JEE Exam  >  JEE Notes  >  JEE Main & Advanced Mock Test Series  >  JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | JEE Main & Advanced Mock Test Series 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
366 docs|219 tests
366 docs|219 tests
Download as PDF
Explore Courses for JEE exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Download the FREE EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

MCQs

,

video lectures

,

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | JEE Main & Advanced Mock Test Series

,

study material

,

Extra Questions

,

Previous Year Questions with Solutions

,

Semester Notes

,

Exam

,

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | JEE Main & Advanced Mock Test Series

,

mock tests for examination

,

Objective type Questions

,

ppt

,

JEE Main 2021 Mathematics March 17 Shift 1 Paper & Solutions | JEE Main & Advanced Mock Test Series

,

Important questions

,

Viva Questions

,

Sample Paper

,

practice quizzes

,

pdf

,

past year papers

,

Summary

,

Free

,

shortcuts and tricks

;