Straight Lines: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


JEE Mains Previous Year Questions 
(2021-2024): Straight Lines 
2024 
Q1: Let ?? ?? ?? be an isosceles triangle in which ?? is at ( - ?? , ?? ) , ? ?? =
?? ?? ?? , ???? = ???? and ?? is on the 
positve ?? -axis. If ???? = ?? v ?? and the line ???? intersects the line ?? = ?? + ?? at ( ?? , ?? ) , then 
?? ?? ?? ?? is      [JEE 
Main 2024 (Online) 1st February Evening Shift] 
Ans: 36 
 
c
sin ? 30
°
=
4 v 3
sin ? 120
°
[ By sine rule ]
2 ?? = 8 ? ?? = 4
AB = | ( b + 1 ) | = 4
 b = 3 , m
AB
= 0
 m
BC
=
- 1
v 3
BC : - y =
- 1
v 3
( x - 3 )
v 3 y + x = 3
 
Point of intersection : ?? = ?? + 3 , v 3 ?? + ?? = 3 
? ( v 3 + 1 ) ?? = 6
?? =
6
v 3 + 1
?? =
6
v 3 + 1
- 3
? =
6 - 3 v 3 - 3
v 3 + 1
 
Page 2


JEE Mains Previous Year Questions 
(2021-2024): Straight Lines 
2024 
Q1: Let ?? ?? ?? be an isosceles triangle in which ?? is at ( - ?? , ?? ) , ? ?? =
?? ?? ?? , ???? = ???? and ?? is on the 
positve ?? -axis. If ???? = ?? v ?? and the line ???? intersects the line ?? = ?? + ?? at ( ?? , ?? ) , then 
?? ?? ?? ?? is      [JEE 
Main 2024 (Online) 1st February Evening Shift] 
Ans: 36 
 
c
sin ? 30
°
=
4 v 3
sin ? 120
°
[ By sine rule ]
2 ?? = 8 ? ?? = 4
AB = | ( b + 1 ) | = 4
 b = 3 , m
AB
= 0
 m
BC
=
- 1
v 3
BC : - y =
- 1
v 3
( x - 3 )
v 3 y + x = 3
 
Point of intersection : ?? = ?? + 3 , v 3 ?? + ?? = 3 
? ( v 3 + 1 ) ?? = 6
?? =
6
v 3 + 1
?? =
6
v 3 + 1
- 3
? =
6 - 3 v 3 - 3
v 3 + 1
 
 
Q2: The lines ?? ?? , ?? ?? , … , ?? ????
 are distinct. For ?? = ?? , ?? , ?? , … , ???? all the lines ?? ???? - ?? are parallel to each 
other and all the lines ?? ?? ?? pass through a given point ?? . The maximum number of points of 
intersection of pairs of lines from the set { ?? ?? , ?? ?? , … , ?? ????
} is equal to      [JEE Main 2024 (Online) 1st 
February Evening Shift] 
Ans: 101  
To find the maximum number of points of intersection of pairs of lines from the given set, we need to 
consider how the lines are arranged based on the given conditions. 
Firstly, there are 10 lines ( ?? 1
, ?? 3
, … , ?? 19
) that are parallel to each other. Since parallel lines do not 
intersect with each other, these 10 lines will not contribute to the number of intersection points among 
themselves. 
Secondly, there are 10 lines ( ?? 2
, ?? 4
, … , ?? 20
) that all pass through a given point ?? . Although these lines 
intersect at ?? , they only contribute one unique point of intersection to the total count. 
To calculate the maximum number of intersection points, we need to consider the total number of ways 
to pick pairs of lines from the 20 lines available without restrictions, then subtract the combinations that 
do not result in intersections, which includes the combinations of parallel lines among themselves and 
the concurrent lines through point ?? . 
This calculation is represented as: 
Total = ?
20
?? 2
- ?
10
?? 2
- ?
10
?? 2
+ 1 
Here, ?
20
C
2
 calculates the total number of ways to pick any two lines out of 20 , which includes 
intersecting and non-intersecting lines. ?
10
C
2
 is subtracted twice: once for the set of parallel lines 
( ?? 1
, ?? 3
, … , ?? 19
) that don't intersect among themselves and once more for the set of concurrent lines 
( ?? 2
, ?? 4
, … , ?? 20
) intersecting only at point ?? . Since all the concurrent lines intersect at the same point, 
we add 1 back to include this intersection point. 
Carrying out this calculation gives us the total number of distinct intersection points as 101 . 
Q3: Let ?? ( - ?? , - ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of a parallelogram ???? ?? ?? . If the point 
?? lies on ?? ?? - ?? = ?? and the point ?? lies on ?? ?? - ?? ?? = ?? , then the value of | ?? + ?? + ?? + ?? | is equal 
to    [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: 32          
Page 3


