JEE Main 2024 January 30 Shift 2 Paper & Solutions | JEE Main & Advanced Mock Test Series 2025 PDF Download

 Page 1


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Tuesday 30
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
SECTION-A 
1. Consider the system of linear equations  
 x + y + z = 5, x + 2y + ?? 2
z = 9,  
 x + 3y + ?? z = ?? , where ?? , ?? ?? R. Then, which of 
the following statement is NOT correct? 
 (1) System has infinite number of solution if ?? ?]?@ 1 
     and ?? =13 
 (2) System is  inconsistent if ?? ?@ ?] ?@ 1 and ??  ?? 13 
 (3) System is  consistent if ?? ?@?? ?@ 1 and ?? ?@?] 13 
 (4) System has unique solution if ?? ?@?? ?@ 1 and ?? ?? 13 
 Ans. (4) 
Sol. ???]
?? 2
1 1 1
1 2 0
1 3
 
 ?? 2 ?? 2
 – ?? – 1 = 0 
 
1
1,
2
?? ?] ?M  
 ?? ?] ?? ?? ?] ????
2
1 1 5
2 9 0 13
3
 
 Infinite solution ?? = 1 & ?? = 13 
 For unique sol
n
 ?? ?@ ?? 1 
 For no sol
n
 ?? = 1 & ?? ?? 13 
 If ?? ?? 1 and ?? ?@?? 13  
 Considering the case when 
1
2
?? ?] ?M and 13 ???? this 
will generate no solution case   
2. For 
?? ????
?? ?? ?? ????
????
, 0,
2
, let ?? ?K ?? ?] ?? ?M ?? 3sin( ) 2sin( ) and a 
real number k be such that ?? ?] ?? tan k tan . Then the 
value of k is equal to : 
 (1)  ?M 2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ?? cos ?? +  3sin ?? cos ??  
 = 2sin ?? cos ?? – 2sin ?? cos ?? 
 5sin ?? cos ?? = –sin ?? cos ?? 
 ?? ?] ?M ?? 1
tan tan
5
 
 tan ?? = –5tan ?? 
3. Let A( ?? , 0) and B(0, ?? ) be the points on the line  
5x + 7y = 50. Let the point P divide the line 
segment AB internally in the ratio 7 : 3. Let 3x – 
25 = 0 be a directrix of the ellipse ?K?]
2 2
2 2
x y
E : 1
a b
 
and the corresponding focus be S. If from S, the 
perpendicular on the x-axis passes through P, then 
the length of the latus rectum of E is equal to    
 (1) 
25
3
 (2) 
32
9
 
 (3) 
25
9
  (4) 
32
5
 
 Ans. (4 ) 
Sol. 
A (10, 0)
P (3, 5)
50
B 0,
7
?] ?? ?? ?] ?? ????
?]???? ?? ???? ?? 
x =
S
25
3
 
 ae = 3        
 ?] a 25
e 3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a 5
?]?] 
Page 2


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Tuesday 30
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
SECTION-A 
1. Consider the system of linear equations  
 x + y + z = 5, x + 2y + ?? 2
z = 9,  
 x + 3y + ?? z = ?? , where ?? , ?? ?? R. Then, which of 
the following statement is NOT correct? 
 (1) System has infinite number of solution if ?? ?]?@ 1 
     and ?? =13 
 (2) System is  inconsistent if ?? ?@ ?] ?@ 1 and ??  ?? 13 
 (3) System is  consistent if ?? ?@?? ?@ 1 and ?? ?@?] 13 
 (4) System has unique solution if ?? ?@?? ?@ 1 and ?? ?? 13 
 Ans. (4) 
Sol. ???]
?? 2
1 1 1
1 2 0
1 3
 
 ?? 2 ?? 2
 – ?? – 1 = 0 
 
1
1,
2
?? ?] ?M  
 ?? ?] ?? ?? ?] ????
2
1 1 5
2 9 0 13
3
 
 Infinite solution ?? = 1 & ?? = 13 
 For unique sol
n
 ?? ?@ ?? 1 
 For no sol
n
 ?? = 1 & ?? ?? 13 
 If ?? ?? 1 and ?? ?@?? 13  
 Considering the case when 
1
2
?? ?] ?M and 13 ???? this 
will generate no solution case   
2. For 
?? ????
?? ?? ?? ????
????
, 0,
2
, let ?? ?K ?? ?] ?? ?M ?? 3sin( ) 2sin( ) and a 
real number k be such that ?? ?] ?? tan k tan . Then the 
value of k is equal to : 
 (1)  ?M 2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ?? cos ?? +  3sin ?? cos ??  
 = 2sin ?? cos ?? – 2sin ?? cos ?? 
 5sin ?? cos ?? = –sin ?? cos ?? 
 ?? ?] ?M ?? 1
tan tan
5
 
 tan ?? = –5tan ?? 
3. Let A( ?? , 0) and B(0, ?? ) be the points on the line  
5x + 7y = 50. Let the point P divide the line 
segment AB internally in the ratio 7 : 3. Let 3x – 
25 = 0 be a directrix of the ellipse ?K?]
2 2
2 2
x y
E : 1
a b
 
and the corresponding focus be S. If from S, the 
perpendicular on the x-axis passes through P, then 
the length of the latus rectum of E is equal to    
 (1) 
25
3
 (2) 
32
9
 
 (3) 
25
9
  (4) 
32
5
 
 Ans. (4 ) 
Sol. 
A (10, 0)
P (3, 5)
50
B 0,
7
?] ?? ?? ?] ?? ????
?]???? ?? ???? ?? 
x =
S
25
3
 
 ae = 3        
 ?] a 25
e 3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a 5
?]?] 
4. Let ?] ?K ?? ?K ?? ?? ?? ?? ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?? 4
 and 
2
b 6 ?] , 
If ?] a.b 3 2 , then the value of 
?H ?I ?? ?K ?? ?? 2
2 2
a b is 
equal to  
 (1) 90 (2) 75 
 (3) 95  (4) 85 
 Ans. (1) 
Sol. 
?? ?] 2
| b | 6 ; 
????
???] | a || b | cos 3 2 
 
2 2 2
| a | | b | cos 18
????
???] 
 
