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Page 1
FINAL JEE –MAIN EXAMINATION – JANUARY, 2024
(Held On Wednesday 31
st
January, 2024) TIME : 3 : 00 PM to 6 : 00 PM
MATHEMATICS TEST PAPER WITH SOLUTION
SECTION-A
1. The number of ways in which 21 identical apples
can be distributed among three children such that
each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136
Ans. (4)
Sol. After giving 2 apples to each child 15 apples left
now 15 apples can be distributed in
15 3 1 17
22
CC
??
?
ways
17 16
136
2
?
??
2. Let A (a, b), B(3, 4) and (–6, –8) respectively
denote the centroid, circumcentre and orthocentre
of a triangle. Then, the distance of the point
P(2a + 3, 7b + 5) from the line 2x + 3y – 4 = 0
measured parallel to the line x – 2y – 1 = 0 is
(1)
15 5
7
(2)
17 5
6
(3)
17 5
7
(4)
5
17
Ans. (3)
Sol. A(a,b), B(3,4), C(-6, -8)
2 : 1
C
(-6, -8)
A
B
(3, 4)
(a, b)
? ? a 0, b = 0 P 3,5 ? ? ?
Distance from P measured along x – 2y – 1 = 0
x 3 rcos , y = 5+rsin ? ? ? ? ?
Where
1
tan
2
??
? ? r 2cos 3sin 17
17 5 17 5
r
77
? ? ? ? ?
?
? ? ?
3. Let z
1
and z
2
be two complex number such that z
1
+ z
2
= 5 and
33
12
z z 20 15i ? ? ? . Then
44
12
zz ?
equals-
(1) 30 3
(2) 75
(3) 15 15
(4) 25 3
Ans. (2)
Sol.-
12
z z 5 ??
33
12
z z 20 15i ? ? ?
? ? ? ?
3
33
1 2 1 2 1 2 12
z z z z 3z z z z ? ? ? ? ?
? ?
33
12 12
z z 125 3z . z 5 ? ? ?
12
20 15i 125 15z z ? ? ? ?
12
3z z 25 4 3i ? ? ? ?
12
3z z 21 3i ? ? ?
12
z .z 7 i ? ? ?
? ?
2
12
z z 25 ? ? ?
? ?
22
12
z z 25 2 7 i ? ? ? ? ?
11 2i ??
? ?
2
22
12
z z 121 4 44i ? ? ? ?
? ? ?
2
44
12
z z 2 7 i 117 44i ? ? ? ? ?
? ? ?
44
12
z z 117 44i 2 49 1 14i ? ? ? ? ? ?
?
44
12
z z 75 ??
Page 2
FINAL JEE –MAIN EXAMINATION – JANUARY, 2024
(Held On Wednesday 31
st
January, 2024) TIME : 3 : 00 PM to 6 : 00 PM
MATHEMATICS TEST PAPER WITH SOLUTION
SECTION-A
1. The number of ways in which 21 identical apples
can be distributed among three children such that
each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136
Ans. (4)
Sol. After giving 2 apples to each child 15 apples left
now 15 apples can be distributed in
15 3 1 17
22
CC
??
?
ways
17 16
136
2
?
??
2. Let A (a, b), B(3, 4) and (–6, –8) respectively
denote the centroid, circumcentre and orthocentre
of a triangle. Then, the distance of the point
P(2a + 3, 7b + 5) from the line 2x + 3y – 4 = 0
measured parallel to the line x – 2y – 1 = 0 is
(1)
15 5
7
(2)
17 5
6
(3)
17 5
7
(4)
5
17
Ans. (3)
Sol. A(a,b), B(3,4), C(-6, -8)
2 : 1
C
(-6, -8)
A
B
(3, 4)
(a, b)
? ? a 0, b = 0 P 3,5 ? ? ?
Distance from P measured along x – 2y – 1 = 0
x 3 rcos , y = 5+rsin ? ? ? ? ?
Where
1
tan
2
??
? ? r 2cos 3sin 17
17 5 17 5
r
77
? ? ? ? ?
?
? ? ?
3. Let z
1
and z
2
be two complex number such that z
1
+ z
2
= 5 and
33
12
z z 20 15i ? ? ? . Then
44
12
zz ?
equals-
(1) 30 3
(2) 75
(3) 15 15
(4) 25 3
Ans. (2)
Sol.-
12
z z 5 ??
33
12
z z 20 15i ? ? ?
? ? ? ?
3
33
1 2 1 2 1 2 12
z z z z 3z z z z ? ? ? ? ?
? ?
33
12 12
z z 125 3z . z 5 ? ? ?