JEE Mains Previous Year Questions 
(2021-2024): Straight Lines 
2024 
Q1: Let ?? ?? ?? be an isosceles triangle in which ?? is at ( - ?? , ?? ) , ? ?? =
?? ?? ?? , ???? = ???? and ?? is on the 
positve ?? -axis. If ???? = ?? v ?? and the line ???? intersects the line ?? = ?? + ?? at ( ?? , ?? ) , then 
?? ?? ?? ?? is      [JEE 
Main 2024 (Online) 1st February Evening Shift] 
Ans: 36 
 
c
sin ? 30
°
=
4 v 3
sin ? 120
°
[ By sine rule ]
2 ?? = 8 ? ?? = 4
AB = | ( b + 1 ) | = 4
 b = 3 , m
AB
= 0
 m
BC
=
- 1
v 3
BC : - y =
- 1
v 3
( x - 3 )
v 3 y + x = 3
 
Point of intersection : ?? = ?? + 3 , v 3 ?? + ?? = 3 
? ( v 3 + 1 ) ?? = 6
?? =
6
v 3 + 1
?? =
6
v 3 + 1
- 3
? =
6 - 3 v 3 - 3
v 3 + 1
 
 
Q2: The lines ?? ?? , ?? ?? , … , ?? ????
 are distinct. For ?? = ?? , ?? , ?? , … , ???? all the lines ?? ???? - ?? are parallel to each 
other and all the lines ?? ?? ?? pass through a given point ?? . The maximum number of points of 
intersection of pairs of lines from the set { ?? ?? , ?? ?? , … , ?? ????
} is equal to      [JEE Main 2024 (Online) 1st 
February Evening Shift] 
Ans: 101  
To find the maximum number of points of intersection of pairs of lines from the given set, we need to 
consider how the lines are arranged based on the given conditions. 
Firstly, there are 10 lines ( ?? 1
, ?? 3
, … , ?? 19
) that are parallel to each other. Since parallel lines do not 
intersect with each other, these 10 lines will not contribute to the number of intersection points among 
themselves. 
Secondly, there are 10 lines ( ?? 2
, ?? 4
, … , ?? 20
) that all pass through a given point ?? . Although these lines 
intersect at ?? , they only contribute one unique point of intersection to the total count. 
To calculate the maximum number of intersection points, we need to consider the total number of ways 
to pick pairs of lines from the 20 lines available without restrictions, then subtract the combinations that 
do not result in intersections, which includes the combinations of parallel lines among themselves and 
the concurrent lines through point ?? . 
This calculation is represented as: 
Total = ?
20
?? 2
- ?
10
?? 2
- ?
10
?? 2
+ 1 
Here, ?
20
C
2
 calculates the total number of ways to pick any two lines out of 20 , which includes 
intersecting and non-intersecting lines. ?
10
C
2
 is subtracted twice: once for the set of parallel lines 
( ?? 1
, ?? 3
, … , ?? 19
) that don't intersect among themselves and once more for the set of concurrent lines 
( ?? 2
, ?? 4
, … , ?? 20
) intersecting only at point ?? . Since all the concurrent lines intersect at the same point, 
we add 1 back to include this intersection point. 
Carrying out this calculation gives us the total number of distinct intersection points as 101 . 
Q3: Let ?? ( - ?? , - ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of a parallelogram ???? ?? ?? . If the point 
?? lies on ?? ?? - ?? = ?? and the point ?? lies on ?? ?? - ?? ?? = ?? , then the value of | ?? + ?? + ?? + ?? | is equal 
to    [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: 32          
 