2
| a | 6
?? ?] 
 Also 1 + ?? 2
 + ?? 2
 = 6  
 ?? 2
 + ?? 2
 = 5 
 to find  
 ( ?? 2
 + ?? 2
) 
2 2 2
| a | | b | sin
????
?? 
 =
????
????
????
1
(5)(6)(6)
2
 
 =  90 
5. Let ?] ?K ?M ?? ?M 2 3
f(x) (x 3) (x 2) ,x [ 4, 4] . If M  and m are 
the maximum and minimum values of f, 
respectively in [–4, 4], then the value of M – m is : 
 (1) 600 (2) 392  
 (3) 608  (4) 108 
 Ans. (3) 
Sol. f'(x) = (x + 3)
2
 . 3(x – 2)
2  
+ (x –2)
3 
2(x + 3)  
 = 5(x + 3) (x – 2)
2
 (x + 1) 
 f'(x) = 0, x = –3, –1, 2 
–3 –1 2
+ – + +
 
 f(–4) = –216 
 f(–3) = 0, f(4) = 49 × 8 = 392 
 M = 392, m = –216 
 M – m = 392 + 216 = 608  
 Ans = '3' 
6. Let a and b be be two distinct positive real 
numbers. Let 11
th
 term of a GP, whose first term is 
a and third term is b, is equal to p
th
 term of another 
GP, whose first term is a and fifth term is b. Then p 
is equal to  
 (1) 20 (2) 25 
 (3) 21  (4) 24 
 Ans. (3) 
Sol. 1
st
 GP ?? t
1
 = a, t
3
 = b = ar
2
 ?? r
2
 = 
b
a
 
  t
11
 = ar
10
 = a(r
2
)
5
 = 
????
?? ????
????
5
b
a
a
 
 2
nd
 G.P. ?? T
1
 = a, T
5
 = ar
4
 = b  
     ?? 
?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 1/4
4
b b
r r
a a
 
  T
p
 = ar
p –1
 
p 1
4
b
a
a
?M ????
?] ????
????
 
  
p 1
5
4
11 p
b b
t T a a
a a
?M ?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 
 ??  
p 1
5 p 21
4
?M ?] ?? ?] 
7. If x
2
 – y
2
 + 2hxy + 2gx + 2fy + c = 0 is the locus of 
a point, which moves such that it is always 
equidistant from the lines x + 2y + 7 = 0 and 2x – y 
+ 8 = 0, then the value of g + c + h – f equals   
 (1) 14 (2) 6 
 (3) 8  (4) 29 
 Ans. (1) 
Sol. Cocus of point P(x, y) whose distance from  
 Gives  
 X + 2y + 7 = 0 & 2x – y + 8 = 0 are equal is 
?K ?K ?M ?K ?]??
x 2y 7 2x y 8
5 5
 
 (x + 2y + 7)
2
 – 
(2x – y + 8)
2
 
= 0
 
  
Page 3


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Tuesday 30
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
SECTION-A 
1. Consider the system of linear equations  
 x + y + z = 5, x + 2y + ?? 2
z = 9,  
 x + 3y + ?? z = ?? , where ?? , ?? ?? R. Then, which of 
the following statement is NOT correct? 
 (1) System has infinite number of solution if ?? ?]?@ 1 
     and ?? =13 
 (2) System is  inconsistent if ?? ?@ ?] ?@ 1 and ??  ?? 13 
 (3) System is  consistent if ?? ?@?? ?@ 1 and ?? ?@?] 13 
 (4) System has unique solution if ?? ?@?? ?@ 1 and ?? ?? 13 
 Ans. (4) 
Sol. ???]
?? 2
1 1 1
1 2 0
1 3
 
 ?? 2 ?? 2
 – ?? – 1 = 0 
 
1
1,
2
?? ?] ?M  
 ?? ?] ?? ?? ?] ????
2
1 1 5
2 9 0 13
3
 
 Infinite solution ?? = 1 & ?? = 13 
 For unique sol
n
 ?? ?@ ?? 1 
 For no sol
n
 ?? = 1 & ?? ?? 13 
 If ?? ?? 1 and ?? ?@?? 13  
 Considering the case when 
1
2
?? ?] ?M and 13 ???? this 
will generate no solution case   
2. For 
?? ????
?? ?? ?? ????
????
, 0,
2
, let ?? ?K ?? ?] ?? ?M ?? 3sin( ) 2sin( ) and a 
real number k be such that ?? ?] ?? tan k tan . Then the 
value of k is equal to : 
 (1)  ?M 2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ?? cos ?? +  3sin ?? cos ??  
 = 2sin ?? cos ?? – 2sin ?? cos ?? 
 5sin ?? cos ?? = –sin ?? cos ?? 
 ?? ?] ?M ?? 1
tan tan
5
 
 tan ?? = –5tan ?? 
3. Let A( ?? , 0) and B(0, ?? ) be the points on the line  
5x + 7y = 50. Let the point P divide the line 
segment AB internally in the ratio 7 : 3. Let 3x – 
25 = 0 be a directrix of the ellipse ?K?]
2 2
2 2
x y
E : 1
a b
 
and the corresponding focus be S. If from S, the 
perpendicular on the x-axis passes through P, then 
the length of the latus rectum of E is equal to    
 (1) 
25
3
 (2) 
32
9
 
 (3) 
25
9
  (4) 
32
5
 
 Ans. (4 ) 
Sol. 
A (10, 0)
P (3, 5)
50
B 0,
7
?] ?? ?? ?] ?? ????
?]???? ?? ???? ?? 
x =
S
25
3
 
 ae = 3        
 ?] a 25
e 3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a 5
?]?] 
4. Let ?] ?K ?? ?K ?? ?? ?? ?? ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?? 4
 and 
2
b 6 ?] , 
If ?] a.b 3 2 , then the value of 
?H ?I ?? ?K ?? ?? 2
2 2
a b is 
equal to  
 (1) 90 (2) 75 
 (3) 95  (4) 85 
 Ans. (1) 
Sol. 
?? ?] 2
| b | 6 ; 
????
???] | a || b | cos 3 2 
 
2 2 2
| a | | b | cos 18
????
???] 
 
