JEE Exam  >  JEE Notes  >  Mock Tests for JEE Main and Advanced 2025  >  JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025 PDF Download

Download, print and study this document offline
Please wait while the PDF view is loading
 Page 1


 
 
17
th
 March. 2021 | Shift 1
SECTION – A 
 
1. Which of the following is true for y(x) that satisfies the differential equation  
dy
dx
= xy-1+x-y;y(0) = 0 : 
 (1) y(1) = 1 
(2)  y(1) = 
1
2
e 1 ? 
(3) y(1) = 
1 1
2 2
e e
?
? 
(4) y(1) = 
?
?
1
2
e 1
 
Ans. (4) 
 
Sol. ? ? ? ?
dy
x 1 y x 1
dx
? ? ? ? 
 ? ? ? ?
dy
x 1 y 1
dx
? ? ? 
 
? ?
dy
x 1 dx
y 1
? ?
?
 
 
? ?
? ? ? ? ?
2
x
n y 1 x c
2
 
 X = 0, y = 0 
 c 0 ? ? 
 
? ?
2
x
n y 1 x
2
? ? ? ? ? 
 putting x = 1, ? n(y + 1) =
1 1
1
2 2
? ? ? 
 
1
2
y 1 e
?
? ? 
 
1
2
y e 1
?
? ? 
 
? ?
?
? ? ?
1
2
y 1 e 1 
  
 
 
Page 2


 
 
17
th
 March. 2021 | Shift 1
SECTION – A 
 
1. Which of the following is true for y(x) that satisfies the differential equation  
dy
dx
= xy-1+x-y;y(0) = 0 : 
 (1) y(1) = 1 
(2)  y(1) = 
1
2
e 1 ? 
(3) y(1) = 
1 1
2 2
e e
?
? 
(4) y(1) = 
?
?
1
2
e 1
 
Ans. (4) 
 
Sol. ? ? ? ?
dy
x 1 y x 1
dx
? ? ? ? 
 ? ? ? ?
dy
x 1 y 1
dx
? ? ? 
 
? ?
dy
x 1 dx
y 1
? ?
?
 
 
? ?
? ? ? ? ?
2
x
n y 1 x c
2
 
 X = 0, y = 0 
 c 0 ? ? 
 
? ?
2
x
n y 1 x
2
? ? ? ? ? 
 putting x = 1, ? n(y + 1) =
1 1
1
2 2
? ? ? 
 
1
2
y 1 e
?
? ? 
 
1
2
y e 1
?
? ? 
 
? ?
?
? ? ?
1
2
y 1 e 1 
  
 
 
 
 
2. The system of equations kx + y + z=1, x + ky + z=k and x + y + zk =k
2
 has no solution if k is 
equal to: 
 (1) -2 
 
(2)-1 
 (3)1 
 
(4) 0 
Ans. (1)  
Sol. ? ?
k 1 1
D 1 k 1 0
1 1 k
 
 ? k(k
2
 – 1) – (k – 1) + (1 – k) = 0 
? (k – 1) (k
2
 + k – 1 – 1) = 0 
? (k – 1) (k
2
 + k – 2) = 0 
? (k –1) (k –1) (k + 2) = 0 
?k = 1, k = 2 
for k = 1 equation identical so k = –2 for no solution. 
   
3. The value of 
1
4
1
5
1
4
1
5
4 .......
?
?
?
?
? ?
     is: 
 (1) 
4
2 30
5
?  
 
(2) 
4
4 30
5
? 
 (3) 
2
2 30
5
? 
 
(4) 
2
5 0
5
3 ? 
Ans. (3)  
Sol. 
1
y 4
5
y
? ?
?
?
 
 ? 
y
y 4
5y 1
? ?
?
 
Page 3


 
 
17
th
 March. 2021 | Shift 1
SECTION – A 
 
1. Which of the following is true for y(x) that satisfies the differential equation  
dy
dx
= xy-1+x-y;y(0) = 0 : 
 (1) y(1) = 1 
(2)  y(1) = 
1
2
e 1 ? 
(3) y(1) = 
1 1
2 2
e e
?
? 
(4) y(1) = 
?
?
1
2
e 1
 
Ans. (4) 
 
Sol. ? ? ? ?
dy
x 1 y x 1
dx
? ? ? ? 
 ? ? ? ?
dy
x 1 y 1
dx
? ? ? 
 
