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Page 1
18
th
March. 2021 | Shift 1
SECTION – A
1. If the functions are defined as f(x) = x and g(x) = ? 1 x, then what is the common domain of the following
functions : f + g, f – g, f/g, g/f, g – f where (f ± g) (x) = f(x) ± g(x), (f/g) (x) =
f(x)
g(x)
(1) 0 < x ? 1
(2) 0 ? x ? 1
(3) 0 ? x ? 1
(4) 0 < x < 1
Ans. (4)
Sol. f + g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f – g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f/g =
?
x
1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1)
g/f =
? 1 x
x
? ? ? ? ? ? 1 x 0 & x 0 x (0,1]
g – f = ? ? 1 x x
? ? ? ? ? ? 1 x 0 & x 0 x [0,1]
? ? x (0,1)
2. Let ?, ?, ? be the roots of the equations, x
3
+ ax
2
+ bx + c = 0, (a, b, c ? R and a, b and a, b ? 0). If the system
of the equations (in u, v, w) given by ?u + ?v + ?w = 0; ?u + ?v + ?w = 0; ?u + ?v + ?w = 0 has non-trivial
solutions, then the value of
2
a
b
is
(1) 5
(2) 1
(3) 0
(4) 3
Ans. (4)
Page 2
18
th
March. 2021 | Shift 1
SECTION – A
1. If the functions are defined as f(x) = x and g(x) = ? 1 x, then what is the common domain of the following
functions : f + g, f – g, f/g, g/f, g – f where (f ± g) (x) = f(x) ± g(x), (f/g) (x) =
f(x)
g(x)
(1) 0 < x ? 1
(2) 0 ? x ? 1
(3) 0 ? x ? 1
(4) 0 < x < 1
Ans. (4)
Sol. f + g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f – g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f/g =
?
x
1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1)
g/f =
? 1 x
x
? ? ? ? ? ? 1 x 0 & x 0 x (0,1]
g – f = ? ? 1 x x
? ? ? ? ? ? 1 x 0 & x 0 x [0,1]
? ? x (0,1)
2. Let ?, ?, ? be the roots of the equations, x
3
+ ax
2
+ bx + c = 0, (a, b, c ? R and a, b and a, b ? 0). If the system
of the equations (in u, v, w) given by ?u + ?v + ?w = 0; ?u + ?v + ?w = 0; ?u + ?v + ?w = 0 has non-trivial
solutions, then the value of
2
a
b
is
(1) 5
(2) 1
(3) 0
(4) 3
Ans. (4)
Sol. x
3
+ ax
2
+ bx + c = 0
For non-trivial solutions,
? ? ?
? ? ? ?
? ? ?
0
?
3
+ ?
3
+ ?
3
– 3 ? ? ? = 0
? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ?? ?
? ?
? ?
2
3 0
(–a) [a
2
– 3b] = 0
a
2
= 3b ? ? ? a 0 ?
? ?
2
a
3
b
3. If the equation ? ? ? ? ? ?
2
a| z| z z d 0 represents a circle where a, d are real constants, then which of the
following condition is correct?
(1) ? ? ?
2
| | ad 0
(2) ? ? ? ? ?
2
| | ad 0anda R {0}
(3)
?
? ? ? 0,a,d R
(4) ? ? ? ?
2
| | ad 0and a R
Ans. (2)
Sol. ? ? ? ? ? ?
2
a| z| z z d 0
? ?
? ?
? ? ? ?
? ? ?
? ? ? ? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ?
d
zz z z 0
a a a
Centre = –
?
a
r =
?
?
2
d
a a
?
? ?
2
d
a a
? ? ?
2
ad
?
? ?
?
Page 3
18
th
March. 2021 | Shift 1
SECTION – A
1. If the functions are defined as f(x) = x and g(x) = ? 1 x, then what is the common domain of the following
functions : f + g, f – g, f/g, g/f, g – f where (f ± g) (x) = f(x) ± g(x), (f/g) (x) =
f(x)
g(x)
(1) 0 < x ? 1
(2) 0 ? x ? 1
(3) 0 ? x ? 1
(4) 0 < x < 1
Ans. (4)
Sol. f + g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f – g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f/g =
?
x
1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1)
g/f =
? 1 x
x
? ? ? ? ? ? 1 x 0 & x 0 x (0,1]
g – f = ? ? 1 x x
? ? ? ? ? ? 1 x 0 & x 0 x [0,1]
? ? x (0,1)
2. Let ?, ?, ? be the roots of the equations, x
3
+ ax
2
+ bx + c = 0, (a, b, c ? R and a, b and a, b ? 0). If the system
of the equations (in u, v, w) given by ?u + ?v + ?w = 0; ?u + ?v + ?w = 0; ?u + ?v + ?w = 0 has non-trivial
solutions, then the value of
2
a
b
is
(1) 5
(2) 1
(3) 0
(4) 3
Ans. (4)
Sol. x
3
+ ax
2
+ bx + c = 0
For non-trivial solutions,
? ? ?