2
| a | 6
?? ?] 
 Also 1 + ?? 2
 + ?? 2
 = 6  
 ?? 2
 + ?? 2
 = 5 
 to find  
 ( ?? 2
 + ?? 2
) 
2 2 2
| a | | b | sin
????
?? 
 =
????
????
????
1
(5)(6)(6)
2
 
 =  90 
5. Let ?] ?K ?M ?? ?M 2 3
f(x) (x 3) (x 2) ,x [ 4, 4] . If M  and m are 
the maximum and minimum values of f, 
respectively in [–4, 4], then the value of M – m is : 
 (1) 600 (2) 392  
 (3) 608  (4) 108 
 Ans. (3) 
Sol. f'(x) = (x + 3)
2
 . 3(x – 2)
2  
+ (x –2)
3 
2(x + 3)  
 = 5(x + 3) (x – 2)
2
 (x + 1) 
 f'(x) = 0, x = –3, –1, 2 
–3 –1 2
+ – + +
 
 f(–4) = –216 
 f(–3) = 0, f(4) = 49 × 8 = 392 
 M = 392, m = –216 
 M – m = 392 + 216 = 608  
 Ans = '3' 
6. Let a and b be be two distinct positive real 
numbers. Let 11
th
 term of a GP, whose first term is 
a and third term is b, is equal to p
th
 term of another 
GP, whose first term is a and fifth term is b. Then p 
is equal to  
 (1) 20 (2) 25 
 (3) 21  (4) 24 
 Ans. (3) 
Sol. 1
st
 GP ?? t
1
 = a, t
3
 = b = ar
2
 ?? r
2
 = 
b
a
 
  t
11
 = ar
10
 = a(r
2
)
5
 = 
????
?? ????
????
5
b
a
a
 
 2
nd
 G.P. ?? T
1
 = a, T
5
 = ar
4
 = b  
     ?? 
?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 1/4
4
b b
r r
a a
 
  T
p
 = ar
p –1
 
p 1
4
b
a
a
?M ????
?] ????
????
 
  
p 1
5
4
11 p
b b
t T a a
a a
?M ?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 
 ??  
p 1
5 p 21
4
?M ?] ?? ?] 
7. If x
2
 – y
2
 + 2hxy + 2gx + 2fy + c = 0 is the locus of 
a point, which moves such that it is always 
equidistant from the lines x + 2y + 7 = 0 and 2x – y 
+ 8 = 0, then the value of g + c + h – f equals   
 (1) 14 (2) 6 
 (3) 8  (4) 29 
 Ans. (1) 
Sol. Cocus of point P(x, y) whose distance from  
 Gives  
 X + 2y + 7 = 0 & 2x – y + 8 = 0 are equal is 
?K ?K ?M ?K ?]??
x 2y 7 2x y 8
5 5
 
 (x + 2y + 7)
2
 – 
(2x – y + 8)
2
 
= 0
 
  
 
 Combined equation of lines 
 (x – 3y + 1) (3x + y + 15) = 0 
 3x
2
 – 3y
2
 – 8xy + 18x – 44y + 15 = 0 
 x
2
 – y
2
 – ?K ?M ?K ?] 8 44
xy 6x y 5 0
3 3
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ?] ?] ?] ?M ?] 4 22
h , g 3, f , c 5
3 3
 
 ?K ?K ?M ?] ?K ?M ?K ?] ?K ?] 4 22
g c h f 3 5 8 6 14
3 3
 
8. Let a and b be two vectors such that 
?] ?? ?] | b | 1 and | b a | 2 . Then 
2
(b a) b ???M is equal 
to  
 (1) 3  
 (2) 5 
 (3) 1     
 (4) 4 
 Ans. (2) 
Sol. 
?? ?? ?? ?] ?? ?] | b | 1 & | b a | 2 
 
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?] ?? ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? b a b b b a 0 
 
?? ?? ?? ???? ?? ?? ?M ?] ?? ?K 2 2 2
(b a) b b a b 
 = 4 + 1 = 5 
9. Let ?] y f(x) be a thrice differentiable function in  
(–5, 5). Let the tangents to the curve y=f(x) at  
(1, f(1)) and (3, f(3)) make angles 
?? 6
 and  
?? 4
, 
respectively with positive x-axis. If  
?H ?I ?H ?I ?? ?? ?? ?K ?] ?? ?K ?? ?? 3
2
1
27 f (t) 1 f (t)dt 3 where ?? , ?? ?@ are 
integers, then the value of ?? + ?? equals 
 (1) –14  
 (2) 26  
 (3)  –16   
 (4) 36  
 Ans. (2) 
 
Sol. y = f(x) ?? ?] dy
f '(x)
dx
 
 
?? ?? ?] ?] ?] ?? ?] ?? ?? (1,f(1))
dy 1 1
f '(1) tan f '(1)
dx 6
3 3
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
?? ?? ?] ?] ?] ?? ?] ?? ?? 
 ?H ?I ?H ?I ?K ?] ?? ?K ?? ?? 3
2
1
27 f '(t) 1 f "(t)dt 3 
 ?H ?I ?H ?I ?]?K
?? 3
2
1
I f '(t) 1 f "(t)dt 
 f'(t) = z ?? f"(t) dt = dz 
 z = f'(3) = 1 
 z = f'(1) = 
1
3
 
 
????
?] ?K ?] ?K????
????
?? 1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
???? ????
?] ?K ?M ?? ?K???? ????
????????
1 1 1 1
1
3 3
3 3 3
 