? ?
dy
x 1 dx
y 1
? ?
?
 
 
? ?
? ? ? ? ?
2
x
n y 1 x c
2
 
 X = 0, y = 0 
 c 0 ? ? 
 
? ?
2
x
n y 1 x
2
? ? ? ? ? 
 putting x = 1, ? n(y + 1) =
1 1
1
2 2
? ? ? 
 
1
2
y 1 e
?
? ? 
 
1
2
y e 1
?
? ? 
 
? ?
?
? ? ?
1
2
y 1 e 1 
  
 
 
 
 
2. The system of equations kx + y + z=1, x + ky + z=k and x + y + zk =k
2
 has no solution if k is 
equal to: 
 (1) -2 
 
(2)-1 
 (3)1 
 
(4) 0 
Ans. (1)  
Sol. ? ?
k 1 1
D 1 k 1 0
1 1 k
 
 ? k(k
2
 – 1) – (k – 1) + (1 – k) = 0 
? (k – 1) (k
2
 + k – 1 – 1) = 0 
? (k – 1) (k
2
 + k – 2) = 0 
? (k –1) (k –1) (k + 2) = 0 
?k = 1, k = 2 
for k = 1 equation identical so k = –2 for no solution. 
   
3. The value of 
1
4
1
5
1
4
1
5
4 .......
?
?
?
?
? ?
     is: 
 (1) 
4
2 30
5
?  
 
(2) 
4
4 30
5
? 
 (3) 
2
2 30
5
? 
 
(4) 
2
5 0
5
3 ? 
Ans. (3)  
Sol. 
1
y 4
5
y
? ?
?
?
 
 ? 
y
y 4
5y 1
? ?
?
 
 
 
17
th
 March. 2021 | Shift 1
 ? 5y
2
-20y-4 = 0 
 ? 
20 400 80
y
10
? ?
? 
 ?
20 4 30
y ,y 0
10
?
? ? 
 
10 2 30
y
5
?
? 
4. If the Boolean expression (p
?
q) ? (q * ( ? P)) is a tautology, then the Boolean expression p * 
( ~q) is equivalent to: 
 (1) ? ? p q 
 (2) ? p q 
 (3) ? q p 
 (4) ? ? q p 
Ans. (3)  
Sol. (p ?q) ? (q* ? P) 
 P q p ?q        q* ? P         ? q       ? q ? p    ? ( ? q ? p) 
 T T T  T  F  F  T 
 T F F  F  T  T  F  
 F T T  T  F  F  T 
 F F T  T  T  F  T 
 ? ? ( ? q ? p)  = q ? ? p 
   = ? p ? q 
 ? * is equivalent to ? 
 ? p * ? q = p ? ? q 
  = ? q ? p  
  = q ? ?p    
 
5. Choose the incorrect statement about the two circles whose equations are given below: 
 X
2
+y
2
-10x-10y+41=0 and  
 X
2
+y
2
-16x-10y+80=0  
(1) Distance between two centres is the average of radii of both the circles. 
(2) Circles have two intersection points. 
(3) Both circles’ centres lie inside region of one another. 
(4) Both circles pass through the centre of each other. 
Ans. (3) 
Sol. C1 (5, 5), C2 (8, 5) 
position of C1(5, 5) in S2 = 0 
= 25 + 25 – 80 – 50 + 80 
= 0 
position of C2(8, 5) in S1 = 0 
? 64 + 25 – 80 – 50 + 41 
= 0 
Page 4


 
 
17
th
 March. 2021 | Shift 1
SECTION – A 
 
1. Which of the following is true for y(x) that satisfies the differential equation  
dy
dx
= xy-1+x-y;y(0) = 0 : 
 (1) y(1) = 1 
(2)  y(1) = 
1
2
e 1 ? 
(3) y(1) = 
1 1
2 2
e e
?
? 
(4) y(1) = 
?
?
1
2
e 1
 
Ans. (4) 
 
Sol. ? ? ? ?
dy
x 1 y x 1
dx
? ? ? ? 
 ? ? ? ?
dy
x 1 y 1
dx
? ? ? 
 
? ?
dy
x 1 dx
y 1
? ?
?
 