P = (
?? - 2
2
,
?? - 1
2
) = (
?? + 1
2
,
?? 2
)
?? - 2
2
=
?? + 1
2
 and 
?? - 1
2
=
?? 2
? ? ?? - ?? = 3 … ( 1 ) , ?? - ?? = 1 … … (
 
Also, ( ?? , ?? ) lies on 3 ?? - 2 ?? = 6 
3 ?? - 2 ?? = 6 
and ( ?? , ?? ) lies on 2 ?? - ?? = 5 
? 2 ?? - ?? = 5.  
Solving (1), (2), (3), (4) 
?? = - 3 , ?? = - 11 , ?? = - 6 , ?? = - 12
? | ?? + ?? + ?? + ?? | = 32
 
Q4: If the sum of squares of all real values of ?? , for which the lines ?? ?? - ?? + ?? = ?? , ?? ?? + ?? ?? + ?? = ?? 
and ???? + ?? ?? - ?? = ?? do not form a triangle is ?? , then the greatest integer less than or equal to ?? is     
[JEE Main 2024 (Online) 27th January Evening Shift] 
Ans: 32 
Explanation: Currently no explanation available 
Q5: Let ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( - ?? , - ?? ) respectively denote the centroid, circumcentre and 
orthocentre of a triangle. Then, the distance of the point ?? ( ?? ?? + ?? , ?? ?? + ?? ) from the line ?? ?? + ?? ?? -
?? = ?? measured parallel to the line ?? - ?? ?? - ?? = ?? is 
A. 
???? v ?? ?? 
B. 
???? v ?? ?? 
Page 4


JEE Mains Previous Year Questions 
(2021-2024): Straight Lines 
2024 
Q1: Let ?? ?? ?? be an isosceles triangle in which ?? is at ( - ?? , ?? ) , ? ?? =
?? ?? ?? , ???? = ???? and ?? is on the 
positve ?? -axis. If ???? = ?? v ?? and the line ???? intersects the line ?? = ?? + ?? at ( ?? , ?? ) , then 
?? ?? ?? ?? is      [JEE 
Main 2024 (Online) 1st February Evening Shift] 
Ans: 36 
 
c
sin ? 30
°
=
4 v 3
sin ? 120
°
[ By sine rule ]
2 ?? = 8 ? ?? = 4
AB = | ( b + 1 ) | = 4
 b = 3 , m
AB
= 0
 m
BC
=
- 1
v 3
BC : - y =
- 1
v 3
( x - 3 )
v 3 y + x = 3
 
Point of intersection : ?? = ?? + 3 , v 3 ?? + ?? = 3 
? ( v 3 + 1 ) ?? = 6
?? =
6
v 3 + 1
?? =
6
v 3 + 1
- 3
? =
6 - 3 v 3 - 3
v 3 + 1
 