12
20 15i 125 15z z ? ? ? ?
12
3z z 25 4 3i ? ? ? ?
12
3z z 21 3i ? ? ?
12
z .z 7 i ? ? ?
? ?
2
12
z z 25 ? ? ?
? ?
22
12
z z 25 2 7 i ? ? ? ? ?
11 2i ??
? ?
2
22
12
z z 121 4 44i ? ? ? ?
? ? ?
2
44
12
z z 2 7 i 117 44i ? ? ? ? ?
? ? ?
44
12
z z 117 44i 2 49 1 14i ? ? ? ? ? ?
?
44
12
z z 75 ??
4. Let a variable line passing through the centre of the
circle x
2
+ y
2
– 16x – 4y = 0, meet the positive
co-ordinate axes at the point A and B. Then the
minimum value of OA + OB, where O is the
origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24
Ans. (2)
Sol.- ? ? ? ? y 2 m x 8 ? ? ?
? x-intercept
?
2
8
m
???
?
??
??
? y-intercept
? ? ? 8m 2 ??
2
OA OB 8 8m 2
m
?
? ? ? ? ? ?
? ?
2
2
f ' m 8 0
m
? ? ?
2
1
m
4
??
1
m
2
?
??
1
f 18
2
? ??
??
??
??
? Minimum = 18
5. Let f,g :(0, ) R ?? be two functions defined by
? ?
2 x
2t
x
f (x) t t e dt
?
?
??
?
and
2
x 1
t
2
0
g(x) t e dt
?
?
?
.
Then the value of
? ? ? ? ? ? ee
f log 9 g log 9 ? is
equal to
(1) 6
(2) 9
(3) 8
(4) 10
Ans. (3)
Sol.-
? ? ? ?
? ? ? ? ? ?
? ?
? ? ? ?
? ? ? ?
2
2
2
2
2 2 2
x
2t
x
2x
x 1
t
2
0
x
x 2 x 2 x
f x t t e dt
f x 2. x x e ............ 1
g x t e dt
g x xe 2x 0
f x g x 2xe 2x e 2x e
?
?
?
?
?
? ? ?
??
? ? ? ?
?
???
?? ? ? ? ?
?
?
Integrating both sides w.r.t.x
? ? ? ?
? ?
? ?
? ?
2
1
e
x
0
2
tt
0
0
log 9 1
f x g x 2xe dx
xt
e dt e
e
1
9 f (x) g(x) 1 9 8
9
?
?
?
?
?
??
?
??
?
?? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
?
6. Let
? ? ,, ? ? ? be mirror image of the point (2, 3, 5)
in the line
x 1 y 2 z 3
2 3 4
? ? ?
?? .
Then 2 3 4 ? ? ? ? ? is equal to
(1) 32
(2) 33
(3) 31
(4) 34
Ans. (2)
Sol.
P(2,3,5)
? ? R , , ? ? ?
? ?
? ?
? ? ? ?
PR 2,3, 4
PR. 2,3, 4 0
2, 3, 5 . 2,3, 4 0
2 3 4 4 9 20 33
?
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
Page 3
FINAL JEE –MAIN EXAMINATION – JANUARY, 2024
(Held On Wednesday 31
st
January, 2024) TIME : 3 : 00 PM to 6 : 00 PM
MATHEMATICS TEST PAPER WITH SOLUTION
SECTION-A
1. The number of ways in which 21 identical apples
can be distributed among three children such that
each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136
Ans. (4)
Sol. After giving 2 apples to each child 15 apples left
now 15 apples can be distributed in
15 3 1 17
22
CC
??
?
ways
17 16
136
2
?
??
2. Let A (a, b), B(3, 4) and (–6, –8) respectively
denote the centroid, circumcentre and orthocentre
of a triangle. Then, the distance of the point
P(2a + 3, 7b + 5) from the line 2x + 3y – 4 = 0
measured parallel to the line x – 2y – 1 = 0 is
(1)
15 5
7
(2)
17 5
6
(3)
17 5
7
(4)
5
17
Ans. (3)
Sol. A(a,b), B(3,4), C(-6, -8)
2 : 1
C
(-6, -8)
A
B
(3, 4)
(a, b)
? ? a 0, b = 0 P 3,5 ? ? ?
Distance from P measured along x – 2y – 1 = 0
x 3 rcos , y = 5+rsin ? ? ? ? ?
Where
1
tan
2
??
? ? r 2cos 3sin 17
17 5 17 5
r
77
? ? ? ? ?
?
? ? ?
