options:
A. 27
B. 17
C. 25
D. 24
Ans: C
Solution:
To find the sum of the fourth powers of the roots ?? ?? and ?? ?? of the quadratic equation 2 ?? 2
+
( c os ? ?? ) ?? - 1 = 0, we start analyzing the expression ?? ?? 4
+ ?? ?? 4
.
Page 3
JEE Main Previous Year Questions
(2025): Linear Inequalities
Q1: If the equation ?? ( ?? - ?? ) ?? ?? + ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = ?? has equal roots, where
?? + ?? = ???? and ?? =
????
?? , then ?? ?? + ?? ?? is equal to _ _ _ _
JEE Main 2025 (Online) 23rd January Morning Shift
Ans: 117
Solution:
To solve the given problem, we start with the quadratic equation:
?? ( ?? - ?? ) ?? 2
+ ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = 0
Given that the roots are equal (let's assume both roots are 1 ), we know the sum of the roots,
?? + ?? , is twice the value of one root, which leads us to:
?? + ?? = 2
Using the formula for the sum of roots for a quadratic equation, ?? + ?? = -
?? ( ?? - ?? )
?? ( ?? - ?? )
, we set this
equal to 2 :
-
?? ( ?? - ?? )
?? ( ?? - ?? )
= 2
Solving for this:
- ???? + ???? = 2 ???? - 2 ???? 2 ???? = ???? + ???? 2 ???? = ?? ( ?? + ?? )
Given that ?? + ?? = 15 and ?? =
36
5
, substitute these into the equation:
2 ???? = 15 ?? 2 ???? = 15 ×
36
5
= 108 ???? = 54
Now, using the equation ?? + ?? = 15 and ???? = 54, find ?? 2
+ ?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 15
2
- 2 × 54 ?? 2
+ ?? 2
= 225 - 108 = 117
Therefore, ?? 2
+ ?? 2
is equal to 117 .
Q2: If the set of all ?? ? ?? - { ?? }, for which the roots of the equation ( ?? - ?? ) ?? ?? +
?? ( ?? - ?? ) ?? + ?? = ?? are positive is ( - 8 , - ?? ] ? [ ?? , ?? ) , then ?? ?? + ?? + ?? is equal to
JEE Main 2025 (Online) 2nd April Evening Shift
Ans: 7
Solution:
?? ( ?? ) = ( 1 - ?? ) ?? 2
+ 2 ( ?? - 3 ) ?? + 9 , ?? ( 0 ) = 9 > 0
?? = 0 ? 4 ( ?? - 3 )
2
= 4 ( 1 - ?? ) · 9
? ?? ? ( - 8 , - 3 ] ? [ 0 , 8 ) ( ?? )
?? 1
+ ?? 2
=
- 2 ( ?? - 3 )
1 - ?? , ?? 1
?? 2
=
9
1 - ??
?? 1
+ ?? 2
> 0 ?
?? - 3
?? - 1
> 0 ? ?? ? ( - 8 , 1 ) ? ( 3 , 8 ) … ( ???? )
?? 1
?? 2
> 0 ? 1 - ?? > 0 ? ?? ? ( - 8 , 1 ) ( ?????? )
? Interaction of (i), (ii) and (iii)
?? ? ( - 8 , - 3 ] ? [ 0 , 1 )
? ?? = 3 , ?? = 0 , ?? = 1 ? 2 ?? + ?? + ?? = 7
Q3: Let ?? ?? and ?? ?? be the distinct roots of ?? ?? ?? + ( ?? ?? ?? ? ?? ) ?? - ?? = ?? , ?? ? ( ?? , ?? ?? ) . If ??
and ?? are the minimum and the maximum values of ?? ?? ?? + ?? ?? ?? , then ???? ( ?? + ?? )
equals :
JEE Main 2025 (Online) 22nd January Evening Shift
Options:
A. 27
B. 17
C. 25
D. 24
Ans: C
Solution:
To find the sum of the fourth powers of the roots ?? ?? and ?? ?? of the quadratic equation 2 ?? 2
+
( c os ? ?? ) ?? - 1 = 0, we start analyzing the expression ?? ?? 4
+ ?? ?? 4
.