 
Q2: The lines ?? ?? , ?? ?? , … , ?? ????
 are distinct. For ?? = ?? , ?? , ?? , … , ???? all the lines ?? ???? - ?? are parallel to each 
other and all the lines ?? ?? ?? pass through a given point ?? . The maximum number of points of 
intersection of pairs of lines from the set { ?? ?? , ?? ?? , … , ?? ????
} is equal to      [JEE Main 2024 (Online) 1st 
February Evening Shift] 
Ans: 101  
To find the maximum number of points of intersection of pairs of lines from the given set, we need to 
consider how the lines are arranged based on the given conditions. 
Firstly, there are 10 lines ( ?? 1
, ?? 3
, … , ?? 19
) that are parallel to each other. Since parallel lines do not 
intersect with each other, these 10 lines will not contribute to the number of intersection points among 
themselves. 
Secondly, there are 10 lines ( ?? 2
, ?? 4
, … , ?? 20
) that all pass through a given point ?? . Although these lines 
intersect at ?? , they only contribute one unique point of intersection to the total count. 
To calculate the maximum number of intersection points, we need to consider the total number of ways 
to pick pairs of lines from the 20 lines available without restrictions, then subtract the combinations that 
do not result in intersections, which includes the combinations of parallel lines among themselves and 
the concurrent lines through point ?? . 
This calculation is represented as: 
Total = ?
20
?? 2
- ?
10
?? 2
- ?
10
?? 2
+ 1 
Here, ?
20
C
2
 calculates the total number of ways to pick any two lines out of 20 , which includes 
intersecting and non-intersecting lines. ?
10
C
2
 is subtracted twice: once for the set of parallel lines 
( ?? 1
, ?? 3
, … , ?? 19
) that don't intersect among themselves and once more for the set of concurrent lines 
( ?? 2
, ?? 4
, … , ?? 20
) intersecting only at point ?? . Since all the concurrent lines intersect at the same point, 
we add 1 back to include this intersection point. 
Carrying out this calculation gives us the total number of distinct intersection points as 101 . 
Q3: Let ?? ( - ?? , - ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of a parallelogram ???? ?? ?? . If the point 
?? lies on ?? ?? - ?? = ?? and the point ?? lies on ?? ?? - ?? ?? = ?? , then the value of | ?? + ?? + ?? + ?? | is equal 
to    [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: 32          
 
P = (
?? - 2
2
,
?? - 1
2
) = (
?? + 1
2
,
?? 2
)
?? - 2
2
=
?? + 1
2
 and 
?? - 1
2
=
?? 2
? ? ?? - ?? = 3 … ( 1 ) , ?? - ?? = 1 … … (
 
Also, ( ?? , ?? ) lies on 3 ?? - 2 ?? = 6 
3 ?? - 2 ?? = 6 
and ( ?? , ?? ) lies on 2 ?? - ?? = 5 
? 2 ?? - ?? = 5.  
Solving (1), (2), (3), (4) 
?? = - 3 , ?? = - 11 , ?? = - 6 , ?? = - 12
? | ?? + ?? + ?? + ?? | = 32
 
Q4: If the sum of squares of all real values of ?? , for which the lines ?? ?? - ?? + ?? = ?? , ?? ?? + ?? ?? + ?? = ?? 
and ???? + ?? ?? - ?? = ?? do not form a triangle is ?? , then the greatest integer less than or equal to ?? is     
[JEE Main 2024 (Online) 27th January Evening Shift] 
Ans: 32 
Explanation: Currently no explanation available 
Q5: Let ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( - ?? , - ?? ) respectively denote the centroid, circumcentre and 
orthocentre of a triangle. Then, the distance of the point ?? ( ?? ?? + ?? , ?? ?? + ?? ) from the line ?? ?? + ?? ?? -
?? = ?? measured parallel to the line ?? - ?? ?? - ?? = ?? is 
A. 
???? v ?? ?? 
B. 
???? v ?? ?? 
C. 
???? v ?? ?? 
D. 
v ?? ????
                  [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (c) 
 
 
? a = 0 , b = 0 ? ? P ( 3 , 5 ) 
Distance from P measured along x - 2y - 1 = 0 
? ?? = 3 + ?? c o s ? ?? , ? ?? = 5 + ?? sin ? ?? 
Where t a n ? ?? =
1
2
 
r ( 2 c o s ? ?? + 3 s i n ? ?? ) = - 17
? ? r = |
- 17 v 5
7
| =
17 v 5
7
 
Q6: Let ?? , ?? , ?? , ?? ? Z and let ?? ( ?? , ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of parallelogram 
?? ?? ???? . If ???? = v ???? and the points ?? and ?? lie on the line ?? ?? = ?? ?? + ?? , then ?? ( ?? + ?? + ?? + ?? ) is 
equal to 
A. 8 
B. 5 
C. 12 
D. 10     [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (a) 
Page 5


