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Page 1
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 71
Date : 4
th
September 2020
Time : 02 : 00 pm - 05 : 00 pm
Subject : Maths
Q.1 Suppose the vectors x
1
, x
2
and x
3
are the solutions of the system of linear equations,
Ax=b when the vector b on the right side is equal to b
1
, b
2
and b
3
respectively. if
1 2 3 1 2 3
1 0 0 1 0 0
x 1 , x 2 , x 0 , b 0 ,b 2 and b 0
1 1 1 0 0 2
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
, then the determinant of A is
equal to
(1) 2 (2)
1
2
(3)
3
2
(4) 4
Sol. (1)
Using AX = B
1 2 3
4 5 6
7 8 9
3 3 ?
? ?
? ?
?
? ?
? ?
? ?
a a a
A a a a
a a a
1 2 3 2 3
4 5 6 5 6
7 8 9 8 9
1 2 0
0 2 2
0 2 0
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
a a a a a
a a a a a
a a a a a
3 6 9
0, 0, 2 ? ? ? a a a
8 5 2
1, 1, 0 ? ? ? ? ? a a a
1 4 7
1, 1, 1 ? ? ? ? ? ? a a a
A =
1 0 0
1 1 0
1 1 2
? ?
? ?
?
? ?
? ?
? ?
? ?
|A| = 2(1) = 2
Q.2 If a and b are real numbers such that
4
(2 ) a b ? ? ? ? ? , where
1 i 3
2
? ?
? ? then a+b
is equal to:
(1) 33 (2) 57 (3) 9 (4) 24
Sol. (3)
4
(2 ) ? ? ? a b ? ?
4
3 1
2
2
? ?
?
? ? ?
? ?
? ?
? ?
i
a b ?
Page 2
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 71
Date : 4
th
September 2020
Time : 02 : 00 pm - 05 : 00 pm
Subject : Maths
Q.1 Suppose the vectors x
1
, x
2
and x
3
are the solutions of the system of linear equations,
Ax=b when the vector b on the right side is equal to b
1
, b
2
and b
3
respectively. if
1 2 3 1 2 3
1 0 0 1 0 0
x 1 , x 2 , x 0 , b 0 ,b 2 and b 0
1 1 1 0 0 2
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
, then the determinant of A is
equal to
(1) 2 (2)
1
2
(3)
3
2
(4) 4
Sol. (1)
Using AX = B
1 2 3
4 5 6
7 8 9
3 3 ?
? ?
? ?
?
? ?
? ?
? ?
a a a
A a a a
a a a
1 2 3 2 3
4 5 6 5 6
7 8 9 8 9
1 2 0
0 2 2
0 2 0
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
a a a a a
a a a a a
a a a a a
3 6 9
0, 0, 2 ? ? ? a a a
8 5 2
1, 1, 0 ? ? ? ? ? a a a
1 4 7
1, 1, 1 ? ? ? ? ? ? a a a
A =
1 0 0
1 1 0
1 1 2
? ?
? ?
?
? ?
? ?
? ?
? ?
|A| = 2(1) = 2
Q.2 If a and b are real numbers such that
4
(2 ) a b ? ? ? ? ? , where
1 i 3
2
? ?
? ? then a+b
is equal to:
(1) 33 (2) 57 (3) 9 (4) 24
Sol. (3)
4
(2 ) ? ? ? a b ? ?
4
3 1
2
2
? ?
?
? ? ?
? ?
? ?
? ?
i
a b ?
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 72
4 4
3 3 3
9
2 2 2
? ? ? ?
?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
i i
? ?
4
/6
9
i
e
?
=
2 /3
9
i
e
?
=
1 3
9
2 2
? ?
?
?
? ?
? ?
? ?
i
=
9 9 3
2 2
?
? i
9 9 3
2 2
? ? i =
1 3
2 2
? ?
?
? ?
? ?
? ?
? ?
i
a b
=
3
2 2
? ?
b bi
a
? ?