 ?] ?M ?] ?M 4 10 4 10
3
3 3 27
9 3
 
 
????
?? ?K ?? ?] ?M ?] ?M ????
????
4 10
3 27 3 36 10 3
3 27
 
 ?? = 36, ?? = – 10 
 ?? + ?? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ?M?]
2 2
x y
H : 1
9 4
, 
in the first quadrant such that the area of triangle 
formed by P and the two foci of H is 2 13 . Then, 
the square of the distance of P from the origin is  
 (1) 18  
 (2) 26 
 (3) 22   
 (4) 20 
 Ans. (3) 
Page 4


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Tuesday 30
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
SECTION-A 
1. Consider the system of linear equations  
 x + y + z = 5, x + 2y + ?? 2
z = 9,  
 x + 3y + ?? z = ?? , where ?? , ?? ?? R. Then, which of 
the following statement is NOT correct? 
 (1) System has infinite number of solution if ?? ?]?@ 1 
     and ?? =13 
 (2) System is  inconsistent if ?? ?@ ?] ?@ 1 and ??  ?? 13 
 (3) System is  consistent if ?? ?@?? ?@ 1 and ?? ?@?] 13 
 (4) System has unique solution if ?? ?@?? ?@ 1 and ?? ?? 13 
 Ans. (4) 
Sol. ???]
?? 2
1 1 1
1 2 0
1 3
 
 ?? 2 ?? 2
 – ?? – 1 = 0 
 
1
1,
2
?? ?] ?M  
 ?? ?] ?? ?? ?] ????
2
1 1 5
2 9 0 13
3
 
 Infinite solution ?? = 1 & ?? = 13 
 For unique sol
n
 ?? ?@ ?? 1 
 For no sol
n
 ?? = 1 & ?? ?? 13 
 If ?? ?? 1 and ?? ?@?? 13  
 Considering the case when 
1
2
?? ?] ?M and 13 ???? this 
will generate no solution case   
2. For 
?? ????
?? ?? ?? ????
????
, 0,
2
, let ?? ?K ?? ?] ?? ?M ?? 3sin( ) 2sin( ) and a 
real number k be such that ?? ?] ?? tan k tan . Then the 
value of k is equal to : 
 (1)  ?M 2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ?? cos ?? +  3sin ?? cos ??  
 = 2sin ?? cos ?? – 2sin ?? cos ?? 
 5sin ?? cos ?? = –sin ?? cos ?? 
 ?? ?] ?M ?? 1
tan tan
5
 
 tan ?? = –5tan ?? 
3. Let A( ?? , 0) and B(0, ?? ) be the points on the line  
5x + 7y = 50. Let the point P divide the line 
segment AB internally in the ratio 7 : 3. Let 3x – 
25 = 0 be a directrix of the ellipse ?K?]
2 2
2 2
x y
E : 1
a b
 
and the corresponding focus be S. If from S, the 
perpendicular on the x-axis passes through P, then 
the length of the latus rectum of E is equal to    
 (1) 
25
3
 (2) 
32
9
 
 (3) 
25
9
  (4) 
32
5
 
 Ans. (4 ) 
Sol. 
A (10, 0)
P (3, 5)
50
B 0,
7
?] ?? ?? ?] ?? ????
?]???? ?? ???? ?? 
x =
S
25
3
 
 ae = 3        
 ?] a 25
e 3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a 5
?]?] 
4. Let ?] ?K ?? ?K ?? ?? ?? ?? ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?? 4
 and 
2
b 6 ?] , 
If ?] a.b 3 2 , then the value of 
?H ?I ?? ?K ?? ?? 2
2 2
a b is 
equal to  
 (1) 90 (2) 75 
 (3) 95  (4) 85 
 Ans. (1) 
Sol. 
?? ?] 2
| b | 6 ; 
????
???] | a || b | cos 3 2 
 
2 2 2
| a | | b | cos 18
????
???] 
 
2
| a | 6
?? ?] 
 Also 1 + ?? 2
 + ?? 2
 = 6  
 ?? 2
 + ?? 2
 = 5 
 to find  
 ( ?? 2
 + ?? 2
) 
2 2 2
| a | | b | sin
????
?? 
 =
????
????
????
1
(5)(6)(6)
2
 
 =  90 
5. Let ?] ?K ?M ?? ?M 2 3
f(x) (x 3) (x 2) ,x [ 4, 4] . If M  and m are 
the maximum and minimum values of f, 
respectively in [–4, 4], then the value of M – m is : 
 (1) 600 (2) 392  
 (3) 608  (4) 108 
 Ans. (3) 
Sol. f'(x) = (x + 3)
2
 . 3(x – 2)
2  
+ (x –2)
3 
2(x + 3)  
 = 5(x + 3) (x – 2)
2
 (x + 1) 
 f'(x) = 0, x = –3, –1, 2 
–3 –1 2
+ – + +
 
 f(–4) = –216 
 f(–3) = 0, f(4) = 49 × 8 = 392 
 M = 392, m = –216 
 M – m = 392 + 216 = 608  
 Ans = '3' 
6. Let a and b be be two distinct positive real 
numbers. Let 11
th
 term of a GP, whose first term is 
a and third term is b, is equal to p
th
 term of another 
GP, whose first term is a and fifth term is b. Then p 
is equal to  
 (1) 20 (2) 25 
 (3) 21  (4) 24 
 Ans. (3) 
Sol. 1
st
 GP ?? t
1
 = a, t
3
 = b = ar
2
 ?? r
2
 = 
b
a
 
  t
11
 = ar
10
 = a(r
2
)
5
 = 
????
?? ????
????
5
b
a
a
 
 2
nd
 G.P. ?? T
1
 = a, T
5
 = ar
4
 = b  
     ?? 
?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 1/4
4
b b
r r
a a
 
  T
p
 = ar
p –1
 
p 1
4
b
a
a
?M ????
?] ????
????
 