3. Let z
1
and z
2
be two complex number such that z
1
+ z
2
= 5 and
33
12
z z 20 15i ? ? ? . Then
44
12
zz ?
equals-
(1) 30 3
(2) 75
(3) 15 15
(4) 25 3
Ans. (2)
Sol.-
12
z z 5 ??
33
12
z z 20 15i ? ? ?
? ? ? ?
3
33
1 2 1 2 1 2 12
z z z z 3z z z z ? ? ? ? ?
? ?
33
12 12
z z 125 3z . z 5 ? ? ?
12
20 15i 125 15z z ? ? ? ?
12
3z z 25 4 3i ? ? ? ?
12
3z z 21 3i ? ? ?
12
z .z 7 i ? ? ?
? ?
2
12
z z 25 ? ? ?
? ?
22
12
z z 25 2 7 i ? ? ? ? ?
11 2i ??
? ?
2
22
12
z z 121 4 44i ? ? ? ?
? ? ?
2
44
12
z z 2 7 i 117 44i ? ? ? ? ?
? ? ?
44
12
z z 117 44i 2 49 1 14i ? ? ? ? ? ?
?
44
12
z z 75 ??
4. Let a variable line passing through the centre of the
circle x
2
+ y
2
– 16x – 4y = 0, meet the positive
co-ordinate axes at the point A and B. Then the
minimum value of OA + OB, where O is the
origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24
Ans. (2)
Sol.- ? ? ? ? y 2 m x 8 ? ? ?
? x-intercept
?
2
8
m
???
?
??
??
? y-intercept
? ? ? 8m 2 ??
2
OA OB 8 8m 2
m
?
? ? ? ? ? ?
? ?
2
2
f ' m 8 0
m
? ? ?
2
1
m
4
??
1
m
2
?
??
1
f 18
2
? ??
??
??
??
? Minimum = 18
5. Let f,g :(0, ) R ?? be two functions defined by
? ?
2 x
2t
x
f (x) t t e dt
?
?
??
?
and
2
x 1
t
2
0
g(x) t e dt
?
?
?
.
Then the value of
? ? ? ? ? ? ee
f log 9 g log 9 ? is
equal to
(1) 6
(2) 9
(3) 8
(4) 10
Ans. (3)
Sol.-
? ? ? ?
? ? ? ? ? ?
? ?
? ? ? ?
? ? ? ?
2
2
2
2
2 2 2
x
2t
x
2x
x 1
t
2
0
x
x 2 x 2 x
f x t t e dt
f x 2. x x e ............ 1
g x t e dt
g x xe 2x 0
f x g x 2xe 2x e 2x e
?
?
?
?
?
? ? ?
??
? ? ? ?
?
???
?? ? ? ? ?
?
?
Integrating both sides w.r.t.x
? ? ? ?
? ?
? ?
? ?
2
1
e
x
0
2
tt
0
0
log 9 1
f x g x 2xe dx
xt
e dt e
e
1
9 f (x) g(x) 1 9 8
9
?
?
?
?
?
??
?
??
?
?? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
?
6. Let
? ? ,, ? ? ? be mirror image of the point (2, 3, 5)
in the line
x 1 y 2 z 3
2 3 4
? ? ?
?? .
Then 2 3 4 ? ? ? ? ? is equal to
(1) 32
(2) 33
(3) 31
(4) 34
Ans. (2)
Sol.
P(2,3,5)
? ? R , , ? ? ?
? ?
? ?
? ? ? ?
PR 2,3, 4
PR. 2,3, 4 0
2, 3, 5 . 2,3, 4 0
2 3 4 4 9 20 33
?
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
7. Let P be a parabola with vertex (2, 3) and directrix
2x + y = 6. Let an ellipse
22
22
xy
E : 1,a b
ab
? ? ? of
eccentricity
1
2
pass through the focus of the
parabola P. Then the square of the length of the
latus rectum of E, is
(1)
385
8
(2)
347
8
(3)
512
25
(4)
656
25
Ans. (4)
Sol.-
Focur
axis
(2, 3)
( , ) ? ?
(1.6, 2.8)
Slope of axis
1
2
?
? ?
1
y 3 x 2
2
? ? ?
2y 6 x 2 ? ? ? ?
2y x 4 0 ? ? ? ?
2x y 6 0 ? ? ?
4x 2y 12 0 ? ? ?
1.6 4 2.4 ? ? ? ? ? ?
2.8 6 3.2 ? ? ? ? ? ?
Ellipse passes through (2.4, 3.2)
22
22
24 32
10 10
1
ab
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ……….(1)
Also
22
22
b 1 b 1
1
a 2 a 2
? ? ? ?
22
a 2b ??
Put in (1)
2
328
b
25
??