3 9 3
9
2 2
? ? ?
b
b
a = 0 ? ?a + b = 9
Q.3 The distance of the point (1, –2, 3) from the plane x–y+z=5 measured parallel to he
line
x y z
2 3 6
? ?
?
is:
(1)
1
7
(2) 7 (3)
7
5
(4) 1
Sol. (4)
Equation of line through (1,-2,3) whose
d.r.s. are (2,3,-6)
1 2 3
2 3 6
? ? ?
? ? ?
?
x y z
?
any point on line (2 1,3 2, 6 3) ? ? ? ? ? ? ?
put in ( 5)
2 1 3 2 6 3 5
7 1
1
7
? ? ?
? ? ? ? ? ?
? ? ?
?
x y z
? ? ?
?
?
distance
2 2 2
(2 ) (3 ) (6 ) ? ? ? ? ? ?
2 2 2
4 9 36 7 1 ? ? ? ? ? ? ? ?
unit
Page 3
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 71
Date : 4
th
September 2020
Time : 02 : 00 pm - 05 : 00 pm
Subject : Maths
Q.1 Suppose the vectors x
1
, x
2
and x
3
are the solutions of the system of linear equations,
Ax=b when the vector b on the right side is equal to b
1
, b
2
and b
3
respectively. if
1 2 3 1 2 3
1 0 0 1 0 0
x 1 , x 2 , x 0 , b 0 ,b 2 and b 0
1 1 1 0 0 2
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
, then the determinant of A is
equal to
(1) 2 (2)
1
2
(3)
3
2
(4) 4
Sol. (1)
Using AX = B
1 2 3
4 5 6
7 8 9
3 3 ?
? ?
? ?
?
? ?
? ?
? ?
a a a
A a a a
a a a
1 2 3 2 3
4 5 6 5 6
7 8 9 8 9
1 2 0
0 2 2
0 2 0
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
a a a a a
a a a a a
a a a a a
3 6 9
0, 0, 2 ? ? ? a a a
8 5 2
1, 1, 0 ? ? ? ? ? a a a
1 4 7
1, 1, 1 ? ? ? ? ? ? a a a
A =
1 0 0
1 1 0
1 1 2
? ?
? ?
?
? ?
? ?
? ?
? ?
|A| = 2(1) = 2
Q.2 If a and b are real numbers such that
4
(2 ) a b ? ? ? ? ? , where
1 i 3
2
? ?
? ? then a+b
is equal to:
(1) 33 (2) 57 (3) 9 (4) 24
Sol. (3)
4
(2 ) ? ? ? a b ? ?
4
3 1
2
2
? ?
?
? ? ?
? ?
? ?
? ?
i
a b ?
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 72
4 4
3 3 3
9
2 2 2
? ? ? ?
?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
i i
? ?
4
/6
9
i
e
?
=
2 /3
9
i
e
?
=
1 3
9
2 2
? ?
?
?
? ?
? ?
? ?
i
=
9 9 3
2 2
?
? i
9 9 3
2 2
? ? i =
1 3
2 2
? ?
?
? ?
? ?
? ?
? ?
i
a b
=
3
2 2
? ?
b bi
a
? ?
3 9 3
9
2 2
? ? ?
b
b
a = 0 ? ?a + b = 9
Q.3 The distance of the point (1, –2, 3) from the plane x–y+z=5 measured parallel to he
line
x y z
2 3 6
? ?
?
is:
(1)
1
7
(2) 7 (3)
7
5
(4) 1
Sol. (4)
Equation of line through (1,-2,3) whose
d.r.s. are (2,3,-6)
1 2 3
2 3 6
? ? ?
? ? ?
?
x y z
?
any point on line (2 1,3 2, 6 3) ? ? ? ? ? ? ?
put in ( 5)
2 1 3 2 6 3 5
7 1
1
7
? ? ?
? ? ? ? ? ?
? ? ?
?
x y z
? ? ?
?