  
p 1
5
4
11 p
b b
t T a a
a a
?M ?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 
 ??  
p 1
5 p 21
4
?M ?] ?? ?] 
7. If x
2
 – y
2
 + 2hxy + 2gx + 2fy + c = 0 is the locus of 
a point, which moves such that it is always 
equidistant from the lines x + 2y + 7 = 0 and 2x – y 
+ 8 = 0, then the value of g + c + h – f equals   
 (1) 14 (2) 6 
 (3) 8  (4) 29 
 Ans. (1) 
Sol. Cocus of point P(x, y) whose distance from  
 Gives  
 X + 2y + 7 = 0 & 2x – y + 8 = 0 are equal is 
?K ?K ?M ?K ?]??
x 2y 7 2x y 8
5 5
 
 (x + 2y + 7)
2
 – 
(2x – y + 8)
2
 
= 0
 
  
 
 Combined equation of lines 
 (x – 3y + 1) (3x + y + 15) = 0 
 3x
2
 – 3y
2
 – 8xy + 18x – 44y + 15 = 0 
 x
2
 – y
2
 – ?K ?M ?K ?] 8 44
xy 6x y 5 0
3 3
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ?] ?] ?] ?M ?] 4 22
h , g 3, f , c 5
3 3
 
 ?K ?K ?M ?] ?K ?M ?K ?] ?K ?] 4 22
g c h f 3 5 8 6 14
3 3
 
8. Let a and b be two vectors such that 
?] ?? ?] | b | 1 and | b a | 2 . Then 
2
(b a) b ???M is equal 
to  
 (1) 3  
 (2) 5 
 (3) 1     
 (4) 4 
 Ans. (2) 
Sol. 
?? ?? ?? ?] ?? ?] | b | 1 & | b a | 2 
 
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?] ?? ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? b a b b b a 0 
 
?? ?? ?? ???? ?? ?? ?M ?] ?? ?K 2 2 2
(b a) b b a b 
 = 4 + 1 = 5 
9. Let ?] y f(x) be a thrice differentiable function in  
(–5, 5). Let the tangents to the curve y=f(x) at  
(1, f(1)) and (3, f(3)) make angles 
?? 6
 and  
?? 4
, 
respectively with positive x-axis. If  
?H ?I ?H ?I ?? ?? ?? ?K ?] ?? ?K ?? ?? 3
2
1
27 f (t) 1 f (t)dt 3 where ?? , ?? ?@ are 
integers, then the value of ?? + ?? equals 
 (1) –14  
 (2) 26  
 (3)  –16   
 (4) 36  
 Ans. (2) 
 
Sol. y = f(x) ?? ?] dy
f '(x)
dx
 
 
?? ?? ?] ?] ?] ?? ?] ?? ?? (1,f(1))
dy 1 1
f '(1) tan f '(1)
dx 6
3 3
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
?? ?? ?] ?] ?] ?? ?] ?? ?? 
 ?H ?I ?H ?I ?K ?] ?? ?K ?? ?? 3
2
1
27 f '(t) 1 f "(t)dt 3 
 ?H ?I ?H ?I ?]?K
?? 3
2
1
I f '(t) 1 f "(t)dt 
 f'(t) = z ?? f"(t) dt = dz 
 z = f'(3) = 1 
 z = f'(1) = 
1
3
 
 
????
?] ?K ?] ?K????
????
?? 1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
???? ????
?] ?K ?M ?? ?K???? ????
????????
1 1 1 1
1
3 3
3 3 3
 
 ?] ?M ?] ?M 4 10 4 10
3
3 3 27
9 3
 
 
????
?? ?K ?? ?] ?M ?] ?M ????
????
4 10
3 27 3 36 10 3
3 27
 
 ?? = 36, ?? = – 10 
 ?? + ?? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ?M?]
2 2
x y
H : 1
9 4
, 
in the first quadrant such that the area of triangle 
formed by P and the two foci of H is 2 13 . Then, 
the square of the distance of P from the origin is  
 (1) 18  
 (2) 26 
 (3) 22   
 (4) 20 
 Ans. (3) 
 
Sol. 
 
y
p
O
s
1
s
2
x
( , ) ?? ?? 
 
2 2
x y
1
9 4
?M?] 
 a
2
 = 9, b
2
 = 4 
 
2
2 2 2 2
2
b
b a (e 1) e 1
a
?] ?M ?? ?] ?K 
 
2
4 13
e 1
9 9
?] ?K ?] 
 
1 2
13 13
e s s 2ae 2 3 2 13
3 3
?] ?? ?] ?] ?? ?? ?] 
 Area of 
1 2 1 2
1
PS S s s 2 13
2
?d ?] ?? ?? ?? ?] 
 
1
(2 13) 2 13 2
2
?? ?? ?? ?? ?] ?? ?? ?] 
 
2 2 2
2
1 1 1 18 3 2
9 4 9
?? ?? ?? ?M ?] ?? ?M ?] ?? ?? ?] ?? ?? ?] 
 Distance of P from origin = 
2 2
?? ?K ?? 
                                        = 18 4 22 ?K?] 
11. Bag A contains 3 white, 7 red balls and bag B 
contains 3 white, 2 red balls. One bag is selected at 
random and a ball is drawn from it. The probability 
of drawing the ball from the bag A, if the ball 
drawn in white, is : 
 (1) 
1
4
 (2) 
1
9
  
 (3) 
1
3
  (4) 
3
10
  
 Ans. (3) 
Sol. E
1
 : A is selected 
A
3W
7R
    
B
3W
2R
 
 E
2
 : B is selected   
 E : white ball is drawn  
 P (E
1
/E) =  
 
?S ?? ?] ?K ?? ?K ?? 1
1 1 2 2
1
P(E).P(E / E )
2 10
1 3 1 3
P(E ). P(E / E ) P(E ). P(E / E )
2 10 2 5
 
 = ?] ?K 3 1
3 6 3
 
 
12. Let f : R ?? R be defined ?] ?K ?K 2x x
f(x) ae be cx . If 
?]?M f(0) 1 , ?? ?] e
f (log 2) 21 and 
  ?H ?I e
log 4
0
39
f(x) cx dx
2
?M?]
?? , then the value of |a+b+c| 
equals : 
 (1) 16 (2) 10  
 (3) 12  (4) 8  
 Ans. (4) 
Sol. f(x) = ae
2x
 + be
x
 + cx   f(0) = –1 
    a + b = –1    
 f ?? (x) = 2ae
2x
 + be
x
 + c     f ?? (ln 2) = 21 
     8a + 2 b + c = 21  
 ?K?]
?? ln 4
2x x
0
39
(ae be )dx
2
 