The equation can be rewritten with its roots using:
?? + ?? = -
c os ? ?? 2
, ???? = -
1
2
We need to calculate ?? 2
+ ?? 2
and ?? 2
?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = ( -
c os ? ?? 2
)
2
- 2 ( -
1
2
) =
c os
2
? ?? 4
+ 1
?? 2
?? 2
= ( ???? )
2
= ( -
1
2
)
2
=
1
4
Substitute these into:
?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2 ?? 2
?? 2
= (
c o s
2
? ?? 4
+ 1 )
2
-
1
2
Maximize and minimize (
c os
2
? ?? 4
+ 1 )
2
:
Zero of c os ? ?? leads to:
(
0
2
4
+ 1 )
2
= 1
Max value c os
2
? ?? = 1 :
(
1
4
+ 1 )
2
= (
5
4
)
2
=
25
16
Substitute back:
Max:
25
16
-
1
2
=
25
16
-
8
16
=
17
16
Min: 1 -
1
2
=
1
2
Finally, compute 16 ( ?? + ?? ) :
16 (
17
16
+
1
2
) = 16 (
17
16
+
8
16
) = 16 ×
25
16
= 25
Q4: The product of all the rational roots of the equation ( ?? ?? - ?? ?? + ???? )
?? - ( ?? -
?? ) ( ?? - ?? ) = ?? , is equal to
JEE Main 2025 (Online) 24th January Morning Shift
Options:
A. 7
B. 21
C. 28
D. 14
Ans: D
Solution:
To solve the given equation, start by rewriting the expression:
( ?? 2
- 9 ?? + 11 )
2
- ( ?? - 4 ) ( ?? - 5 ) = 3
First, simplify the second part of the expression:
( ?? - 4 ) ( ?? - 5 ) = ?? 2
- 9 ?? + 20
Page 4
JEE Main Previous Year Questions
(2025): Linear Inequalities
Q1: If the equation ?? ( ?? - ?? ) ?? ?? + ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = ?? has equal roots, where
?? + ?? = ???? and ?? =
????
?? , then ?? ?? + ?? ?? is equal to _ _ _ _
JEE Main 2025 (Online) 23rd January Morning Shift
Ans: 117
Solution:
To solve the given problem, we start with the quadratic equation:
?? ( ?? - ?? ) ?? 2
+ ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = 0
Given that the roots are equal (let's assume both roots are 1 ), we know the sum of the roots,
?? + ?? , is twice the value of one root, which leads us to:
?? + ?? = 2
Using the formula for the sum of roots for a quadratic equation, ?? + ?? = -
?? ( ?? - ?? )
?? ( ?? - ?? )
, we set this
equal to 2 :
-
?? ( ?? - ?? )
?? ( ?? - ?? )
= 2
Solving for this:
- ???? + ???? = 2 ???? - 2 ???? 2 ???? = ???? + ???? 2 ???? = ?? ( ?? + ?? )
Given that ?? + ?? = 15 and ?? =
36
5
, substitute these into the equation:
2 ???? = 15 ?? 2 ???? = 15 ×
36
5
= 108 ???? = 54
Now, using the equation ?? + ?? = 15 and ???? = 54, find ?? 2
+ ?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 15
2
- 2 × 54 ?? 2
+ ?? 2
= 225 - 108 = 117
Therefore, ?? 2
+ ?? 2
is equal to 117 .
Q2: If the set of all ?? ? ?? - { ?? }, for which the roots of the equation ( ?? - ?? ) ?? ?? +
?? ( ?? - ?? ) ?? + ?? = ?? are positive is ( - 8 , - ?? ] ? [ ?? , ?? ) , then ?? ?? + ?? + ?? is equal to
JEE Main 2025 (Online) 2nd April Evening Shift
Ans: 7
Solution:
?? ( ?? ) = ( 1 - ?? ) ?? 2
+ 2 ( ?? - 3 ) ?? + 9 , ?? ( 0 ) = 9 > 0
?? = 0 ? 4 ( ?? - 3 )
2
= 4 ( 1 - ?? ) · 9
? ?? ? ( - 8 , - 3 ] ? [ 0 , 8 ) ( ?? )
?? 1
+ ?? 2
=
- 2 ( ?? - 3 )
1 - ?? , ?? 1
?? 2
=
9
1 - ??
