JEE Main 2024 January 30 Shift 2 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025 PDF Download

 Page 1


  
                        
 
  
 
 
 
 
 
 
 
 
 
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
13
 
 ? 2 ?
2
 – ? – 1 = 0 
 
1
1,
2
? ? ?  
 ? ? ? ? ?
??
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
? ? ? and 13 ?? this 
will generate no solution case   
2. For 
? ??
? ? ?
??
??
, 0,
2
, let ? ? ? ? ? ? ? 3sin( ) 2sin( ) and a 
real number k be such that ? ? ? tan k tan . Then the 
value of k is equal to : 
 (1)  ?
2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ? cos ? +  3sin ? cos ?  
 = 2sin ? cos ? – 2sin ? cos ? 
 5sin ? cos ? = –sin ? cos ? 
 ? ? ? ?
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 ??
22
22
xy
E : 1
ab
 
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
e3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a5
?? 
Page 2


  
                        
 
  
 
 
 
 
 
 
 
 
 
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
13
 
 ? 2 ?
2
 – ? – 1 = 0 
 
1
1,
2
? ? ?  
 ? ? ? ? ?
??
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
? ? ? and 13 ?? this 
will generate no solution case   
2. For 
? ??
? ? ?
??
??
, 0,
2
, let ? ? ? ? ? ? ? 3sin( ) 2sin( ) and a 
real number k be such that ? ? ? tan k tan . Then the 
value of k is equal to : 
 (1)  ?
2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ? cos ? +  3sin ? cos ?  
 = 2sin ? cos ? – 2sin ? cos ? 
 5sin ? cos ? = –sin ? cos ? 
 ? ? ? ?
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 ??
22
22
xy
E : 1
ab
 
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
e3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a5
?? 
 
 
4. Let ? ? ? ? ? ? ? ?
ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?
4
 and 
2
b6 ? , 
If ? a.b 3 2 , then the value of 
? ?
? ? ? ?
2
22
ab 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 ? ? ? ? ?
23
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
bb
rr
aa
 
  T
p
 = ar
p –1
 
p1
4
b
a
a
?
??
?
??
??
 
  
p1
5
4
11 p
bb
t T a a
aa
?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 ?  
p1
5 p 21
4
?
? ? ? 
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 
? ? ? ?
??
x 2y 7 2x y 8
55
 
 (x + 2y + 7)
2
 – 
(2x – y + 8)
2
 
= 0
 
  
Page 3


  
                        
 
  
 
 
 
 
 
 
 
 
 
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
13
 
 ? 2 ?
2
 – ? – 1 = 0 
 
1
1,
2
? ? ?  
 ? ? ? ? ?
??
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
? ? ? and 13 ?? this 
will generate no solution case   
2. For 
? ??
? ? ?
??
??
, 0,
2
, let ? ? ? ? ? ? ? 3sin( ) 2sin( ) and a 
real number k be such that ? ? ? tan k tan . Then the 
value of k is equal to : 
 (1)  ?
2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ? cos ? +  3sin ? cos ?  
 = 2sin ? cos ? – 2sin ? cos ? 
 5sin ? cos ? = –sin ? cos ? 
 ? ? ? ?
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 ??
22
22
xy
E : 1
ab
 
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
e3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a5
?? 
 
 
4. Let ? ? ? ? ? ? ? ?
ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?
4
 and 
2
b6 ? , 
If ? a.b 3 2 , then the value of 
? ?
? ? ? ?
2
22
ab 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 ? ? ? ? ?
23
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
bb
rr
aa
 
  T
p
 = ar
p –1
 
p1
4
b
a
a
?
??
?
??
??
 
  
p1
5
4
11 p
bb
t T a a
aa
?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 ?  
p1
5 p 21
4
?
? ? ? 
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 
? ? ? ?
??
x 2y 7 2x y 8
55
 
 (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
 – ? ? ? ?
8 44
xy 6x y 5 0
33
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ? ? ? ? ?
4 22
h , g 3, f , c 5
33
 
 ? ? ? ? ? ? ? ? ? ?
4 22
g c h f 3 5 8 6 14
33
 
8. Let a and b be two vectors such that 
? ? ? | b | 1 and | b a | 2 . Then 
2
(b a) b ?? 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 
 
? ? ? ?? ?
? ? ? ? ?
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  
? ?
? ?
? ? ? ? ? ? ? ?
?
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
33
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
? ?
? ? ? ? ?
?
?
 