2
22
2
2
2b 4b 1 328 656
b4
a a 2 25 25
??
? ? ? ? ? ? ?
??
??
8. The temperature T(t) of a body at time t = 0 is 160
o
F and it decreases continuously as per the
differential equation
dT
K(T 80)
dt
? ? ? , where K
is positive constant. If T(15) = 120
o
F, then T(45) is
equal to
(1) 85
o
F
(2) 95
o
F
(3) 90
o
F
(4) 80
o
F
Ans. (3)
Sol.-
? ?
? ?
? ?
? ?
Tt
160 0
T
160
kt
k.15
k15
k.45
3
k.15
dT
k T 80
dt
dT
Kdt
T 80
ln T 80 kt
ln T 80 ln 80 kt
T 80
ln kt
80
T 80 80e
120 80 80e
40 1
e
80 2
T 45 80 80e
80 80 e
1
80 80
8
90
?
?
?
?
?
? ? ?
??
?
? ? ? ? ?
??
? ? ? ?
?
??
??
??
??
? ? ?
??
? ? ?
?
??
Page 4
FINAL JEE –MAIN EXAMINATION – JANUARY, 2024
(Held On Wednesday 31
st
January, 2024) TIME : 3 : 00 PM to 6 : 00 PM
MATHEMATICS TEST PAPER WITH SOLUTION
SECTION-A
1. The number of ways in which 21 identical apples
can be distributed among three children such that
each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136
Ans. (4)
Sol. After giving 2 apples to each child 15 apples left
now 15 apples can be distributed in
15 3 1 17
22
CC
??
?
ways
17 16
136
2
?
??
2. Let A (a, b), B(3, 4) and (–6, –8) respectively
denote the centroid, circumcentre and orthocentre
of a triangle. Then, the distance of the point
P(2a + 3, 7b + 5) from the line 2x + 3y – 4 = 0
measured parallel to the line x – 2y – 1 = 0 is
(1)
15 5
7
(2)
17 5
6
(3)
17 5
7
(4)
5
17
Ans. (3)
Sol. A(a,b), B(3,4), C(-6, -8)
2 : 1
C
(-6, -8)
A
B
(3, 4)
(a, b)
? ? a 0, b = 0 P 3,5 ? ? ?
Distance from P measured along x – 2y – 1 = 0
x 3 rcos , y = 5+rsin ? ? ? ? ?
Where
1
tan
2
??
? ? r 2cos 3sin 17
17 5 17 5
r
77
? ? ? ? ?
?
? ? ?
3. Let z
1
and z
2
be two complex number such that z
1
+ z
2
= 5 and
33
12
z z 20 15i ? ? ? . Then
44
12
zz ?
equals-
(1) 30 3
(2) 75
(3) 15 15
(4) 25 3
Ans. (2)
Sol.-
12
z z 5 ??
33
12
z z 20 15i ? ? ?
? ? ? ?
3
33
1 2 1 2 1 2 12
z z z z 3z z z z ? ? ? ? ?
? ?
33
12 12
z z 125 3z . z 5 ? ? ?
12
20 15i 125 15z z ? ? ? ?
12
3z z 25 4 3i ? ? ? ?
12
3z z 21 3i ? ? ?
12
z .z 7 i ? ? ?
? ?
2
12
z z 25 ? ? ?
? ?
22
12
z z 25 2 7 i ? ? ? ? ?
11 2i ??
? ?
2
22
12
z z 121 4 44i ? ? ? ?
? ? ?
2
44
12
z z 2 7 i 117 44i ? ? ? ? ?
? ? ?
44
12
z z 117 44i 2 49 1 14i ? ? ? ? ? ?
?
44
12
z z 75 ??
4. Let a variable line passing through the centre of the
circle x
2
+ y
2
– 16x – 4y = 0, meet the positive
co-ordinate axes at the point A and B. Then the
minimum value of OA + OB, where O is the
origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24
Ans. (2)
Sol.- ? ? ? ? y 2 m x 8 ? ? ?
? x-intercept
?
2
8
m
???
?
??
??
? y-intercept
? ? ? 8m 2 ??
2
OA OB 8 8m 2
m
?
? ? ? ? ? ?
? ?
2
2
f ' m 8 0
m
? ? ?
2
1
m
4
??
1
m
2
?
??
1
f 18
2
? ??
??
??
??
? Minimum = 18
5. Let f,g :(0, ) R ?? be two functions defined by
? ?
2 x
2t
x
f (x) t t e dt
?
?
??
?
and
2
x 1
t
2
0
g(x) t e dt
?
?
?
.