?
distance
2 2 2
(2 ) (3 ) (6 ) ? ? ? ? ? ?
2 2 2
4 9 36 7 1 ? ? ? ? ? ? ? ?
unit
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 73
Q.4 Let f : (0, ) (0, ) ? ? ? be a differentiable function such that f(1) = e and
2 2 2 2
t x
t f (x) x f (t)
lim 0
t x
?
?
?
?
. If f(x)=1,then x is equal to :
(1) e (2) 2e (3)
1
e
(4)
1
2e
Sol. (3)
(1) ? f e ...(1)
2 2 2 2
( ) ( )
lim
?
?
?
t x
t f x x f t
t x
= 0
L’ Hospital
? ? ? ?
2 2
lim 2 ( ) 2 ( ) ( )
?
?
? ?
t x
tf x x f t f t
= 0
2 2
2 ( ) 2 ( ) ( ) 0
?
? ? ? ? xf x x f x f x
? ? ? ?
2 ( ) ( ) ( ) 0
?
? ? xf x f x xf x
? ?
? ?
( ) 1
?
?
f x
f x x
ln ( ) ln ln ? ? f x x c
? ( ) ? f x cx ...(2)
if x = 1
f(1) = c(1)
f(1) = c
From eq.(1) & (2)
c = e …(3)
From eq.(3)
f(x) = ex
? ?y = ex or y = cx
? if f(x) = 1 ? ? x =
1
e
Q.5 Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:
(1) If a function f is not continuous at a, then it is not differentiable at a.
(2) If a function f is continuous at a, then it is differentiable at a.
(3) If a function f is continuous at a, then it is not differentiable at a.
(4) If a function f is not continuous at a, then it is differentiable at a.
Sol. (1)
Contrapositive of p ? q = ~q ? ~p
Page 4
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 71
Date : 4
th
September 2020
Time : 02 : 00 pm - 05 : 00 pm
Subject : Maths
Q.1 Suppose the vectors x
1
, x
2
and x
3
are the solutions of the system of linear equations,
Ax=b when the vector b on the right side is equal to b
1
, b
2
and b
3
respectively. if
1 2 3 1 2 3
1 0 0 1 0 0
x 1 , x 2 , x 0 , b 0 ,b 2 and b 0
1 1 1 0 0 2
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
, then the determinant of A is
equal to
(1) 2 (2)
1
2
(3)
3
2
(4) 4
Sol. (1)
Using AX = B
1 2 3
4 5 6
7 8 9
3 3 ?
? ?
? ?
?
? ?
? ?
? ?
a a a
A a a a
a a a
1 2 3 2 3
4 5 6 5 6
7 8 9 8 9
1 2 0
0 2 2
0 2 0
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
a a a a a
a a a a a
a a a a a
3 6 9
0, 0, 2 ? ? ? a a a
8 5 2
1, 1, 0 ? ? ? ? ? a a a
1 4 7
1, 1, 1 ? ? ? ? ? ? a a a
A =
1 0 0
1 1 0
1 1 2
? ?
? ?
?
? ?
? ?
? ?
? ?
|A| = 2(1) = 2
Q.2 If a and b are real numbers such that
4
(2 ) a b ? ? ? ? ? , where
1 i 3
2
? ?
? ? then a+b
is equal to:
(1) 33 (2) 57 (3) 9 (4) 24
Sol. (3)
4
(2 ) ? ? ? a b ? ?
4
3 1
2
2
? ?
?
? ? ?
? ?
? ?
? ?
i
a b ?
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 72
4 4
3 3 3
9
2 2 2
? ? ? ?
?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
i i
? ?
4
/6
9
i
e
?
=
2 /3
9
i
e
?
=
1 3
9
2 2
? ?
?
?
? ?
? ?
? ?
i
=
9 9 3
2 2
?
? i
9 9 3
2 2
? ? i =
1 3
2 2
? ?
?
? ?
? ?
? ?
? ?
i
a b
=
3
2 2
? ?
b bi
a
? ?