 
????
?K?]
????
????
ln 4
2x
x
0
ae 39
be
2 2
 ??  8a + 4b – ?M?]
a 39
b
2 2
 
  15a + 6b = 39  
  15 a – 6a – 6 =  39  
  9a = 45  ??  a = 5  
  b = - 6  
   c = 21 – 40 + 12 = –7    
  a + b + c – 8  
  |a + b + c| = 8  
Page 5


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Tuesday 30
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
 
SECTION-A 
1. Consider the system of linear equations  
 x + y + z = 5, x + 2y + ?? 2
z = 9,  
 x + 3y + ?? z = ?? , where ?? , ?? ?? R. Then, which of 
the following statement is NOT correct? 
 (1) System has infinite number of solution if ?? ?]?@ 1 
     and ?? =13 
 (2) System is  inconsistent if ?? ?@ ?] ?@ 1 and ??  ?? 13 
 (3) System is  consistent if ?? ?@?? ?@ 1 and ?? ?@?] 13 
 (4) System has unique solution if ?? ?@?? ?@ 1 and ?? ?? 13 
 Ans. (4) 
Sol. ???]
?? 2
1 1 1
1 2 0
1 3
 
 ?? 2 ?? 2
 – ?? – 1 = 0 
 
1
1,
2
?? ?] ?M  
 ?? ?] ?? ?? ?] ????
2
1 1 5
2 9 0 13
3
 
 Infinite solution ?? = 1 & ?? = 13 
 For unique sol
n
 ?? ?@ ?? 1 
 For no sol
n
 ?? = 1 & ?? ?? 13 
 If ?? ?? 1 and ?? ?@?? 13  
 Considering the case when 
1
2
?? ?] ?M and 13 ???? this 
will generate no solution case   
2. For 
?? ????
?? ?? ?? ????
????
, 0,
2
, let ?? ?K ?? ?] ?? ?M ?? 3sin( ) 2sin( ) and a 
real number k be such that ?? ?] ?? tan k tan . Then the 
value of k is equal to : 
 (1)  ?M 2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ?? cos ?? +  3sin ?? cos ??  
 = 2sin ?? cos ?? – 2sin ?? cos ?? 
 5sin ?? cos ?? = –sin ?? cos ?? 
 ?? ?] ?M ?? 1
tan tan
5
 
 tan ?? = –5tan ?? 
3. Let A( ?? , 0) and B(0, ?? ) be the points on the line  
5x + 7y = 50. Let the point P divide the line 
segment AB internally in the ratio 7 : 3. Let 3x – 
25 = 0 be a directrix of the ellipse ?K?]
2 2
2 2
x y
E : 1
a b
 
and the corresponding focus be S. If from S, the 
perpendicular on the x-axis passes through P, then 
the length of the latus rectum of E is equal to    
 (1) 
25
3
 (2) 
32
9
 
 (3) 
25
9
  (4) 
32
5
 
 Ans. (4 ) 
Sol. 
A (10, 0)
P (3, 5)
50
B 0,
7
?] ?? ?? ?] ?? ????
?]???? ?? ???? ?? 
x =
S
25
3
 
 ae = 3        
 ?] a 25
e 3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a 5
?]?] 
4. Let ?] ?K ?? ?K ?? ?? ?? ?? ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?? 4
 and 
2
b 6 ?] , 
If ?] a.b 3 2 , then the value of 
?H ?I ?? ?K ?? ?? 2
2 2
a b is 
equal to  
 (1) 90 (2) 75 
 (3) 95  (4) 85 
 Ans. (1) 
Sol. 
?? ?] 2
| b | 6 ; 
????
???] | a || b | cos 3 2 
 
2 2 2
| a | | b | cos 18
????
???] 
 
2
| a | 6
?? ?] 
 Also 1 + ?? 2
 + ?? 2
 = 6  
 ?? 2
 + ?? 2
 = 5 
 to find  
 ( ?? 2
 + ?? 2
) 
2 2 2
| a | | b | sin
????
?? 
 =
????
????
????
1
(5)(6)(6)
2
 
 =  90 
5. Let ?] ?K ?M ?? ?M 2 3
f(x) (x 3) (x 2) ,x [ 4, 4] . If M  and m are 
the maximum and minimum values of f, 
respectively in [–4, 4], then the value of M – m is : 
 (1) 600 (2) 392  
 (3) 608  (4) 108 
 Ans. (3) 
Sol. f'(x) = (x + 3)
2
 . 3(x – 2)
2  
+ (x –2)
3 
2(x + 3)  
 = 5(x + 3) (x – 2)
2
 (x + 1) 
 f'(x) = 0, x = –3, –1, 2 
–3 –1 2
+ – + +
 
 f(–4) = –216 
 f(–3) = 0, f(4) = 49 × 8 = 392 
 M = 392, m = –216 
 M – m = 392 + 216 = 608  
 Ans = '3' 
6. Let a and b be be two distinct positive real 
numbers. Let 11
th
 term of a GP, whose first term is 
a and third term is b, is equal to p
th
 term of another 
GP, whose first term is a and fifth term is b. Then p 
is equal to  
 (1) 20 (2) 25 
 (3) 21  (4) 24 
 Ans. (3) 
Sol. 1
st
 GP ?? t
1
 = a, t
3
 = b = ar
2
 ?? r
2
 = 
b
a
 
  t
11
 = ar
10
 = a(r
2
)
5
 = 
????
?? ????
????
5
b
a
a
 
 2
nd
 G.P. ?? T
1
 = a, T
5
 = ar
4
 = b  
     ?? 
?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 1/4
4
b b
r r
a a
 
