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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 are they different from linear equations? |  |
Ans.Linear inequalities are mathematical statements that express a relationship between two expressions, where one expression is greater than, less than, or not equal to the other. Unlike linear equations, which have a specific solution where two expressions are equal, linear inequalities allow for a range of solutions and are represented graphically by shaded regions on a coordinate plane. For example, the inequality 2x + 3 < 7="" represents="" all="" values="" of="" x="" that="" satisfy="" this="" condition,="" while="" the="" equation="" 2x="" +="" 3="7" has="" a="" single="" solution.=""
| 2.="" how="" do="" you="" solve="" a="" system="" of="" linear="" inequalities?="" | |
="" ans.to="" solve="" a="" system="" of="" linear="" inequalities,="" you="" follow="" these="" steps:="" first,="" graph="" each="" inequality="" on="" the="" same="" coordinate="" plane.="" for="" each="" inequality,="" the="" boundary="" line="" (where="" the="" inequality="" becomes="" an="" equation)="" is="" drawn,="" and="" the="" region="" that="" satisfies="" the="" inequality="" is="" shaded.="" if="" the="" inequality="" is="" strict="" (using="">< or="">), the line is dashed; if it includes equality (using ≤ or ≥), the line is solid. The solution to the system is where the shaded regions overlap, representing all the points that satisfy all inequalities in the system.
| 3. What methods can be used to graph linear inequalities? | |
Ans.Linear inequalities can be graphed using the following methods: 1. <b>Graphing by hand</b>: Plot the boundary line by converting the inequality to an equation. Determine whether to use a solid or dashed line based on the type of inequality. Then, choose a test point to decide which side of the line to shade. 2. <b>Using graphing calculators or software</b>: Input the inequalities into a graphing tool, which will automatically generate the graph and shaded regions. 3. <b>Table of values</b>: Generate a table of values to find points that satisfy the inequality, then plot these points to visualize the region.
| 4. What are common applications of linear inequalities in real-life scenarios? | |
Ans.Linear inequalities are useful in various real-life situations such as: 1. <b>Budgeting</b>: Determining the maximum amount of money that can be spent based on income and expenditures. 2. <b>Resource allocation</b>: Deciding how to distribute limited resources (e.g., materials, time) in production processes while meeting certain constraints. 3. <b>Optimization problems</b>: Finding the best outcome in scenarios involving various constraints, such as maximizing profit or minimizing cost while adhering to specified limits. 4. <b>Statistical analysis</b>: Analyzing data sets to determine ranges of values that meet certain criteria.
| 5. How do linear inequalities relate to linear programming? | |
Ans.Linear inequalities are fundamental to linear programming, which is a mathematical method for finding the best outcome in a given situation with constraints. In linear programming, the objective function is subject to a set of linear inequalities that define the feasible region. The solutions to these inequalities help identify the vertices of the feasible region, where the objective function can be optimized. Essentially, linear programming utilizes linear inequalities to model constraints and seek optimal solutions within those limits.
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