3 9 3
9
2 2
? ? ?
b
b
a = 0 ? ?a + b = 9
Q.3 The distance of the point (1, –2, 3) from the plane x–y+z=5 measured parallel to he
line
x y z
2 3 6
? ?
?
is:
(1)
1
7
(2) 7 (3)
7
5
(4) 1
Sol. (4)
Equation of line through (1,-2,3) whose
d.r.s. are (2,3,-6)
1 2 3
2 3 6
? ? ?
? ? ?
?
x y z
?
any point on line (2 1,3 2, 6 3) ? ? ? ? ? ? ?
put in ( 5)
2 1 3 2 6 3 5
7 1
1
7
? ? ?
? ? ? ? ? ?
? ? ?
?
x y z
? ? ?
?
?
distance
2 2 2
(2 ) (3 ) (6 ) ? ? ? ? ? ?
2 2 2
4 9 36 7 1 ? ? ? ? ? ? ? ?
unit
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 73
Q.4 Let f : (0, ) (0, ) ? ? ? be a differentiable function such that f(1) = e and
2 2 2 2
t x
t f (x) x f (t)
lim 0
t x
?
?
?
?
. If f(x)=1,then x is equal to :
(1) e (2) 2e (3)
1
e
(4)
1
2e
Sol. (3)
(1) ? f e ...(1)
2 2 2 2
( ) ( )
lim
?
?
?
t x
t f x x f t
t x
= 0
L’ Hospital
? ? ? ?
2 2
lim 2 ( ) 2 ( ) ( )
?
?
? ?
t x
tf x x f t f t
= 0
2 2
2 ( ) 2 ( ) ( ) 0
?
? ? ? ? xf x x f x f x
? ? ? ?
2 ( ) ( ) ( ) 0
?
? ? xf x f x xf x
? ?
? ?
( ) 1
?
?
f x
f x x
ln ( ) ln ln ? ? f x x c
? ( ) ? f x cx ...(2)
if x = 1
f(1) = c(1)
f(1) = c
From eq.(1) & (2)
c = e …(3)
From eq.(3)
f(x) = ex
? ?y = ex or y = cx
? if f(x) = 1 ? ? x =
1
e
Q.5 Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:
(1) If a function f is not continuous at a, then it is not differentiable at a.
(2) If a function f is continuous at a, then it is differentiable at a.
(3) If a function f is continuous at a, then it is not differentiable at a.
(4) If a function f is not continuous at a, then it is differentiable at a.
Sol. (1)
Contrapositive of p ? q = ~q ? ~p
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 74
Q.6 The minimum value of 2
sinx
+2
cosx
is:
(1)
1 2
2
?
(2)
1
1
2
2
?
(3)
1 2
2
? ?
(4)
1
1
2
2
? ?
Sol. (2)
Using A.M. ? G.M.
y = 2
sinx
+ 2
cosx
sinx cosx
sinx cosx
2 2
2
2
?
?
?
2
sinx
+ 2
cosx
?
sin x co s x
1
2
2 .2
?
2
sinx
+ 2
cosx
2 sinx cosx
2
2
? ?
?
? (2
Sinx
+ 2
cosx
)
minimum
=
2 2
2
2
?
=
1
1
2
2
?
?
Q.7 If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3)
has y-intercept equal to –4, then a value of k is:
(1) –2 (2)
15
(3)
14
(4) –4
Sol. (4)
PQ
4 3
m m k 1
1 k
?
?
? ? ? ?
?
k 1 7
midpoint of PQ ,
2 2
? ? ?
?
? ?
? ?
equation of perpendicular bisector
7 k 1
y (k 1) x
2 2
? ? ?
? ? ? ?
? ?
? ?
for y intercept put x = 0
2
2
7 k 1
4
2 2
k 1 15
k 4
2 2
y
? ? ?
? ? ? ?
? ?
? ?
?
? ? ? ?