?? 1
+ ?? 2
> 0 ?
?? - 3
?? - 1
> 0 ? ?? ? ( - 8 , 1 ) ? ( 3 , 8 ) … ( ???? )
?? 1
?? 2
> 0 ? 1 - ?? > 0 ? ?? ? ( - 8 , 1 ) ( ?????? )
? Interaction of (i), (ii) and (iii)
?? ? ( - 8 , - 3 ] ? [ 0 , 1 )
? ?? = 3 , ?? = 0 , ?? = 1 ? 2 ?? + ?? + ?? = 7
Q3: Let ?? ?? and ?? ?? be the distinct roots of ?? ?? ?? + ( ?? ?? ?? ? ?? ) ?? - ?? = ?? , ?? ? ( ?? , ?? ?? ) . If ??
and ?? are the minimum and the maximum values of ?? ?? ?? + ?? ?? ?? , then ???? ( ?? + ?? )
equals :
JEE Main 2025 (Online) 22nd January Evening Shift
Options:
A. 27
B. 17
C. 25
D. 24
Ans: C
Solution:
To find the sum of the fourth powers of the roots ?? ?? and ?? ?? of the quadratic equation 2 ?? 2
+
( c os ? ?? ) ?? - 1 = 0, we start analyzing the expression ?? ?? 4
+ ?? ?? 4
.
The equation can be rewritten with its roots using:
?? + ?? = -
c os ? ?? 2
, ???? = -
1
2
We need to calculate ?? 2
+ ?? 2
and ?? 2
?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = ( -
c os ? ?? 2
)
2
- 2 ( -
1
2
) =
c os
2
? ?? 4
+ 1
?? 2
?? 2
= ( ???? )
2
= ( -
1
2
)
2
=
1
4
Substitute these into:
?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2 ?? 2
?? 2
= (
c o s
2
? ?? 4
+ 1 )
2
-
1
2
Maximize and minimize (
c os
2
? ?? 4
+ 1 )
2
:
Zero of c os ? ?? leads to:
(
0
2
4
+ 1 )
2
= 1
Max value c os
2
? ?? = 1 :
(
1
4
+ 1 )
2
= (
5
4
)
2
=
25
16
Substitute back:
Max:
25
16
-
1
2
=
25
16
-
8
16
=
17
16
Min: 1 -
1
2
=
1
2
Finally, compute 16 ( ?? + ?? ) :
16 (
17
16
+
1
2
) = 16 (
17
16
+
8
16
) = 16 ×
25
16
= 25
Q4: The product of all the rational roots of the equation ( ?? ?? - ?? ?? + ???? )
?? - ( ?? -
?? ) ( ?? - ?? ) = ?? , is equal to
JEE Main 2025 (Online) 24th January Morning Shift
Options:
A. 7
B. 21
C. 28
D. 14
Ans: D
Solution:
To solve the given equation, start by rewriting the expression:
( ?? 2
- 9 ?? + 11 )
2
- ( ?? - 4 ) ( ?? - 5 ) = 3
First, simplify the second part of the expression:
( ?? - 4 ) ( ?? - 5 ) = ?? 2
- 9 ?? + 20
Now the equation becomes:
( ?? 2
- 9 ?? + 11 )
2
- ( ?? 2
- 9 ?? + 20 ) = 3
Introduce a substitution for simplification:
Let ?? = ?? 2
- 9 ??