? ? ? ?
? ? ?
0
?
3
+ ?
3
+ ?
3
– 3 ? ? ? = 0
? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ?? ?
? ?
? ?
2
3 0
(–a) [a
2
– 3b] = 0
a
2
= 3b ? ? ? a 0 ?
? ?
2
a
3
b
3. If the equation ? ? ? ? ? ?
2
a| z| z z d 0 represents a circle where a, d are real constants, then which of the
following condition is correct?
(1) ? ? ?
2
| | ad 0
(2) ? ? ? ? ?
2
| | ad 0anda R {0}
(3)
?
? ? ? 0,a,d R
(4) ? ? ? ?
2
| | ad 0and a R
Ans. (2)
Sol. ? ? ? ? ? ?
2
a| z| z z d 0
? ?
? ?
? ? ? ?
? ? ?
? ? ? ? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ?
d
zz z z 0
a a a
Centre = –
?
a
r =
?
?
2
d
a a
?
? ?
2
d
a a
? ? ?
2
ad
?
? ?
?
18
th
March. 2021 | Shift 1
4. ? ? ? ?
? ? ? ?
2 2 2 2
1 1 1 1
.....
3 1 5 1 7 1 (201) 1
is equal to:
(1)
101
404
(2)
101
408
(3)
99
400
(4)
25
101
Ans. (4)
Sol.
? ?
? ?
? ? ? ?
? ?
100 100
2
r 1 r 1
1 1
S
(2r 2) 2(r) (2r 1) 1
?
? ?
? ? ? ? ?
? ?
?
? ?
?
100
r 1
1 1 1
S
4 r r 1
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
1 1 1 1 1 1 1 1
S 1 ....
4 2 2 3 3 4 100 101
? ?
? ? ? ? ?
? ?
? ?
1 100 25
S
4 101 101
5. The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx
+ 1 is also an integer, is:
(1) 3
(2) 2
(3) 1
(4) 0
Ans. (2)
Sol. 3x + 4(mx + 1) = 9
x(3 + 4m) = 5
?
?
5
x
(3 4m)
(3 + 4m) = ±1, ±5
4m = – 3 ± 1, –3 ± 5
4m = – 4, –2, –8, 2
m = –1, ?
1
2
, – 2,
1
2
Two integral value of m
Page 4
18
th
March. 2021 | Shift 1
SECTION – A
1. If the functions are defined as f(x) = x and g(x) = ? 1 x, then what is the common domain of the following
functions : f + g, f – g, f/g, g/f, g – f where (f ± g) (x) = f(x) ± g(x), (f/g) (x) =
f(x)
g(x)
(1) 0 < x ? 1
(2) 0 ? x ? 1
(3) 0 ? x ? 1
(4) 0 < x < 1
Ans. (4)
Sol. f + g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f – g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f/g =
?
x
1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1)
g/f =
? 1 x
x
? ? ? ? ? ? 1 x 0 & x 0 x (0,1]
g – f = ? ? 1 x x
? ? ? ? ? ? 1 x 0 & x 0 x [0,1]
? ? x (0,1)
2. Let ?, ?, ? be the roots of the equations, x
3
+ ax
2
+ bx + c = 0, (a, b, c ? R and a, b and a, b ? 0). If the system
of the equations (in u, v, w) given by ?u + ?v + ?w = 0; ?u + ?v + ?w = 0; ?u + ?v + ?w = 0 has non-trivial
solutions, then the value of
2
a
b
is
(1) 5
(2) 1
(3) 0
(4) 3
Ans. (4)
Sol. x
3
+ ax
2
+ bx + c = 0
For non-trivial solutions,
? ? ?
? ? ? ?
? ? ?
0
?
3
+ ?
3
+ ?
3
– 3 ? ? ? = 0
? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ?? ?
? ?
? ?
2
3 0
(–a) [a
2
– 3b] = 0
a
2
= 3b ? ? ? a 0 ?
? ?
2
a
3
b
3. If the equation ? ? ? ? ? ?
2
a| z| z z d 0 represents a circle where a, d are real constants, then which of the
following condition is correct?
(1) ? ? ?
2
| | ad 0
(2) ? ? ? ? ?
2
| | ad 0anda R {0}
(3)
?
? ? ? 0,a,d R
(4) ? ? ? ?
2
| | ad 0and a R
Ans. (2)
Sol. ? ? ? ? ? ?
2
a| z| z z d 0
? ?
? ?
? ? ? ?
? ? ?
? ? ? ? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ?
d
zz z z 0
a a a
Centre = –
?
a
r =
?
?
2
d
a a
?
? ?
2
d
a a
? ? ?
2
ad
?
? ?
?
18
th
March. 2021 | Shift 1
4. ? ? ? ?
? ? ? ?
2 2 2 2
1 1 1 1
.....
3 1 5 1 7 1 (201) 1
is equal to:
(1)
101
404
(2)
101
408
(3)
99
400
(4)
25
101
Ans. (4)
Sol.
? ?
? ?
? ? ? ?
? ?
100 100
2
r 1 r 1
1 1
S
(2r 2) 2(r) (2r 1) 1
?
? ?
? ? ? ? ?
? ?
?
? ?
?
100
r 1
1 1 1
S
4 r r 1
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
1 1 1 1 1 1 1 1
S 1 ....
4 2 2 3 3 4 100 101
? ?
? ? ? ? ?
? ?
? ?
1 100 25
S
4 101 101
5. The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx
+ 1 is also an integer, is:
(1) 3
(2) 2
(3) 1
(4) 0
Ans. (2)
Sol. 3x + 4(mx + 1) = 9
x(3 + 4m) = 5
?
?
5
x
(3 4m)
(3 + 4m) = ±1, ±5
4m = – 3 ± 1, –3 ± 5
4m = – 4, –2, –8, 2
m = –1, ?
1
2
, – 2,
1
2
Two integral value of m
6. The solutions of the equation
?
? ? ? ? ?
?
2 2 2
2 2 2
1 sin x sin x sin x
cos x 1 cos x cos x 0,(0 x ),
4sin2x 4sin2x 1 4sin2x
are:
(1)
? ? 5
,
6 6
(2)
? ? 7 11
,
12 12
(3)
? ? 5 7
,
12 12
(4)
? ?
,
12 6
Ans. (2)
Sol. R
1
? R
1
+ R
2
? ?
?
2 2 2
2 2 1
cos x 1 cos x cos x 0
4sin2x 4sin2x 1 4sin2x
C
1
? C
1
– C
2
? ? ?
?
2 2
0 2 1
1 1 cos x cos x 0
0 4sin2x 1 4sin2x
? 2 + 8sin2x – 4sin2x = 0
? sin2x = ?
1
2
? x =
? ? 7 11
,
12 12
7. If
?
?
?
?
?
?
?
?
?
?
? ? ?
? ?
2
1
; | x | 1
f(x) | x |
ax b ; | x | 1
is differentiable at every point of the domain, then the values of a and b are
respectively:
(1) ?
5 3
,
2 2
(2) ?
1 3
,
2 2
(3)
1 1
,
2 2
(4) ?
1 3
,
2 2
Ans. (2)
Sol. f(x) is continuous at x = 1 ? 1 = a + b
f(x) is differentiable at x = 1 ? –1 = 2a
? ? ? ? ?
1 3
a b
2 2
Page 5
18
th
March. 2021 | Shift 1
SECTION – A
1. If the functions are defined as f(x) = x and g(x) = ? 1 x, then what is the common domain of the following
functions : f + g, f – g, f/g, g/f, g – f where (f ± g) (x) = f(x) ± g(x), (f/g) (x) =
f(x)
g(x)
(1) 0 < x ? 1
(2) 0 ? x ? 1
(3) 0 ? x ? 1
(4) 0 < x < 1
Ans. (4)
Sol. f + g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f – g = ? ? x 1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1]
f/g =
?
x
1 x
? ? ? ? ? ? x 0 & 1 x 0 x [0,1)
g/f =
? 1 x
x
? ? ? ? ? ? 1 x 0 & x 0 x (0,1]
g – f = ? ? 1 x x
? ? ? ? ? ? 1 x 0 & x 0 x [0,1]
? ? x (0,1)
2. Let ?, ?, ? be the roots of the equations, x
3
+ ax
2
+ bx + c = 0, (a, b, c ? R and a, b and a, b ? 0). If the system
of the equations (in u, v, w) given by ?u + ?v + ?w = 0; ?u + ?v + ?w = 0; ?u + ?v + ?w = 0 has non-trivial
solutions, then the value of
2
a
b
is
(1) 5
(2) 1
(3) 0
(4) 3
Ans. (4)
Sol. x
3
+ ax
2
+ bx + c = 0
For non-trivial solutions,
? ? ?
? ? ? ?
? ? ?
0
?
3
+ ?
3
+ ?
3
– 3 ? ? ? = 0
? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ?? ?
? ?
? ?
2
3 0
(–a) [a
2
– 3b] = 0
a
2
= 3b ? ? ? a 0 ?
? ?