 ? ?
? ?
? ? ? ? ?
?
3
2
1
27 f '(t) 1 f "(t)dt 3 
 ? ?
? ?
??
?
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
 
 
??
? ? ? ?
??
??
?
1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
?? ??
? ? ? ? ?
?? ??
????
1 1 1 1
1
33
3 3 3
 
 ? ? ? ?
4 10 4 10
3
3 3 27
93
 
 
??
? ? ? ? ? ? ?
??
??
4 10
3 27 3 36 10 3
3 27
 
 ?= 36, ? = – 10 
 ? + ? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ??
22
xy
H : 1
94
, 
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


  
                        
 
  
 
 
 
 
 
 
 
 
 
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
13
 
 ? 2 ?
2
 – ? – 1 = 0 
 
1
1,
2
? ? ?  
 ? ? ? ? ?
??
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
? ? ? and 13 ?? this 
will generate no solution case   
2. For 
? ??
? ? ?
??
??
, 0,
2
, let ? ? ? ? ? ? ? 3sin( ) 2sin( ) and a 
real number k be such that ? ? ? tan k tan . Then the 
value of k is equal to : 
 (1)  ?
2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ? cos ? +  3sin ? cos ?  
 = 2sin ? cos ? – 2sin ? cos ? 
 5sin ? cos ? = –sin ? cos ? 
 ? ? ? ?
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 ??
22
22
xy
E : 1
ab
 
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
e3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a5
?? 
 
 
4. Let ? ? ? ? ? ? ? ?
ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?
4
 and 
2
b6 ? , 
If ? a.b 3 2 , then the value of 
? ?
? ? ? ?
2
22
ab 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 ? ? ? ? ?
23
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
bb
rr
aa
 
  T
p
 = ar
p –1
 
p1
4
b
a
a
?
??
?
??
??
 
  
p1
5
4
11 p
bb
t T a a
aa
?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 ?  
p1
5 p 21
4
?
? ? ? 
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 
? ? ? ?
??
x 2y 7 2x y 8
55
 
 (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
 – ? ? ? ?
8 44
xy 6x y 5 0
33
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ? ? ? ? ?
4 22
h , g 3, f , c 5
33
 
 ? ? ? ? ? ? ? ? ? ?
4 22
g c h f 3 5 8 6 14
33
 
8. Let a and b be two vectors such that 
? ? ? | b | 1 and | b a | 2 . Then 
2
(b a) b ?? 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 
 
? ? ? ?? ?
? ? ? ? ?
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  
? ?
? ?
? ? ? ? ? ? ? ?
?
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
33
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
? ?
? ? ? ? ?
?
?
 
 ? ?
? ?
? ? ? ? ?
?
3
2
1
27 f '(t) 1 f "(t)dt 3 
 ? ?
? ?
??
?
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
 
 
??
? ? ? ?
??
??
?
1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
?? ??
? ? ? ? ?
?? ??
????
1 1 1 1
1
33
3 3 3
 
 ? ? ? ?
4 10 4 10
3
3 3 27
93
 
 
??
? ? ? ? ? ? ?
??
??
4 10
3 27 3 36 10 3
3 27
 
 ?= 36, ? = – 10 
 ? + ? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ??
22
xy
H : 1
94
, 
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
( , ) ? ?
 
 
22
xy
1
94
?? 
 a
2
 = 9, b
2
 = 4 
 
2
2 2 2 2
2
b
b a (e 1) e 1
a
? ? ? ? ? 
 
2
4 13
e1
99
? ? ? 
 
12
13 13
e s s 2ae 2 3 2 13
33
? ? ? ? ? ? ? 
 Area of 
1 2 1 2
1
PS S s s 2 13
2
? ? ? ? ? ? 
 
1
(2 13) 2 13 2
2
? ? ? ? ? ? ? ? 
 
2 2 2
2
1 1 1 18 3 2
9 4 9
? ? ?
? ? ? ? ? ? ? ? ? ? ? 
 Distance of P from origin = 
22
? ? ? 
                                        = 18 4 22 ?? 
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) =  
 
?
?
?
?
? ? ?
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
 
 = ?
?
31
3 6 3
 
 
12. Let f : R ? R be defined ? ? ?
2x x
f(x) ae be cx . If 
?? f(0) 1 , ? ?
e
f (log 2) 21 and 
  ? ?
e
log 4
0
39
f(x) cx dx
2
??
?
, 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  
 ??
?
ln 4
2x x
0
39
(ae be )dx
2
 
 
??
??
??
??
ln 4
2x
x
0
ae 39
be
22
 ?  8a + 4b – ??
a 39
b
22
 
  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


  
                        
 
  
 
 
 
 
 
 
 
 
 
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
13
 
 ? 2 ?
2
 – ? – 1 = 0 
 
1
1,
2
? ? ?  
 ? ? ? ? ?
??
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
? ? ? and 13 ?? this 
will generate no solution case   
2. For 
? ??
? ? ?
??
??
, 0,
2
, let ? ? ? ? ? ? ? 3sin( ) 2sin( ) and a 
real number k be such that ? ? ? tan k tan . Then the 
value of k is equal to : 
 (1)  ?
2
3
 (2) –5   
 (3) 
2
3
  (4) 5 
 Ans. (2 ) 
Sol. 3sin ? cos ? +  3sin ? cos ?  
 = 2sin ? cos ? – 2sin ? cos ? 
 5sin ? cos ? = –sin ? cos ? 
 ? ? ? ?
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 ??
22
22
xy
E : 1
ab
 
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
e3
 
 a = 5 
 b = 4 
 
2
2b 32
Length of LR
a5
?? 
 