Then the value of
? ? ? ? ? ? ee
f log 9 g log 9 ? is
equal to
(1) 6
(2) 9
(3) 8
(4) 10
Ans. (3)
Sol.-
? ? ? ?
? ? ? ? ? ?
? ?
? ? ? ?
? ? ? ?
2
2
2
2
2 2 2
x
2t
x
2x
x 1
t
2
0
x
x 2 x 2 x
f x t t e dt
f x 2. x x e ............ 1
g x t e dt
g x xe 2x 0
f x g x 2xe 2x e 2x e
?
?
?
?
?
? ? ?
??
? ? ? ?
?
???
?? ? ? ? ?
?
?
Integrating both sides w.r.t.x
? ? ? ?
? ?
? ?
? ?
2
1
e
x
0
2
tt
0
0
log 9 1
f x g x 2xe dx
xt
e dt e
e
1
9 f (x) g(x) 1 9 8
9
?
?
?
?
?
??
?
??
?
?? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
?
6. Let
? ? ,, ? ? ? be mirror image of the point (2, 3, 5)
in the line
x 1 y 2 z 3
2 3 4
? ? ?
?? .
Then 2 3 4 ? ? ? ? ? is equal to
(1) 32
(2) 33
(3) 31
(4) 34
Ans. (2)
Sol.
P(2,3,5)
? ? R , , ? ? ?
? ?
? ?
? ? ? ?
PR 2,3, 4
PR. 2,3, 4 0
2, 3, 5 . 2,3, 4 0
2 3 4 4 9 20 33
?
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
7. Let P be a parabola with vertex (2, 3) and directrix
2x + y = 6. Let an ellipse
22
22
xy
E : 1,a b
ab
? ? ? of
eccentricity
1
2
pass through the focus of the
parabola P. Then the square of the length of the
latus rectum of E, is
(1)
385
8
(2)
347
8
(3)
512
25
(4)
656
25
Ans. (4)
Sol.-
Focur
axis
(2, 3)
( , ) ? ?
(1.6, 2.8)
Slope of axis
1
2
?
? ?
1
y 3 x 2
2
? ? ?
2y 6 x 2 ? ? ? ?
2y x 4 0 ? ? ? ?
2x y 6 0 ? ? ?
4x 2y 12 0 ? ? ?
1.6 4 2.4 ? ? ? ? ? ?
2.8 6 3.2 ? ? ? ? ? ?
Ellipse passes through (2.4, 3.2)
22
22
24 32
10 10
1
ab
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ……….(1)
Also
22
22
b 1 b 1
1
a 2 a 2
? ? ? ?
22
a 2b ??
Put in (1)
2
328
b
25
??
2
22
2
2
2b 4b 1 328 656
b4
a a 2 25 25
??
? ? ? ? ? ? ?
??
??
8. The temperature T(t) of a body at time t = 0 is 160
o
F and it decreases continuously as per the
differential equation
dT
K(T 80)
dt
? ? ? , where K
is positive constant. If T(15) = 120
o
F, then T(45) is
equal to
(1) 85
o
F
(2) 95
o
F
(3) 90
o
F
(4) 80
o
F
Ans. (3)
Sol.-
? ?
? ?
? ?
? ?
Tt
160 0
T
160
kt
k.15
k15
k.45
3
k.15
dT
k T 80
dt
dT
Kdt
T 80
ln T 80 kt
ln T 80 ln 80 kt
T 80
ln kt
80
T 80 80e
120 80 80e
40 1
e
80 2
T 45 80 80e
80 80 e
1
80 80
8
90
?
?
?
?
?
? ? ?
??
?
? ? ? ? ?
??
? ? ? ?
?
??
??
??
??
? ? ?
??
? ? ?
?
??
9. Let 2
nd
, 8
th
and 44
th
, terms of a non-constant A.P.
be respectively the 1
st
, 2
nd
and 3
rd
terms of G.P. If
the first term of A.P. is 1 then the sum of first
20 terms is equal to-
(1) 980 (2) 960
(3) 990 (4) 970
Ans. (4)
Sol.- 1 + d, 1 + 7d, 1 + 43d are in GP
(1 + 7d)
2
= (1 + d) (1 + 43d)
1 + 49d
2
+ 14d = 1 + 44 d + 43d
2
6d
2
– 30d = 0
d = 5
? ?
? ?
20
20
S 2 1 20 1 5
2
=10 2 95
=970
? ? ? ? ? ??
??
?
10. Let f : R (0, ) ? ? ? be strictly increasing
function such that
x
f (7x)
lim 1
f (x)
??
? . Then, the value
of
x
f (5x)
lim 1
f (x)
??