Q.8 The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the
x-axis and vertices C and D lie on the parabola, y = x
2
– 1 below the x-axis, is:
(1)
2
3 3
(2)
4
3
(3)
1
3 3
(4)
4
3 3
Sol. (4)
Page 5
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 71
Date : 4
th
September 2020
Time : 02 : 00 pm - 05 : 00 pm
Subject : Maths
Q.1 Suppose the vectors x
1
, x
2
and x
3
are the solutions of the system of linear equations,
Ax=b when the vector b on the right side is equal to b
1
, b
2
and b
3
respectively. if
1 2 3 1 2 3
1 0 0 1 0 0
x 1 , x 2 , x 0 , b 0 ,b 2 and b 0
1 1 1 0 0 2
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
, then the determinant of A is
equal to
(1) 2 (2)
1
2
(3)
3
2
(4) 4
Sol. (1)
Using AX = B
1 2 3
4 5 6
7 8 9
3 3 ?
? ?
? ?
?
? ?
? ?
? ?
a a a
A a a a
a a a
1 2 3 2 3
4 5 6 5 6
7 8 9 8 9
1 2 0
0 2 2
0 2 0
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
a a a a a
a a a a a
a a a a a
3 6 9
0, 0, 2 ? ? ? a a a
8 5 2
1, 1, 0 ? ? ? ? ? a a a
1 4 7
1, 1, 1 ? ? ? ? ? ? a a a
A =
1 0 0
1 1 0
1 1 2
? ?
? ?
?
? ?
? ?
? ?
? ?
|A| = 2(1) = 2
Q.2 If a and b are real numbers such that
4
(2 ) a b ? ? ? ? ? , where
1 i 3
2
? ?
? ? then a+b
is equal to:
(1) 33 (2) 57 (3) 9 (4) 24
Sol. (3)
4
(2 ) ? ? ? a b ? ?
4
3 1
2
2
? ?
?
? ? ?
? ?
? ?
? ?
i
a b ?
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 72
4 4
3 3 3
9
2 2 2
? ? ? ?
?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
i i
? ?
4
/6
9
i
e
?
=
2 /3
9
i
e
?
=
1 3
9
2 2
? ?
?
?
? ?
? ?
? ?
i
=
9 9 3
2 2
?
? i
9 9 3
2 2
? ? i =
1 3
2 2
? ?
?
? ?
? ?
? ?
? ?
i
a b
=
3
2 2
? ?
b bi
a
? ?
3 9 3
9
2 2
? ? ?
b
b
a = 0 ? ?a + b = 9
Q.3 The distance of the point (1, –2, 3) from the plane x–y+z=5 measured parallel to he
line
x y z
2 3 6
? ?
?
is:
(1)
1
7
(2) 7 (3)
7
5
(4) 1
Sol. (4)
Equation of line through (1,-2,3) whose
d.r.s. are (2,3,-6)
1 2 3
2 3 6
? ? ?
? ? ?
?
x y z
?
any point on line (2 1,3 2, 6 3) ? ? ? ? ? ? ?
put in ( 5)
2 1 3 2 6 3 5
7 1
1
7
? ? ?
? ? ? ? ? ?
? ? ?
?
x y z
? ? ?
?
?
distance
2 2 2
(2 ) (3 ) (6 ) ? ? ? ? ? ?
2 2 2
4 9 36 7 1 ? ? ? ? ? ? ? ?
unit
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 73
Q.4 Let f : (0, ) (0, ) ? ? ? be a differentiable function such that f(1) = e and
2 2 2 2
t x
t f (x) x f (t)
lim 0
t x
?
?
?
?
. If f(x)=1,then x is equal to :
(1) e (2) 2e (3)
1
e
(4)
1
2e
Sol. (3)
(1) ? f e ...(1)
2 2 2 2
( ) ( )
lim
?
?
?
t x
t f x x f t
t x
= 0
L’ Hospital
? ? ? ?
2 2
lim 2 ( ) 2 ( ) ( )
?