Thus, the equation transforms to:
?? 2
+ 22 ?? + 121 - ?? - 20 - 3 = 0
Simplify further:
?? 2
+ 21 ?? + 98 = 0
Factor the quadratic:
( ?? + 14 ) ( ?? + 7 ) = 0
This gives:
?? = - 7 ? or ? ?? = - 14
Address each case where ?? = ?? 2
- 9 ?? :
?? 2
- 9 ?? = - 7
?? 2
- 9 ?? + 7 = 0
Solving this quadratic equation, we find the roots:
?? =
9 ± v 81 - 4 × 7
2
=
9 ± v 53
2
?? 2
- 9 ?? = - 14
?? 2
- 9 ?? + 14 = 0
Solving this quadratic equation:
?? =
9 ± v 81 - 4 × 14
2
=
9 ± v 25
2
?? =
9 ± 5
2
= 7 ? or ? ?? = 2
The rational roots from the second equation are 7 and 2 . Thus, the product of all the rational
roots is:
7 × 2 = 14
Q5: The number of real solution(s) of the equation
?? ?? + ?? ?? + ?? = ?? ???? { | ?? - ?? | , | ?? + ?? | } is :
JEE Main 2025 (Online) 24th January Evening Shift
Options:
A. 2
B. 3
C. 1
D. 0
Ans: A
Solution:
Page 5
JEE Main Previous Year Questions
(2025): Linear Inequalities
Q1: If the equation ?? ( ?? - ?? ) ?? ?? + ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = ?? has equal roots, where
?? + ?? = ???? and ?? =
????
?? , then ?? ?? + ?? ?? is equal to _ _ _ _
JEE Main 2025 (Online) 23rd January Morning Shift
Ans: 117
Solution:
To solve the given problem, we start with the quadratic equation:
?? ( ?? - ?? ) ?? 2
+ ?? ( ?? - ?? ) ?? + ?? ( ?? - ?? ) = 0
Given that the roots are equal (let's assume both roots are 1 ), we know the sum of the roots,
?? + ?? , is twice the value of one root, which leads us to:
?? + ?? = 2
Using the formula for the sum of roots for a quadratic equation, ?? + ?? = -
?? ( ?? - ?? )
?? ( ?? - ?? )
, we set this
equal to 2 :
-
?? ( ?? - ?? )
?? ( ?? - ?? )
= 2
Solving for this:
- ???? + ???? = 2 ???? - 2 ???? 2 ???? = ???? + ???? 2 ???? = ?? ( ?? + ?? )
Given that ?? + ?? = 15 and ?? =
36
5
, substitute these into the equation:
2 ???? = 15 ?? 2 ???? = 15 ×
36
5
= 108 ???? = 54
Now, using the equation ?? + ?? = 15 and ???? = 54, find ?? 2
+ ?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = 15
2
- 2 × 54 ?? 2
+ ?? 2
= 225 - 108 = 117
Therefore, ?? 2
+ ?? 2
is equal to 117 .
Q2: If the set of all ?? ? ?? - { ?? }, for which the roots of the equation ( ?? - ?? ) ?? ?? +
?? ( ?? - ?? ) ?? + ?? = ?? are positive is ( - 8 , - ?? ] ? [ ?? , ?? ) , then ?? ?? + ?? + ?? is equal to
JEE Main 2025 (Online) 2nd April Evening Shift
Ans: 7
Solution:
?? ( ?? ) = ( 1 - ?? ) ?? 2
+ 2 ( ?? - 3 ) ?? + 9 , ?? ( 0 ) = 9 > 0
?? = 0 ? 4 ( ?? - 3 )
2
= 4 ( 1 - ?? ) · 9
? ?? ? ( - 8 , - 3 ] ? [ 0 , 8 ) ( ?? )
?? 1
+ ?? 2
=
- 2 ( ?? - 3 )
1 - ?? , ?? 1
?? 2
=
9
1 - ??
?? 1
+ ?? 2
> 0 ?