 
? ?
? ? ? ? ?
2
x
n y 1 x c
2
 
 X = 0, y = 0 
 c 0 ? ? 
 
? ?
2
x
n y 1 x
2
? ? ? ? ? 
 putting x = 1, ? n(y + 1) =
1 1
1
2 2
? ? ? 
 
1
2
y 1 e
?
? ? 
 
1
2
y e 1
?
? ? 
 
? ?
?
? ? ?
1
2
y 1 e 1 
  
 
 
 
 
2. The system of equations kx + y + z=1, x + ky + z=k and x + y + zk =k
2
 has no solution if k is 
equal to: 
 (1) -2 
 
(2)-1 
 (3)1 
 
(4) 0 
Ans. (1)  
Sol. ? ?
k 1 1
D 1 k 1 0
1 1 k
 
 ? k(k
2
 – 1) – (k – 1) + (1 – k) = 0 
? (k – 1) (k
2
 + k – 1 – 1) = 0 
? (k – 1) (k
2
 + k – 2) = 0 
? (k –1) (k –1) (k + 2) = 0 
?k = 1, k = 2 
for k = 1 equation identical so k = –2 for no solution. 
   
3. The value of 
1
4
1
5
1
4
1
5
4 .......
?
?
?
?
? ?
     is: 
 (1) 
4
2 30
5
?  
 
(2) 
4
4 30
5
? 
 (3) 
2
2 30
5
? 
 
(4) 
2
5 0
5
3 ? 
Ans. (3)  
Sol. 
1
y 4
5
y
? ?
?
?
 
 ? 
y
y 4
5y 1
? ?
?
 
 
 
17
th
 March. 2021 | Shift 1
 ? 5y
2
-20y-4 = 0 
 ? 
20 400 80
y
10
? ?
? 
 ?
20 4 30
y ,y 0
10
?
? ? 
 
10 2 30
y
5
?
? 
4. If the Boolean expression (p
?
q) ? (q * ( ? P)) is a tautology, then the Boolean expression p * 
( ~q) is equivalent to: 
 (1) ? ? p q 
 (2) ? p q 
 (3) ? q p 
 (4) ? ? q p 
Ans. (3)  
Sol. (p ?q) ? (q* ? P) 
 P q p ?q        q* ? P         ? q       ? q ? p    ? ( ? q ? p) 
 T T T  T  F  F  T 
 T F F  F  T  T  F  
 F T T  T  F  F  T 
 F F T  T  T  F  T 
 ? ? ( ? q ? p)  = q ? ? p 
   = ? p ? q 
 ? * is equivalent to ? 
 ? p * ? q = p ? ? q 
  = ? q ? p  
  = q ? ?p    
 
5. Choose the incorrect statement about the two circles whose equations are given below: 
 X
2
+y
2
-10x-10y+41=0 and  
 X
2
+y
2
-16x-10y+80=0  
(1) Distance between two centres is the average of radii of both the circles. 
(2) Circles have two intersection points. 
(3) Both circles’ centres lie inside region of one another. 
(4) Both circles pass through the centre of each other. 
Ans. (3) 
Sol. C1 (5, 5), C2 (8, 5) 
position of C1(5, 5) in S2 = 0 
= 25 + 25 – 80 – 50 + 80 
= 0 
position of C2(8, 5) in S1 = 0 
? 64 + 25 – 80 – 50 + 41 
= 0 
 
 
 
6. The sum of possible values of x for 
? ?
1 1 1
1 8
tan 1 cot tan
1 31
? ? ?
? ? ? ?
? ? ?
? ? ? ?
?
? ? ? ?
x
x
  is: 
 (1) 
33
4
? 
 (2) 
32
4
? 
 (3) 
31
4
? 
 (4) 
30
4
? 
Ans. (2) 
Sol. Taking tan both sides 
 
? ? ? ?
? ? ? ?
1 x x 1
8
1 1 x x 1 31
? ? ?
?
? ? ?
 
 
2
2x 8
31 2 x
? ?
?
 
 ? 4x
2
+31x-8 = 0 
 
1
x 8,
4
? ? ? 
 but at 
1
x
4
? 
 LHS>
2
?
and RHS<
2
?
 