??
?
??
??
is equal to
(1) 4
(2) 0
(3) 7/5
(4) 1
Ans. (2)
Sol.- f : R (0, ) ??
? ?
? ?
x
f 7x
lim 1
fx
??
?
f is increasing
? ? ? ? ? ? f x f 5x f 7x ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
x
f x f 5x f 7x
f x f x f x
f 5x
1 lim 1
fx
f 5x
1
fx
1 1 0
??
??
??
??
??
??
??
? ? ?
11. The area of the region enclosed by the parabola
y = 4x – x
2
and 3y = (x – 4)
2
is equal to
(1)
32
9
(2) 4
(3) 6
(4)
14
3
Ans. (3)
Sol.-
Area ? ?
? ?
2
4
2
1
x4
4x x dx
3
??
?
?? ? ? ?
??
??
?
Area
? ?
4
3
23
1
x4
4x x
2 3 9
?
? ? ?
64 64 4 1 27
2 3 2 3 9
??
? ? ? ? ?
??
??
? ? 27 21 6 ? ? ?
12. Let the mean and the variance of 6 observation a,
b, 68, 44, 48, 60 be 55 and 194, respectively if
a > b, then a + 3b is
(1) 200
(2) 190
(3) 180
(4) 210
Ans. (3)
Sol.- a, b, 68, 44, 48, 60
Mean = 55 a > b
Variance = 194 a + 3b
a b 68 44 48 60
55
6
220 a b 330
a b 110......(1)
? ? ? ? ?
?
? ? ? ?
? ? ?
Page 5
FINAL JEE –MAIN EXAMINATION – JANUARY, 2024
(Held On Wednesday 31
st
January, 2024) TIME : 3 : 00 PM to 6 : 00 PM
MATHEMATICS TEST PAPER WITH SOLUTION
SECTION-A
1. The number of ways in which 21 identical apples
can be distributed among three children such that
each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136
Ans. (4)
Sol. After giving 2 apples to each child 15 apples left
now 15 apples can be distributed in
15 3 1 17
22
CC
??
?
ways
17 16
136
2
?
??
2. Let A (a, b), B(3, 4) and (–6, –8) respectively
denote the centroid, circumcentre and orthocentre
of a triangle. Then, the distance of the point
P(2a + 3, 7b + 5) from the line 2x + 3y – 4 = 0
measured parallel to the line x – 2y – 1 = 0 is
(1)
15 5
7
(2)
17 5
6
(3)
17 5
7
(4)
5
17
Ans. (3)
Sol. A(a,b), B(3,4), C(-6, -8)
2 : 1
C
(-6, -8)
A
B
(3, 4)
(a, b)
? ? a 0, b = 0 P 3,5 ? ? ?
Distance from P measured along x – 2y – 1 = 0
x 3 rcos , y = 5+rsin ? ? ? ? ?
Where
1
tan
2
??
? ? r 2cos 3sin 17
17 5 17 5
r
77
? ? ? ? ?
?
? ? ?
3. Let z
1
and z
2
be two complex number such that z
1
+ z
2
= 5 and
33
12
z z 20 15i ? ? ? . Then
44
12
zz ?
equals-
(1) 30 3
(2) 75
(3) 15 15
(4) 25 3
Ans. (2)
Sol.-
12
z z 5 ??
33
12
z z 20 15i ? ? ?
? ? ? ?
3
33
1 2 1 2 1 2 12
z z z z 3z z z z ? ? ? ? ?
? ?
33
12 12
z z 125 3z . z 5 ? ? ?
12
20 15i 125 15z z ? ? ? ?
12
3z z 25 4 3i ? ? ? ?
12
3z z 21 3i ? ? ?
12
z .z 7 i ? ? ?
? ?
2
12
z z 25 ? ? ?
? ?
22
12
z z 25 2 7 i ? ? ? ? ?
11 2i ??
? ?
2
22
12
z z 121 4 44i ? ? ? ?
? ? ?
2
44
12
z z 2 7 i 117 44i ? ? ? ? ?
? ? ?
44
12
z z 117 44i 2 49 1 14i ? ? ? ? ? ?
?
44
12
z z 75 ??
4. Let a variable line passing through the centre of the
circle x
2
+ y
2
– 16x – 4y = 0, meet the positive
co-ordinate axes at the point A and B. Then the
minimum value of OA + OB, where O is the
origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24
Ans. (2)
Sol.- ? ? ? ? y 2 m x 8 ? ? ?
? x-intercept
?
2
8
m
???
?
??
??
? y-intercept
? ? ? 8m 2 ??
2
OA OB 8 8m 2
m
?