  T
p
 = ar
p –1
 
p 1
4
b
a
a
?M ????
?] ????
????
 
  
p 1
5
4
11 p
b b
t T a a
a a
?M ?? ?? ?? ?? ?] ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? 
 ??  
p 1
5 p 21
4
?M ?] ?? ?] 
7. If x
2
 – y
2
 + 2hxy + 2gx + 2fy + c = 0 is the locus of 
a point, which moves such that it is always 
equidistant from the lines x + 2y + 7 = 0 and 2x – y 
+ 8 = 0, then the value of g + c + h – f equals   
 (1) 14 (2) 6 
 (3) 8  (4) 29 
 Ans. (1) 
Sol. Cocus of point P(x, y) whose distance from  
 Gives  
 X + 2y + 7 = 0 & 2x – y + 8 = 0 are equal is 
?K ?K ?M ?K ?]??
x 2y 7 2x y 8
5 5
 
 (x + 2y + 7)
2
 – 
(2x – y + 8)
2
 
= 0
 
  
 
 Combined equation of lines 
 (x – 3y + 1) (3x + y + 15) = 0 
 3x
2
 – 3y
2
 – 8xy + 18x – 44y + 15 = 0 
 x
2
 – y
2
 – ?K ?M ?K ?] 8 44
xy 6x y 5 0
3 3
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ?] ?] ?] ?M ?] 4 22
h , g 3, f , c 5
3 3
 
 ?K ?K ?M ?] ?K ?M ?K ?] ?K ?] 4 22
g c h f 3 5 8 6 14
3 3
 
8. Let a and b be two vectors such that 
?] ?? ?] | b | 1 and | b a | 2 . Then 
2
(b a) b ???M is equal 
to  
 (1) 3  
 (2) 5 
 (3) 1     
 (4) 4 
 Ans. (2) 
Sol. 
?? ?? ?? ?] ?? ?] | b | 1 & | b a | 2 
 
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?] ?? ?? ?] ?? ?? ?? ?? ?? ?? ?? ?? b a b b b a 0 
 
?? ?? ?? ???? ?? ?? ?M ?] ?? ?K 2 2 2
(b a) b b a b 
 = 4 + 1 = 5 
9. Let ?] y f(x) be a thrice differentiable function in  
(–5, 5). Let the tangents to the curve y=f(x) at  
(1, f(1)) and (3, f(3)) make angles 
?? 6
 and  
?? 4
, 
respectively with positive x-axis. If  
?H ?I ?H ?I ?? ?? ?? ?K ?] ?? ?K ?? ?? 3
2
1
27 f (t) 1 f (t)dt 3 where ?? , ?? ?@ are 
integers, then the value of ?? + ?? equals 
 (1) –14  
 (2) 26  
 (3)  –16   
 (4) 36  
 Ans. (2) 
 
Sol. y = f(x) ?? ?] dy
f '(x)
dx
 
 
?? ?? ?] ?] ?] ?? ?] ?? ?? (1,f(1))
dy 1 1
f '(1) tan f '(1)
dx 6
3 3
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
?? ?? ?] ?] ?] ?? ?] ?? ?? 
 ?H ?I ?H ?I ?K ?] ?? ?K ?? ?? 3
2
1
27 f '(t) 1 f "(t)dt 3 
 ?H ?I ?H ?I ?]?K
?? 3
2
1
I f '(t) 1 f "(t)dt 
 f'(t) = z ?? f"(t) dt = dz 
 z = f'(3) = 1 
 z = f'(1) = 
1
3
 
 
????
?] ?K ?] ?K????
????
?? 1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
???? ????
?] ?K ?M ?? ?K???? ????
????????
1 1 1 1
1
3 3
3 3 3
 
 ?] ?M ?] ?M 4 10 4 10
3
3 3 27
9 3
 
 
????
?? ?K ?? ?] ?M ?] ?M ????
????
4 10
3 27 3 36 10 3
3 27
 
 ?? = 36, ?? = – 10 
 ?? + ?? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ?M?]
2 2
x y
H : 1
9 4
, 
in the first quadrant such that the area of triangle 
formed by P and the two foci of H is 2 13 . Then, 
the square of the distance of P from the origin is  
 (1) 18  
 (2) 26 
 (3) 22   
 (4) 20 
 Ans. (3) 
 
Sol. 
 
y
p
O
s
1
s
2
x
( , ) ?? ?? 
 
2 2
x y
1
9 4
?M?] 
 a
2
 = 9, b
2
 = 4 
 
2
2 2 2 2
2
b
b a (e 1) e 1
a
?] ?M ?? ?] ?K 
 
2
4 13
e 1
9 9
?] ?K ?] 
 
1 2
13 13
e s s 2ae 2 3 2 13
3 3
?] ?? ?] ?] ?? ?? ?] 
 Area of 
1 2 1 2
1
PS S s s 2 13
2
?d ?] ?? ?? ?? ?] 
 
1
(2 13) 2 13 2
2
?? ?? ?? ?? ?] ?? ?? ?] 
 