?? - 3
?? - 1
> 0 ? ?? ? ( - 8 , 1 ) ? ( 3 , 8 ) … ( ???? )
?? 1
?? 2
> 0 ? 1 - ?? > 0 ? ?? ? ( - 8 , 1 ) ( ?????? )
? Interaction of (i), (ii) and (iii)
?? ? ( - 8 , - 3 ] ? [ 0 , 1 )
? ?? = 3 , ?? = 0 , ?? = 1 ? 2 ?? + ?? + ?? = 7
Q3: Let ?? ?? and ?? ?? be the distinct roots of ?? ?? ?? + ( ?? ?? ?? ? ?? ) ?? - ?? = ?? , ?? ? ( ?? , ?? ?? ) . If ??
and ?? are the minimum and the maximum values of ?? ?? ?? + ?? ?? ?? , then ???? ( ?? + ?? )
equals :
JEE Main 2025 (Online) 22nd January Evening Shift
Options:
A. 27
B. 17
C. 25
D. 24
Ans: C
Solution:
To find the sum of the fourth powers of the roots ?? ?? and ?? ?? of the quadratic equation 2 ?? 2
+
( c os ? ?? ) ?? - 1 = 0, we start analyzing the expression ?? ?? 4
+ ?? ?? 4
.
The equation can be rewritten with its roots using:
?? + ?? = -
c os ? ?? 2
, ???? = -
1
2
We need to calculate ?? 2
+ ?? 2
and ?? 2
?? 2
:
?? 2
+ ?? 2
= ( ?? + ?? )
2
- 2 ???? = ( -
c os ? ?? 2
)
2
- 2 ( -
1
2
) =
c os
2
? ?? 4
+ 1
?? 2
?? 2
= ( ???? )
2
= ( -
1
2
)
2
=
1
4
Substitute these into:
?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2 ?? 2
?? 2
= (
c o s
2
? ?? 4
+ 1 )
2
-
1
2
Maximize and minimize (
c os
2
? ?? 4
+ 1 )
2
:
Zero of c os ? ?? leads to:
(
0
2
4
+ 1 )
2
= 1
Max value c os
2
? ?? = 1 :
(
1
4
+ 1 )
2
= (
5
4
)
2
=
25
16
Substitute back:
Max:
25
16
-
1
2
=
25
16
-
8
16
=
17
16
Min: 1 -
1
2
=
1
2
Finally, compute 16 ( ?? + ?? ) :
16 (
17
16
+
1
2
) = 16 (
17
16
+
8
16
) = 16 ×
25
16
= 25
Q4: The product of all the rational roots of the equation ( ?? ?? - ?? ?? + ???? )
?? - ( ?? -
?? ) ( ?? - ?? ) = ?? , is equal to
JEE Main 2025 (Online) 24th January Morning Shift
Options:
A. 7
B. 21
C. 28
D. 14
Ans: D
Solution:
To solve the given equation, start by rewriting the expression:
( ?? 2
- 9 ?? + 11 )
2
- ( ?? - 4 ) ( ?? - 5 ) = 3
First, simplify the second part of the expression:
( ?? - 4 ) ( ?? - 5 ) = ?? 2
- 9 ?? + 20
Now the equation becomes:
( ?? 2
- 9 ?? + 11 )
2
- ( ?? 2
- 9 ?? + 20 ) = 3
Introduce a substitution for simplification:
Let ?? = ?? 2
- 9 ??
Thus, the equation transforms to:
?? 2
+ 22 ?? + 121 - ?? - 20 - 3 = 0
Simplify further:
?? 2
+ 21 ?? + 98 = 0
Factor the quadratic:
( ?? + 14 ) ( ?? + 7 ) = 0
This gives:
?? = - 7 ? or ? ?? = - 14
Address each case where ?? = ?? 2
- 9 ?? :
?? 2
- 9 ?? = - 7
?? 2
- 9 ?? + 7 = 0
Solving this quadratic equation, we find the roots:
?? =
9 ± v 81 - 4 × 7
2
=
9 ± v 53
2
?? 2
- 9 ?? = - 14
?? 2
- 9 ?? + 14 = 0
Solving this quadratic equation:
?? =
9 ± v 81 - 4 × 14
2
=
9 ± v 25
2
?? =
9 ± 5
2
= 7 ? or ? ?? = 2
The rational roots from the second equation are 7 and 2 . Thus, the product of all the rational
roots is:
7 × 2 = 14
Q5: The number of real solution(s) of the equation
?? ?? + ?? ?? + ?? = ?? ???? { | ?? - ?? | , | ?? + ?? | } is :
JEE Main 2025 (Online) 24th January Evening Shift
Options:
A. 2
B. 3
C. 1
D. 0
Ans: A
Solution:
Only 2 solutions.