? ? ? ? ? ?
? ?
2
2
f ' m 8 0
m
? ? ?
2
1
m
4
??
1
m
2
?
??
1
f 18
2
? ??
??
??
??
? Minimum = 18
5. Let f,g :(0, ) R ?? be two functions defined by
? ?
2 x
2t
x
f (x) t t e dt
?
?
??
?
and
2
x 1
t
2
0
g(x) t e dt
?
?
?
.
Then the value of
? ? ? ? ? ? ee
f log 9 g log 9 ? is
equal to
(1) 6
(2) 9
(3) 8
(4) 10
Ans. (3)
Sol.-
? ? ? ?
? ? ? ? ? ?
? ?
? ? ? ?
? ? ? ?
2
2
2
2
2 2 2
x
2t
x
2x
x 1
t
2
0
x
x 2 x 2 x
f x t t e dt
f x 2. x x e ............ 1
g x t e dt
g x xe 2x 0
f x g x 2xe 2x e 2x e
?
?
?
?
?
? ? ?
??
? ? ? ?
?
???
?? ? ? ? ?
?
?
Integrating both sides w.r.t.x
? ? ? ?
? ?
? ?
? ?
2
1
e
x
0
2
tt
0
0
log 9 1
f x g x 2xe dx
xt
e dt e
e
1
9 f (x) g(x) 1 9 8
9
?
?
?
?
?
??
?
??
?
?? ? ? ?
??
??
??
? ? ? ? ?
??
??
?
?
6. Let
? ? ,, ? ? ? be mirror image of the point (2, 3, 5)
in the line
x 1 y 2 z 3
2 3 4
? ? ?
?? .
Then 2 3 4 ? ? ? ? ? is equal to
(1) 32
(2) 33
(3) 31
(4) 34
Ans. (2)
Sol.
P(2,3,5)
? ? R , , ? ? ?
? ?
? ?
? ? ? ?
PR 2,3, 4
PR. 2,3, 4 0
2, 3, 5 . 2,3, 4 0
2 3 4 4 9 20 33
?
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
7. Let P be a parabola with vertex (2, 3) and directrix
2x + y = 6. Let an ellipse
22
22
xy
E : 1,a b
ab
? ? ? of
eccentricity
1
2
pass through the focus of the
parabola P. Then the square of the length of the
latus rectum of E, is
(1)
385
8
(2)
347
8
(3)
512
25
(4)
656
25
Ans. (4)
Sol.-
Focur
axis
(2, 3)
( , ) ? ?
(1.6, 2.8)
Slope of axis
1
2
?
? ?
1
y 3 x 2
2
? ? ?
2y 6 x 2 ? ? ? ?
2y x 4 0 ? ? ? ?
2x y 6 0 ? ? ?
4x 2y 12 0 ? ? ?
1.6 4 2.4 ? ? ? ? ? ?
2.8 6 3.2 ? ? ? ? ? ?
Ellipse passes through (2.4, 3.2)
22
22
24 32
10 10
1
ab
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ……….(1)
Also
22
22
b 1 b 1
1
a 2 a 2
? ? ? ?
22
a 2b ??
Put in (1)
2
328
b
25
??
2
22
2
2
2b 4b 1 328 656
b4
a a 2 25 25
??
? ? ? ? ? ? ?
??
??
8. The temperature T(t) of a body at time t = 0 is 160
o
F and it decreases continuously as per the
differential equation
dT
K(T 80)
dt
? ? ? , where K
is positive constant. If T(15) = 120
o
F, then T(45) is
equal to
(1) 85
o
F
(2) 95
o
F
(3) 90
o
F
(4) 80
o
F
Ans. (3)
Sol.-
? ?
? ?
? ?
? ?
Tt
160 0
T
160
kt
k.15
k15
k.45
3
k.15
dT
k T 80
dt
dT
Kdt
T 80
ln T 80 kt
ln T 80 ln 80 kt
T 80
ln kt
80
T 80 80e
120 80 80e
40 1
e
80 2
T 45 80 80e
80 80 e
1
80 80
8
90
?
?
?
?
?
? ? ?
??
?
? ? ? ? ?
??
? ? ? ?
?
??
??
??
??
? ? ?
??
? ? ?
?
??
9. Let 2
nd
, 8
th
and 44
th
, terms of a non-constant A.P.
be respectively the 1
st
, 2
nd
and 3
rd
terms of G.P. If
the first term of A.P. is 1 then the sum of first
20 terms is equal to-
(1) 980 (2) 960
(3) 990 (4) 970
Ans. (4)
Sol.- 1 + d, 1 + 7d, 1 + 43d are in GP
(1 + 7d)
2
= (1 + d) (1 + 43d)
1 + 49d
2
+ 14d = 1 + 44 d + 43d
2
6d
2
– 30d = 0
d = 5
? ?