 
4. Let ? ? ? ? ? ? ? ?
ˆˆ ˆ
a i j k, , R . Let a vector b be such 
that the angle between a and b is 
?
4
 and 
2
b6 ? , 
If ? a.b 3 2 , then the value of 
? ?
? ? ? ?
2
22
ab 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 ? ? ? ? ?
23
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
bb
rr
aa
 
  T
p
 = ar
p –1
 
p1
4
b
a
a
?
??
?
??
??
 
  
p1
5
4
11 p
bb
t T a a
aa
?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 ?  
p1
5 p 21
4
?
? ? ? 
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 
? ? ? ?
??
x 2y 7 2x y 8
55
 
 (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
 – ? ? ? ?
8 44
xy 6x y 5 0
33
 
 x
2
 – y
2
 + 2h xy + 2gx 2 + 2fy + c = 0 
 ? ? ? ? ?
4 22
h , g 3, f , c 5
33
 
 ? ? ? ? ? ? ? ? ? ?
4 22
g c h f 3 5 8 6 14
33
 
8. Let a and b be two vectors such that 
? ? ? | b | 1 and | b a | 2 . Then 
2
(b a) b ?? 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 
 
? ? ? ?? ?
? ? ? ? ?
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  
? ?
? ?
? ? ? ? ? ? ? ?
?
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
33
 
 
(3,f(3))
dy
f '(3) tan 1 f '(3) 1
dx 4
? ?
? ? ? ? ?
?
?
 
 ? ?
? ?
? ? ? ? ?
?
3
2
1
27 f '(t) 1 f "(t)dt 3 
 ? ?
? ?
??
?
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
 
 
??
? ? ? ?
??
??
?
1
1 3
2
1/ 3 1/ 3
z
I (z 1)dz z
3
 
 
?? ??
? ? ? ? ?
?? ??
????
1 1 1 1
1
33
3 3 3
 
 ? ? ? ?
4 10 4 10
3
3 3 27
93
 
 
??
? ? ? ? ? ? ?
??
??
4 10
3 27 3 36 10 3
3 27
 
 ?= 36, ? = – 10 
 ? + ? = 36 – 10 = 26 
10. Let P be a point on the hyperbola ??
22
xy
H : 1
94
, 
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
( , ) ? ?
 
 
22
xy
1
94
?? 
 a
2
 = 9, b
2
 = 4 
 
2
2 2 2 2
2
b
b a (e 1) e 1
a
? ? ? ? ? 
 
2
4 13
e1
99
? ? ? 
 
12
13 13
e s s 2ae 2 3 2 13
33
? ? ? ? ? ? ? 
 Area of 
1 2 1 2
1
PS S s s 2 13
2
? ? ? ? ? ? 
 
1
(2 13) 2 13 2
2
? ? ? ? ? ? ? ? 
 
2 2 2
2
1 1 1 18 3 2
9 4 9
? ? ?
? ? ? ? ? ? ? ? ? ? ? 
 Distance of P from origin = 
22
? ? ? 
                                        = 18 4 22 ?? 
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) =  
 
?
?
?
?
? ? ?
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
 
 = ?
?
31
3 6 3
 
 
12. Let f : R ? R be defined ? ? ?
2x x
f(x) ae be cx . If 
?? f(0) 1 , ? ?
e
f (log 2) 21 and 
  ? ?
e
log 4
0
39
f(x) cx dx
2
??
?
, 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  
 ??
?
ln 4
2x x
0
39
(ae be )dx
2
 
 
??
??
??
??
ln 4
2x
x
0
ae 39
be
22
 ?  8a + 4b – ??
a 39
b
22
 
  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 ? ? ? ? ? ? ? ??
1
ˆ ˆ ˆ ˆ ˆˆ
L : r (i j 2k) (i j 2k), R 
 ? ? ? ? ? ? ? ?
2
ˆ ˆ ˆ ˆˆ
L : r ( j k) (3i j pk), R and 
? ? ? ? ? ?
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  
 ? ? ? ? ?
?
ˆˆ ˆ
i j k
ˆˆ ˆ
1 1 2 i 7j 4k
3 1 1
 
  ( , 7 , 4 ) ? ? ? ? ? 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 
? ? ? ?
?
?
??
?
2
x 3x a , x 1
f(x)
bx 2 , x 1
be 
differentiable on R. Then, the value of 
?
?
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 
 
?
? ? ? ?
??
12
2
21
(x 3x 3)dx (5x 2)dx 
 = 
?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
12
3 2 2
21
x 3x 5x
3x 2x
3 2 2
 
 = 
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
1 3 8 5
3 6 6 10 4 2
3 2 3 2
 
 =   ? ? ? ?
35
6 12 17
22
 
15. Let ?? 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 ? ? ?
1985 100
z z 1 0 and 
? ? ? ?
32
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     
? ?lso,  z = w
2
  
? w
3970
 + w
200
 + 1  
? w + w
2
 + 1 = 0  
 Two common root