So, only solution is x = –8 
 
7. Lt 2 3
ˆ ˆ ˆ
4 ? ? ?
?
a i j k and 
ˆ ˆ ˆ
7 6 ?
?
b i + j - k  
 
 If 
? ?
ˆ ˆ ˆ
r a r b,r. i 2j k 3 ? ? ? ? ? ? ?
?
? ? ? ?
, then 
? ?
ˆ ˆ ˆ
r. 2i 3j k ? ?
?
 is equal to : 
(1) 10 
(2) 13 
(3) 12 
(4) 8 
Ans. (3) 
Sol. ? ? 2, 3, 4 ? ?
?
a  , ? ? 7,1, 6 ? ?
?
b 
Page 5


 
 
17
th
 March. 2021 | Shift 1
SECTION – A 
 
1. Which of the following is true for y(x) that satisfies the differential equation  
dy
dx
= xy-1+x-y;y(0) = 0 : 
 (1) y(1) = 1 
(2)  y(1) = 
1
2
e 1 ? 
(3) y(1) = 
1 1
2 2
e e
?
? 
(4) y(1) = 
?
?
1
2
e 1
 
Ans. (4) 
 
Sol. ? ? ? ?
dy
x 1 y x 1
dx
? ? ? ? 
 ? ? ? ?
dy
x 1 y 1
dx
? ? ? 
 
? ?
dy
x 1 dx
y 1
? ?
?
 
 
? ?
? ? ? ? ?
2
x
n y 1 x c
2
 
 X = 0, y = 0 
 c 0 ? ? 
 
? ?
2
x
n y 1 x
2
? ? ? ? ? 
 putting x = 1, ? n(y + 1) =
1 1
1
2 2
? ? ? 
 
1
2
y 1 e
?
? ? 
 
1
2
y e 1
?
? ? 
 
? ?
?
? ? ?
1
2
y 1 e 1 
  
 
 
 
 
2. The system of equations kx + y + z=1, x + ky + z=k and x + y + zk =k
2
 has no solution if k is 
equal to: 
 (1) -2 
 
(2)-1 
 (3)1 
 
(4) 0 
Ans. (1)  
Sol. ? ?
k 1 1
D 1 k 1 0
1 1 k
 
 ? k(k
2
 – 1) – (k – 1) + (1 – k) = 0 
? (k – 1) (k
2
 + k – 1 – 1) = 0 
? (k – 1) (k
2
 + k – 2) = 0 
? (k –1) (k –1) (k + 2) = 0 
?k = 1, k = 2 
for k = 1 equation identical so k = –2 for no solution. 
   
3. The value of 
1
4
1
5
1
4
1
5
4 .......
?
?
?
?
? ?
     is: 
 (1) 
4
2 30
5
?  
 
(2) 
4
4 30
5
? 
 (3) 
2
2 30
5
? 
 
(4) 
2
5 0
5
3 ? 
Ans. (3)  
Sol. 
1
y 4
5
y
? ?
?
?
 
 ? 
y
y 4
5y 1
? ?
?
 
 
 
17
th
 March. 2021 | Shift 1
 ? 5y
2
-20y-4 = 0 
 ? 
20 400 80
y
10
? ?
? 
 ?
20 4 30
y ,y 0
10
?
? ? 
 
10 2 30
y
5
?
? 
4. If the Boolean expression (p
?
q) ? (q * ( ? P)) is a tautology, then the Boolean expression p * 
( ~q) is equivalent to: 
 (1) ? ? p q 
 (2) ? p q 
 (3) ? q p 
 (4) ? ? q p 
Ans. (3)  
Sol. (p ?q) ? (q* ? P) 
 P q p ?q        q* ? P         ? q       ? q ? p    ? ( ? q ? p) 
 T T T  T  F  F  T 
 T F F  F  T  T  F  
 F T T  T  F  F  T 
 F F T  T  T  F  T 
 ? ? ( ? q ? p)  = q ? ? p 
   = ? p ? q 
 ? * is equivalent to ? 
 ? p * ? q = p ? ? q 
  = ? q ? p  
  = q ? ?p    
 