? ?
20
20
S 2 1 20 1 5
2
=10 2 95
=970
? ? ? ? ? ??
??
?
10. Let f : R (0, ) ? ? ? be strictly increasing
function such that
x
f (7x)
lim 1
f (x)
??
? . Then, the value
of
x
f (5x)
lim 1
f (x)
??
??
?
??
??
is equal to
(1) 4
(2) 0
(3) 7/5
(4) 1
Ans. (2)
Sol.- f : R (0, ) ??
? ?
? ?
x
f 7x
lim 1
fx
??
?
f is increasing
? ? ? ? ? ? f x f 5x f 7x ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
x
f x f 5x f 7x
f x f x f x
f 5x
1 lim 1
fx
f 5x
1
fx
1 1 0
??
??
??
??
??
??
??
? ? ?
11. The area of the region enclosed by the parabola
y = 4x – x
2
and 3y = (x – 4)
2
is equal to
(1)
32
9
(2) 4
(3) 6
(4)
14
3
Ans. (3)
Sol.-
Area ? ?
? ?
2
4
2
1
x4
4x x dx
3
??
?
?? ? ? ?
??
??
?
Area
? ?
4
3
23
1
x4
4x x
2 3 9
?
? ? ?
64 64 4 1 27
2 3 2 3 9
??
? ? ? ? ?
??
??
? ? 27 21 6 ? ? ?
12. Let the mean and the variance of 6 observation a,
b, 68, 44, 48, 60 be 55 and 194, respectively if
a > b, then a + 3b is
(1) 200
(2) 190
(3) 180
(4) 210
Ans. (3)
Sol.- a, b, 68, 44, 48, 60
Mean = 55 a > b
Variance = 194 a + 3b
a b 68 44 48 60
55
6
220 a b 330
a b 110......(1)
? ? ? ? ?
?
? ? ? ?
? ? ?
Also,
? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2
i
2 2 2 2
22
22
22
22
22
22
xx
194
n
a 55 b 55 68 55 44 55
48 55 60 55 194 6
a 55 b 55 169 121 49 25 1164
a 55 b 55 1164 364 800
a 3025 110a b 3025 110b 800
a b 800 6050 12100
a b 6850.......(2)
Solve (1) & (2);
a
?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
??
?
? ?
=75,b=35
a 3b 75 3 35 75 105 180 ? ? ? ? ? ? ?
13. If the function f :( , 1] (a,b] ? ? ? ? defined by
3
x 3x 1
f(x) e
??
? is one-one and onto, then the
distance of the point P(2b + 4, a + 2) from the line
x + e
–3
y = 4 is :
(1)
6
2 1 e ? (2)
6
4 1 e ?
(3)
6
3 1 e ? (4)
6
1e ?
Ans. (1)
Sol.- ? ?
3
x 3x 1
f x e
??
?
? ? ? ?
3
x 3x 1 2
f ' x e . 3x 3
??
??
? ? ? ?
3
x 3x 1
e .3 x 1 x 1
??
? ? ?
For ? ? f ' x 0 ?
? ? fx ? is increasing function
? ? a e 0 f
??
? ? ? ? ? ?
? ?
1 3 1 3
b e e f 1
? ? ?
? ? ? ?
P(2b + 4, a + 2)
? ?
3
P 2 e 4,2 ??
x + e y = 4
-3
P
d
? ?
33
6
6
2e 4 2e 4
d 2 1 e
1e
?
?
? ? ?
? ? ?
?
14. Consider the function f :(0, ) R ?? defined by
e
log x
f (x) e
?
? . If m and n be respectively the
number of points at which f is not continuous and f
is not differentiable, then m + n is
(1) 0
(2) 3
(3) 1
(4) 2
Ans. (3)
Sol.-
? ?
? ?
e
log x
f : 0, R
f x e
?
??
?
? ?
ln x
ln x
ln x
1
;0 x 1
1
e
fx
1
e
;x 1
e
?
?
??
?
?
??
?
?
?
?
?
1
x;0 x 1
1
x
1
, x 1
x
?
? ? ?
?
?
?
?
? ?
?
1
0 1
m = 0 (No point at which function is not continuous)
n = 1 (Not differentiable)
? m + n = 1
15. The number of solutions, of the equation
sinx sinx
e 2e 2
?
?? is
(1) 2
(2) more than 2
(3) 1
(4) 0
Ans. (4)
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