JEE Mains Previous Year Questions 
(2021-2024): Straight Lines 
2024 
Q1: Let ?? ?? ?? be an isosceles triangle in which ?? is at ( - ?? , ?? ) , ? ?? =
?? ?? ?? , ???? = ???? and ?? is on the 
positve ?? -axis. If ???? = ?? v ?? and the line ???? intersects the line ?? = ?? + ?? at ( ?? , ?? ) , then 
?? ?? ?? ?? is      [JEE 
Main 2024 (Online) 1st February Evening Shift] 
Ans: 36 
 
c
sin ? 30
°
=
4 v 3
sin ? 120
°
[ By sine rule ]
2 ?? = 8 ? ?? = 4
AB = | ( b + 1 ) | = 4
 b = 3 , m
AB
= 0
 m
BC
=
- 1
v 3
BC : - y =
- 1
v 3
( x - 3 )
v 3 y + x = 3
 
Point of intersection : ?? = ?? + 3 , v 3 ?? + ?? = 3 
? ( v 3 + 1 ) ?? = 6
?? =
6
v 3 + 1
?? =
6
v 3 + 1
- 3
? =
6 - 3 v 3 - 3
v 3 + 1
 
 
Q2: The lines ?? ?? , ?? ?? , … , ?? ????
 are distinct. For ?? = ?? , ?? , ?? , … , ???? all the lines ?? ???? - ?? are parallel to each 
other and all the lines ?? ?? ?? pass through a given point ?? . The maximum number of points of 
intersection of pairs of lines from the set { ?? ?? , ?? ?? , … , ?? ????
} is equal to      [JEE Main 2024 (Online) 1st 
February Evening Shift] 
Ans: 101  
To find the maximum number of points of intersection of pairs of lines from the given set, we need to 
consider how the lines are arranged based on the given conditions. 
Firstly, there are 10 lines ( ?? 1
, ?? 3
, … , ?? 19
) that are parallel to each other. Since parallel lines do not 
intersect with each other, these 10 lines will not contribute to the number of intersection points among 
themselves. 
Secondly, there are 10 lines ( ?? 2
, ?? 4
, … , ?? 20
) that all pass through a given point ?? . Although these lines 
intersect at ?? , they only contribute one unique point of intersection to the total count. 
To calculate the maximum number of intersection points, we need to consider the total number of ways 
to pick pairs of lines from the 20 lines available without restrictions, then subtract the combinations that 
do not result in intersections, which includes the combinations of parallel lines among themselves and 
the concurrent lines through point ?? . 
This calculation is represented as: 
Total = ?
20
?? 2
- ?
10
?? 2
- ?
10
?? 2
+ 1 
Here, ?
20
C
2
 calculates the total number of ways to pick any two lines out of 20 , which includes 
intersecting and non-intersecting lines. ?
10
C
2
 is subtracted twice: once for the set of parallel lines 
( ?? 1
, ?? 3
, … , ?? 19
) that don't intersect among themselves and once more for the set of concurrent lines 
( ?? 2
, ?? 4
, … , ?? 20
) intersecting only at point ?? . Since all the concurrent lines intersect at the same point, 
we add 1 back to include this intersection point. 
Carrying out this calculation gives us the total number of distinct intersection points as 101 . 
Q3: Let ?? ( - ?? , - ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of a parallelogram ???? ?? ?? . If the point 
?? lies on ?? ?? - ?? = ?? and the point ?? lies on ?? ?? - ?? ?? = ?? , then the value of | ?? + ?? + ?? + ?? | is equal 
to    [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: 32          
 
P = (
?? - 2
2
,
?? - 1
2
) = (
?? + 1
2
,
?? 2
)
?? - 2
2
=
?? + 1
2
 and 
?? - 1
2
=
?? 2
? ? ?? - ?? = 3 … ( 1 ) , ?? - ?? = 1 … … (
 
Also, ( ?? , ?? ) lies on 3 ?? - 2 ?? = 6 
3 ?? - 2 ?? = 6 
and ( ?? , ?? ) lies on 2 ?? - ?? = 5 
? 2 ?? - ?? = 5.  
Solving (1), (2), (3), (4) 
?? = - 3 , ?? = - 11 , ?? = - 6 , ?? = - 12
? | ?? + ?? + ?? + ?? | = 32
 