?
? ?
t x
tf x x f t f t
= 0
2 2
2 ( ) 2 ( ) ( ) 0
?
? ? ? ? xf x x f x f x
? ? ? ?
2 ( ) ( ) ( ) 0
?
? ? xf x f x xf x
? ?
? ?
( ) 1
?
?
f x
f x x
ln ( ) ln ln ? ? f x x c
? ( ) ? f x cx ...(2)
if x = 1
f(1) = c(1)
f(1) = c
From eq.(1) & (2)
c = e …(3)
From eq.(3)
f(x) = ex
? ?y = ex or y = cx
? if f(x) = 1 ? ? x =
1
e
Q.5 Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:
(1) If a function f is not continuous at a, then it is not differentiable at a.
(2) If a function f is continuous at a, then it is differentiable at a.
(3) If a function f is continuous at a, then it is not differentiable at a.
(4) If a function f is not continuous at a, then it is differentiable at a.
Sol. (1)
Contrapositive of p ? q = ~q ? ~p
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 74
Q.6 The minimum value of 2
sinx
+2
cosx
is:
(1)
1 2
2
?
(2)
1
1
2
2
?
(3)
1 2
2
? ?
(4)
1
1
2
2
? ?
Sol. (2)
Using A.M. ? G.M.
y = 2
sinx
+ 2
cosx
sinx cosx
sinx cosx
2 2
2
2
?
?
?
2
sinx
+ 2
cosx
?
sin x co s x
1
2
2 .2
?
2
sinx
+ 2
cosx
2 sinx cosx
2
2
? ?
?
? (2
Sinx
+ 2
cosx
)
minimum
=
2 2
2
2
?
=
1
1
2
2
?
?
Q.7 If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3)
has y-intercept equal to –4, then a value of k is:
(1) –2 (2)
15
(3)
14
(4) –4
Sol. (4)
PQ
4 3
m m k 1
1 k
?
?
? ? ? ?
?
k 1 7
midpoint of PQ ,
2 2
? ? ?
?
? ?
? ?
equation of perpendicular bisector
7 k 1
y (k 1) x
2 2
? ? ?
? ? ? ?
? ?
? ?
for y intercept put x = 0
2
2
7 k 1
4
2 2
k 1 15
k 4
2 2
y
? ? ?
? ? ? ?
? ?
? ?
?
? ? ? ?
Q.8 The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the
x-axis and vertices C and D lie on the parabola, y = x
2
– 1 below the x-axis, is:
(1)
2
3 3
(2)
4
3
(3)
1
3 3
(4)
4
3 3
Sol. (4)
JEE Main 2020 Paper
4
th
September 2020 | (Shift-2), Maths Page | 75
A B
(–a,0) (a,0)
y=x -1
2
c(a,a -1)
2
x
y
D
(-a,a -1)
2
? ?
2
3
2
2
2
2
2
max
Area 2a a 1
A 2a 2a
dA
6a 2 0
da
d A 1
12a, at a = –
da 3
d A
–4 3 0
da
1
So Area is maximum at a –
3
2 2 2 6
A sq.units
3 3 3 3 3
? ?
? ?
? ? ?
?
? ?
?
? ? ?
? ? ?
Q.9 The integral
3
3 2 2 2
6
x 3x 2 x 3x 3 x 6x dx
?
?
?
?
/
/
tan .sin ( sec .sin tan .sin ) is equal to:
(1)
9
2
(2)
1
18
?
(3)
1
9
?
(4)
7
18
Sol. (2)
I =
/3
3 2 4
/6
2.tan sec sin 3
?
x x x
?
?
+ 3tan
4
x sin
2
3x. 2sin3xcos3x dx
=
1
2
/3
3 2 4
/6
4 tan sec sin 3
?
x x x
?
?
+ 3.4tan an
4
xsin
3
3xcos3xdx
=
? ?
/3
4 4
/6
1
tan sin 3
2
?
d
x x dx
dx
?
?
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