2 2 2
2
1 1 1 18 3 2
9 4 9
?? ?? ?? ?M ?] ?? ?M ?] ?? ?? ?] ?? ?? ?] 
 Distance of P from origin = 
2 2
?? ?K ?? 
                                        = 18 4 22 ?K?] 
11. Bag A contains 3 white, 7 red balls and bag B 
contains 3 white, 2 red balls. One bag is selected at 
random and a ball is drawn from it. The probability 
of drawing the ball from the bag A, if the ball 
drawn in white, is : 
 (1) 
1
4
 (2) 
1
9
  
 (3) 
1
3
  (4) 
3
10
  
 Ans. (3) 
Sol. E
1
 : A is selected 
A
3W
7R
    
B
3W
2R
 
 E
2
 : B is selected   
 E : white ball is drawn  
 P (E
1
/E) =  
 
?S ?? ?] ?K ?? ?K ?? 1
1 1 2 2
1
P(E).P(E / E )
2 10
1 3 1 3
P(E ). P(E / E ) P(E ). P(E / E )
2 10 2 5
 
 = ?] ?K 3 1
3 6 3
 
 
12. Let f : R ?? R be defined ?] ?K ?K 2x x
f(x) ae be cx . If 
?]?M f(0) 1 , ?? ?] e
f (log 2) 21 and 
  ?H ?I e
log 4
0
39
f(x) cx dx
2
?M?]
?? , then the value of |a+b+c| 
equals : 
 (1) 16 (2) 10  
 (3) 12  (4) 8  
 Ans. (4) 
Sol. f(x) = ae
2x
 + be
x
 + cx   f(0) = –1 
    a + b = –1    
 f ?? (x) = 2ae
2x
 + be
x
 + c     f ?? (ln 2) = 21 
     8a + 2 b + c = 21  
 ?K?]
?? ln 4
2x x
0
39
(ae be )dx
2
 
 
????
?K?]
????
????
ln 4
2x
x
0
ae 39
be
2 2
 ??  8a + 4b – ?M?]
a 39
b
2 2
 
  15a + 6b = 39  
  15 a – 6a – 6 =  39  
  9a = 45  ??  a = 5  
  b = - 6  
   c = 21 – 40 + 12 = –7    
  a + b + c – 8  
  |a + b + c| = 8  
 
13. Let ?] ?M ?K ?K ?? ?M ?K ???? 1
ˆ ˆ ˆ ˆ ˆˆ
L : r (i j 2k) (i j 2k), R 
 ?] ?M ?K ?? ?K ?K ?? ?? 2
ˆ ˆ ˆ ˆˆ
L : r ( j k) (3i j pk), R and 
?] ?? ?K ?K ?? ?? 3
ˆˆ ˆ
L : r ( i mj nk) R 
 Be three lines such that L
1
 is perpendicular to L
2
 
and L
3
 is perpendicular to both L
1
 and L
2
. Then the 
point which lies on L
3
 is  
 (1) (–1, 7, 4)   (2) (–1, –7, 4)  
 (3) (1, 7, –4)  (4) (1, –7, 4)  
 Ans. (1) 
Sol. L
1
 ?~ L
2
 L
3
 ?~ L
1
,  L
2
  
 3 – 1 + 2 P = 0 
 P = – 1  
 ?M ?] ?M ?K ?K ?M ˆˆ ˆ
i j k
ˆˆ ˆ
1 1 2 i 7j 4k
3 1 1
 
  ( , 7 , 4 ) ?| ?M ?? ?? ?? will lie on L
3 
 
For ?? = 1 the point will be (-1, 7, 4)  
14. Let a and b be real constants such that the function 
f defined by 
?? ?K ?K ?? ?] ???K?^
?? 2
x 3x a , x 1
f(x)
bx 2 , x 1
be 
differentiable on R. Then, the value of 
?M ?? 2
2
f(x)dx 
equals  
 (1) 
15
6
 (2) 
19
6
  
 (3) 21  (4) 17 
 Ans. (4) 
Sol. f is continuous  f ?? (x) = 2x + 3  ,  k < 1  
 ?|   4 + a = b + 2              b         ,   x > 1  
       a = b – 2               f is differentiable  
   ?|    b = 5 
  ?|     a = 3 
 
?M ?K ?K ?K ?K ????
1 2
2
2 1
(x 3x 3)dx (5x 2)dx 
 = 
?M ?? ?? ?? ?? ?K ?K ?K ?K ?? ?? ?? ?? ?? ?? ?? ?? 1 2
3 2 2
2 1
x 3x 5x
3x 2x
3 2 2
 
 = 
?M ?? ?? ?? ?? ?? ?? ?K ?K ?M ?K ?M ?K ?K ?M ?M ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 1 3 8 5
3 6 6 10 4 2
3 2 3 2
 
 =   ?K ?K ?M ?] 3 5
6 12 17
2 2
 
15. Let ?M?? f : {0} be a function satisfying 
????
?] ????
????
x f(x)
f
y f(y)
 for all x, y, f(y) ?? 0. If ?? f (1) = 2024, 
then  
 (1) xf ?? (x) – 2024 f(x) = 0  
 (2) xf ?? (x) + 2024f(x) = 0  
 (3) xf ?? (x) +f(x) = 2024  
 (4) xf ?? (x) –2023f(x) = 0  
 Ans. (1) 
Sol. 
x f(x)
f
y f(y)
????
?] ????
????
    
f(1) 2024
f(1) 1
?? ?] ?]  
Partially differentiating w. r. t. x 
x 1 1
f . f(x)
y y f(y)
????
???? ?] ????
????
 
y ?? x 
?? ?? ?] 1 f(x)
f(1).
x f(x)
 
2024f(x) = xf ?? (x)  ???@ xf ?? (x) – 2024 f(x) = 0 
16. If z is a  complex number, then the number of 
common roots of the equation ?K ?K ?] 1985 100
z z 1 0 and 
?K ?K ?K ?] 3 2
z 2z 2z 1 0 ,  is equal to : 
 (1) 1 (2) 2  
 (3) 0  (4) 3  
 Ans. (2) 
Sol. z
1985
 + z
100
 + 1 = 0   &  z
3
 + 2z
2
 + 2z + 1 = 0  
 (z + 1) (z
2
 – z + 1) + 2z(z + 1) = 0 
  (z + 1)  (z
2
 + z + 1) = 0     
?? z = – 1 ,   z = w,
 
w
2
  
 Now putting z = –1 not satisfy 
 Now put z = w   
?? w
1985
 + w
100
 + 1   
?? w
2
 + w + 1 = 0     
?@ ?a lso,  z = w
2
  
?? w
3970
 + w
200
 + 1  
?? w + w
2
 + 1 = 0  
 Two common root