5. Choose the incorrect statement about the two circles whose equations are given below: 
 X
2
+y
2
-10x-10y+41=0 and  
 X
2
+y
2
-16x-10y+80=0  
(1) Distance between two centres is the average of radii of both the circles. 
(2) Circles have two intersection points. 
(3) Both circles’ centres lie inside region of one another. 
(4) Both circles pass through the centre of each other. 
Ans. (3) 
Sol. C1 (5, 5), C2 (8, 5) 
position of C1(5, 5) in S2 = 0 
= 25 + 25 – 80 – 50 + 80 
= 0 
position of C2(8, 5) in S1 = 0 
? 64 + 25 – 80 – 50 + 41 
= 0 
 
 
 
6. The sum of possible values of x for 
? ?
1 1 1
1 8
tan 1 cot tan
1 31
? ? ?
? ? ? ?
? ? ?
? ? ? ?
?
? ? ? ?
x
x
  is: 
 (1) 
33
4
? 
 (2) 
32
4
? 
 (3) 
31
4
? 
 (4) 
30
4
? 
Ans. (2) 
Sol. Taking tan both sides 
 
? ? ? ?
? ? ? ?
1 x x 1
8
1 1 x x 1 31
? ? ?
?
? ? ?
 
 
2
2x 8
31 2 x
? ?
?
 
 ? 4x
2
+31x-8 = 0 
 
1
x 8,
4
? ? ? 
 but at 
1
x
4
? 
 LHS>
2
?
and RHS<
2
?
 
So, only solution is x = –8 
 
7. Lt 2 3
ˆ ˆ ˆ
4 ? ? ?
?
a i j k and 
ˆ ˆ ˆ
7 6 ?
?
b i + j - k  
 
 If 
? ?
ˆ ˆ ˆ
r a r b,r. i 2j k 3 ? ? ? ? ? ? ?
?
? ? ? ?
, then 
? ?
ˆ ˆ ˆ
r. 2i 3j k ? ?
?
 is equal to : 
(1) 10 
(2) 13 
(3) 12 
(4) 8 
Ans. (3) 
Sol. ? ? 2, 3, 4 ? ?
?
a  , ? ? 7,1, 6 ? ?
?
b 
 
 
17
th
 March. 2021 | Shift 1
 0 ? ? ? ?
? ?
?
?
r a r b 
 
? ?
0 ? ? ?
?
?
r a b 
 
? ?
? ?
?
? ?
r a b ?  
 
? ?
? ?
5 4 10 ? ? ? ?
?
?
i j k r ? 
 ? ? 3 ? 2, ,1 ? ? ?
?
r  
if  ? ? 1, 2, 3 1 ? ? ?
?
r 
? ? 3 5 8 10 ? ? ? ? ? ? ? ? = 1 
 ? (-5, -4, 10) . (2, -3, 1) 
  ? –10 + 12 + 10 = 12 
 
8. The equation of the plane which contains the y-axis and passes through the point (1, 2, 3)is: 
 (1) 3x + z = 6 
 (2) 3x –z=0 
 (3) x + 3z=10 
 (4) x + 3z = 0 
Ans. (2) 
Sol. Let the equation of the plane is a(x – 1) + b(y – 2) + c (z – 3) = 0 
Y–axis lies on it D.R.’s of y-axis are 0, 1, 0 
?0.a + 1.b + 0.c = 0 ? b = 0 
? Equation of plane is a(x – 1) + c (z – 3) = 0 
x = 0, z = 0 also satisfy it –a – 3c = 0 ? a = –3c 
–3c (x – 1) + c (z – 3) = 0 
–3x + 3 + z – 3 = 0 
 3x – z = 0 
 
 
9. If 
0 sin
A
sin 0
? ? ?
?
? ?
?
? ?
 and det 
2
1
A I 0
2
? ?
? ?
? ?
? ?
, then  a possible value of ? is: 
 (1)
6
?
 
 (2) 
2
?
 
 (3)
3
?
 
 (4) 
4
?
 
Ans. (4) 
Read More
356 docs|142 tests
Related Searches

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025

,

Viva Questions

,

pdf

,

shortcuts and tricks

,

Free

,

Sample Paper

,

MCQs

,

video lectures

,

mock tests for examination

,

past year papers

,

Important questions

,

Objective type Questions

,

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025

,

Summary

,

Exam

,

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025

,

Semester Notes

,

Extra Questions

,

ppt

,

Previous Year Questions with Solutions

,

practice quizzes

,

study material

;