Q6: The sum, of the squares of all the roots of the equation ?? ?? + | ?? ?? - ?? | - ?? = ?? , is
JEE Main 2025 (Online) 28th January Morning Shift
Options:
A. 6 ( 2 - v 2 )
B. 3 ( 3 - v 2 )
C. 3 ( 2 - v 2 )
D. 6 ( 3 - v 2 )
Ans: A
Solution:
To find the sum of the squares of all the roots of the equation ?? 2
+ | 2 ?? - 3 | - 4 = 0 :
Case I: ?? =
3
2
For ?? =
3
2
, the expression | 2 ?? - 3 | becomes 2 ?? - 3. Thus, the equation becomes:
?? 2
+ 2 ?? - 3 - 4 = 0
Simplifying gives:
?? 2
+ 2 ?? - 7 = 0
Solving this quadratic equation, we find:
?? = 2 v 2 - 1
Case II: ?? <
3
2
For ?? <
3
2
, the expression | 2 ?? - 3 | becomes - ( 2 ?? - 3 ) = - 2 ?? + 3. The equation therefore
becomes:
?? 2
+ 3 - 2 ?? - 4 = 0
Simplifying gives:
Read MoreFAQs on JEE Main Previous Year Questions (2025): Linear Inequalities
| 1. What are linear inequalities and how do they differ from linear equations? |  |
Ans.Linear inequalities are mathematical expressions that show the relationship between two variables where one variable is greater than, less than, greater than or equal to, or less than or equal to another variable. Unlike linear equations, which equate two expressions to find a specific solution, linear inequalities represent a range of solutions on a number line or in a coordinate plane. For example, the linear inequality x + 2 < 5 indicates that x can take on any value less than 3, whereas the equation x + 2 = 5 has a single solution, x = 3.
| 2. How can we graph a linear inequality on a coordinate plane? | |
Ans.To graph a linear inequality, first, convert the inequality into an equation by replacing the inequality sign with an equal sign. For example, if we have the inequality y > 2x + 1, we first graph the line y = 2x + 1. This line will be dashed if the inequality is strict (>, <) and solid if it is inclusive (≥, ≤). Next, choose a test point that is not on the line (commonly the origin (0,0) if it is not on the line) to determine which side of the line to shade. If the test point satisfies the inequality, shade the region containing the point; if not, shade the opposite side.
| 3. What are some common methods to solve systems of linear inequalities? | |
Ans.Common methods to solve systems of linear inequalities include graphing, substitution, and elimination. When using graphing, each inequality is graphed on the same coordinate plane, and the solution is the region where the shaded areas overlap. In substitution, one inequality is solved for one variable, and this expression is substituted into the other inequality. The elimination method involves manipulating the inequalities to eliminate one variable, allowing for the solution of the other variable. The final solution is usually represented as a shaded region on the graph.
| 4. How do we interpret the solutions of a linear inequality? | |
Ans.The solutions of a linear inequality can be interpreted as a range of values that satisfy the condition imposed by the inequality. For example, if we have the inequality x ≤ 5, it indicates that x can take any value from negative infinity up to and including 5. When graphed, this would be represented by a shaded line extending leftward from 5, including 5 itself. Thus, the solutions are not just singular points but rather a continuum of values that meet the inequality's requirements.
| 5. Can you provide a real-world example of linear inequalities? | |
Ans.A real-world example of linear inequalities could be in budgeting. Suppose a person has a monthly budget of $500 for groceries and wants to allocate money between fruits and vegetables. If the cost of fruits is represented by x and vegetables by y, we can formulate the inequality x + y ≤ 500. Here, the solutions represent all possible combinations of x and y that do not exceed the budget, allowing for a range of spending on each category while still adhering to financial constraints.
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Apr 26, 2026 Last updated
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