Q4: If the sum of squares of all real values of ?? , for which the lines ?? ?? - ?? + ?? = ?? , ?? ?? + ?? ?? + ?? = ?? 
and ???? + ?? ?? - ?? = ?? do not form a triangle is ?? , then the greatest integer less than or equal to ?? is     
[JEE Main 2024 (Online) 27th January Evening Shift] 
Ans: 32 
Explanation: Currently no explanation available 
Q5: Let ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( - ?? , - ?? ) respectively denote the centroid, circumcentre and 
orthocentre of a triangle. Then, the distance of the point ?? ( ?? ?? + ?? , ?? ?? + ?? ) from the line ?? ?? + ?? ?? -
?? = ?? measured parallel to the line ?? - ?? ?? - ?? = ?? is 
A. 
???? v ?? ?? 
B. 
???? v ?? ?? 
C. 
???? v ?? ?? 
D. 
v ?? ????
                  [JEE Main 2024 (Online) 31st January Evening Shift] 
Ans: (c) 
 
 
? a = 0 , b = 0 ? ? P ( 3 , 5 ) 
Distance from P measured along x - 2y - 1 = 0 
? ?? = 3 + ?? c o s ? ?? , ? ?? = 5 + ?? sin ? ?? 
Where t a n ? ?? =
1
2
 
r ( 2 c o s ? ?? + 3 s i n ? ?? ) = - 17
? ? r = |
- 17 v 5
7
| =
17 v 5
7
 
Q6: Let ?? , ?? , ?? , ?? ? Z and let ?? ( ?? , ?? ) , ?? ( ?? , ?? ) , ?? ( ?? , ?? ) and ?? ( ?? , ?? ) be the vertices of parallelogram 
?? ?? ???? . If ???? = v ???? and the points ?? and ?? lie on the line ?? ?? = ?? ?? + ?? , then ?? ( ?? + ?? + ?? + ?? ) is 
equal to 
A. 8 
B. 5 
C. 12 
D. 10     [JEE Main 2024 (Online) 31st January Morning Shift] 
Ans: (a) 
 
Let ?? is mid point of diagonals 
?? + ?? 2
=
1 + 1
2
?? + ?? = 2
&
?? + ?? 2
=
2 + 0
2
?? + ?? = 2
2 ( ?? + ?? + ?? + ?? ) = 2 ( 2 + 2 ) = 8
 
Q7: If ?? ?? - ?? ?? + ?? ???? ?? + ?? ???? + ?? ???? + ?? = ?? is the locus of a point, which moves such that it is 
always equidistant from the lines ?? + ?? ?? + ?? = ?? and ?? ?? - ?? + ?? = ?? , then the value of ?? + ?? + ?? -
?? equals 
A. 8 
B. 14 
C. 29 
D. 6      [JEE Main 2024 (Online) 30th January Evening Shift] 
Ans: (c) 
Focus of point P ( x , y ) whose distance from Gives ?? + 2 ?? + 7 = 0 & 2 ?? - ?? + 8 = 0 are equal is 
?? + 2 ?? + 7
v 5
= ±
2 ?? - ?? + 8
v 5
 
( ?? + 2 ?? + 7 )
2
- ( 2 ?? - ?? + 8 )
2
= 0 
Combined equation of lines 
? ( ?? - 3 ?? + 1 ) ( 3 ?? + ?? + 15 ) = 0
3 ?? 2
- 3 ?? 2
- 8 ???? + 18 ?? - 44 ?? + 15 = 0
?? 2
- ?? 2
-
8
3
???? + 6 ?? -
44
3
?? + 5 = 0
?? 2
- ?? 2
+ 2 h ???? + 2 ???? 2 + 2 ???? + ?? = 0
h =
4
3
, ?? = 3 , ?? = -
22
3
, ?? = 5
?? + ?? + h - ?? = 3 + 5 -
4
3
+
22
3
= 8 + 6 = 14