2
a
3
b
3. If the equation ? ? ? ? ? ?
2
a| z| z z d 0 represents a circle where a, d are real constants, then which of the
following condition is correct?
(1) ? ? ?
2
| | ad 0
(2) ? ? ? ? ?
2
| | ad 0anda R {0}
(3)
?
? ? ? 0,a,d R
(4) ? ? ? ?
2
| | ad 0and a R
Ans. (2)
Sol. ? ? ? ? ? ?
2
a| z| z z d 0
? ?
? ?
? ? ? ?
? ? ?
? ? ? ? ? ?
? ?
? ?
? ? ?
? ? ? ?
? ?
d
zz z z 0
a a a
Centre = –
?
a
r =
?
?
2
d
a a
?
? ?
2
d
a a
? ? ?
2
ad
?
? ?
?
18
th
March. 2021 | Shift 1
4. ? ? ? ?
? ? ? ?
2 2 2 2
1 1 1 1
.....
3 1 5 1 7 1 (201) 1
is equal to:
(1)
101
404
(2)
101
408
(3)
99
400
(4)
25
101
Ans. (4)
Sol.
? ?
? ?
? ? ? ?
? ?
100 100
2
r 1 r 1
1 1
S
(2r 2) 2(r) (2r 1) 1
?
? ?
? ? ? ? ?
? ?
?
? ?
?
100
r 1
1 1 1
S
4 r r 1
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
1 1 1 1 1 1 1 1
S 1 ....
4 2 2 3 3 4 100 101
? ?
? ? ? ? ?
? ?
? ?
1 100 25
S
4 101 101
5. The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx
+ 1 is also an integer, is:
(1) 3
(2) 2
(3) 1
(4) 0
Ans. (2)
Sol. 3x + 4(mx + 1) = 9
x(3 + 4m) = 5
?
?
5
x
(3 4m)
(3 + 4m) = ±1, ±5
4m = – 3 ± 1, –3 ± 5
4m = – 4, –2, –8, 2
m = –1, ?
1
2
, – 2,
1
2
Two integral value of m
6. The solutions of the equation
?
? ? ? ? ?
?
2 2 2
2 2 2
1 sin x sin x sin x
cos x 1 cos x cos x 0,(0 x ),
4sin2x 4sin2x 1 4sin2x
are:
(1)
? ? 5
,
6 6
(2)
? ? 7 11
,
12 12
(3)
? ? 5 7
,
12 12
(4)
? ?
,
12 6
Ans. (2)
Sol. R
1
? R
1
+ R
2
? ?
?
2 2 2
2 2 1
cos x 1 cos x cos x 0
4sin2x 4sin2x 1 4sin2x
C
1
? C
1
– C
2
? ? ?
?
2 2
0 2 1
1 1 cos x cos x 0
0 4sin2x 1 4sin2x
? 2 + 8sin2x – 4sin2x = 0
? sin2x = ?
1
2
? x =
? ? 7 11
,
12 12
7. If
?
?
?
?
?
?
?
?
?
?
? ? ?
? ?
2
1
; | x | 1
f(x) | x |
ax b ; | x | 1
is differentiable at every point of the domain, then the values of a and b are
respectively:
(1) ?
5 3
,
2 2
(2) ?
1 3
,
2 2
(3)
1 1
,
2 2
(4) ?
1 3
,
2 2
Ans. (2)
Sol. f(x) is continuous at x = 1 ? 1 = a + b
f(x) is differentiable at x = 1 ? –1 = 2a
? ? ? ? ?
1 3
a b
2 2
18
th
March. 2021 | Shift 1
8. A vector a
?
has components 3p and 1 with respect to a rectangular Cartesian system. This system is rotated
through a certain angle about the origin in the counter clockwise sense. If with respect to new system, a
?
has
components p +1 and 10 , then a value of p is equal to:
(1) 1
(2) –1
(3)
4
5
(4) ?
5
4
Ans. (2)
Sol. ?
old new
a a
? ?
(3p)
2
+ 1 = (P+1)
2
+ 10
9p
2
– p
2
–2p – 10 = 0
8p
2
–2p – 10 = 0
4p
2
– p – 5 = 0
4p
2
– 5p + 4p – 5 = 0
(4p – 5) (p + 1) = 0
p =
5
,
4
–1
9. The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is:
(1) 26664
(2) 122664
(3) 122234
(4) 22264
Ans. (1)
Sol. 1 2 2 3
1 2 3 2
1 3 2 2
3 1 2 2
3 2 1 2
3 2 2 1
2 1 3 2
2 3 1 2
2 2 1 3
2 2 3 1
2 3 2 1
2 1 2 3
2 6 6 6 4
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Additional Information about JEE Main 2021 Mathematics March 18 Shift 1 Paper & Solutions for JEE Preparation
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