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CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 8 - MCQExams.com
CBSE
Class 12 Commerce Applied Mathematics
Applications Of Integrals
Quiz 8
The area of the region
A
=
[
(
x
,
y
)
:
0
≤
y
≤
x
|
x
|
+
1
and
−
1
≤
x
≤
1
]
in sq . units is :
Report Question
0%
2
3
0%
1
3
0%
2
0%
4
3
Explanation
The graph is as follows:
∫
0
−
1
(
−
x
2
+
1
)
d
x
+
∫
1
0
(
x
2
+
1
)
d
x
=
2
If the area enclosed between the curves
y
=
k
x
2
and
x
=
k
y
2
,
(
k
>
0
)
, is
1
square unit. Then
k
is?
Report Question
0%
1
√
3
0%
2
√
3
0%
√
3
2
0%
√
3
Explanation
Area bounded by
y
2
=
4
a
x
&
x
2
=
4
b
y
, a, b
≠
0
is
|
16
a
b
3
|
by using formula:
4
a
=
1
k
=
4
b
,
k
>
0
Area
=
|
16
⋅
1
4
k
⋅
1
4
k
3
|
=
1
⇒
k
2
=
1
3
⇒
k
=
1
√
3
.
The area ehclosed by the curves y = f(x) and y =g(x), where f9x) = max
x
,
x
2
and g(x) = min
x
,
x
2
opver the interval [0,1] is
Report Question
0%
1
6
0%
1
3
0%
1
2
0%
1
The area of the region bounded by the parabolas
y
2
=
a
n
d
x
2
=
y
,
i
s
Report Question
0%
1
3
q.units
0%
8
3
q.units
0%
16
3
q.units
0%
4
3
q.units
The area of the region
{
(
x
,
y
)
:
x
2
+
y
2
≤
1
≤
x
+
y
}
is
Report Question
0%
π
2
5
unit
2
0%
π
2
2
unit
2
0%
π
2
3
unit
2
0%
(
π
4
−
1
2
)
unit
2
The area of the region bounded by the parabola y =
x
2
3x with y 0 is
Report Question
0%
3
0%
3
2
0%
9
2
0%
9
The area of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse,
x
2
9
+
y
2
5
=
1
is
Report Question
0%
27
4
0%
18
0%
27
2
0%
27
Explanation
Given equation of ellipse is
x
2
9
+
y
2
5
=
1
......
(
1
)
∴
a
2
=
9
,
b
2
=
5
⇒
a
=
3
,
b
=
√
5
Now,
e
=
√
1
−
b
2
a
2
=
√
1
−
5
9
=
2
3
Foci
=
(
±
a
e
,
0
)
=
(
±
2
,
0
)
and
b
2
a
=
5
3
∴
Extremities of one of latus rectum are
(
2
,
5
3
)
and
(
2
,
−
5
3
)
∴
Equation of tangent at
(
2
,
5
3
)
is
x
(
2
)
9
+
y
(
5
3
)
5
=
1
or
2
x
+
3
y
=
9
.......
(
2
)
Eqn
(
2
)
intersects
X
and
Y
axes at
(
9
2
,
0
)
and
(
0
,
3
)
respectively.
∴
Area of quadrilateral
4
×
Area of
△
P
O
Q
=
4
×
1
2
×
9
2
×
3
=
27
sq.units.
The area (in sq. units) of the region
{
x
,
y
)
:
y
2
≥
2
x
and
x
2
+
y
2
≤
4
x
,
x
≥
0
,
y
≥
0
}
is :
Report Question
0%
π
−
4
√
2
3
0%
π
2
−
2
√
2
3
0%
π
−
4
3
0%
π
−
8
3
The area bounded by the curve
y
=
e
x
and the lines y = |x - 1|, x = 2 is given by :
Report Question
0%
e
2
+
1
0%
4
e
2
−
1
0%
e
2
−
2
0%
e
−
2
Explanation
Given,
y
=
e
x
,
y
=
|
x
−
1
|
,
x
=
2
→
e
x
=
|
x
−
1
|
→
e
x
−
(
1
−
x
)
=
0
intervals:
|
x
−
1
|
=
0
⇒
x
=
1
x
=
2
Area is given by,
A
=
∫
1
0
(
e
x
−
(
1
−
x
)
)
d
x
+
∫
2
1
(
e
x
−
(
1
−
x
)
)
d
x
=
∫
1
0
(
e
x
+
x
−
1
)
d
x
+
∫
2
1
(
e
x
+
x
−
1
)
d
x
=
[
e
x
+
x
2
2
−
x
]
1
0
+
[
e
x
+
x
2
2
−
x
]
2
1
=
e
+
1
2
−
1
+
(
e
2
−
2
+
2
)
−
(
e
−
1
2
+
1
)
=
e
2
−
2
Area bounded by the curve
y
2
(
2
a
−
x
)
=
x
3
and the line
x
=
2
a
, is
Report Question
0%
3
π
a
2
0%
3
π
a
2
2
0%
3
π
a
2
4
0%
π
a
2
4
Explanation
Given,
y
2
(
2
a
−
x
)
=
x
3
y
=
√
x
3
2
a
−
x
let,
x
=
2
a
sin
2
θ
d
x
=
4
a
sin
θ
cos
θ
d
θ
⇒
A
=
∫
2
a
0
√
x
3
2
a
−
x
d
x
A
=
∫
π
2
0
√
8
a
3
sin
6
θ
2
a
cos
2
θ
4
a
sin
θ
cos
θ
d
θ
=
8
a
2
∫
π
2
0
sin
4
θ
d
θ
=
8
a
2
∫
π
2
0
sin
2
θ
(
1
−
cos
2
θ
)
=
8
a
2
∫
π
2
0
(
1
−
cos
2
θ
2
)
−
1
4
sin
2
2
θ
d
θ
=
8
a
2
[
π
4
−
0
−
1
4
(
π
4
−
0
)
]
=
8
a
2
[
π
4
−
π
16
]
=
3
2
π
a
2
Area bounded by the curve
y
=
s
i
n
−
1
x
,
y
−
a
x
i
s
and
y
=
c
o
s
−
1
x
is equal to
Report Question
0%
(
2
+
√
2
)
0%
(
2
−
√
2
)
0%
(
1
+
√
2
)
0%
(
√
2
−
1
)
Explanation
Given,
y
=
sin
−
1
x
,
y
=
cos
−
1
x
,
y
−
a
x
i
s
⇒
sin
y
=
x
,
cos
y
=
x
⇒
sin
y
=
cos
y
⇒
tan
y
=
1
∴
y
=
tan
−
1
1
=
π
4
A
r
e
a
=
∫
π
4
0
cos
(
y
)
−
sin
(
y
)
d
y
=
∫
π
4
0
cos
(
y
)
d
y
−
∫
π
4
0
sin
(
y
)
d
y
=
1
√
2
−
(
−
1
√
2
+
1
)
=
√
2
−
1
The area bounded by the circles
x
2
+
y
2
=
r
2
,
r
=
1
,
2
and the rays given by
2
x
2
−
3
x
y
−
2
y
2
=
0
,
y
>
0
is
Report Question
0%
π
4
s
q
.
u
n
i
t
s
0%
π
2
s
q
.
u
n
i
t
s
0%
3
π
4
s
q
.
u
n
i
t
s
0%
π
s
q
.
u
n
i
t
s
Area of the region bounded by
y
=
sin
−
1
|
sin
x
|
and
y
=
−
cos
−
1
|
cos
x
|
in the interval
[
0
,
2
π
]
is equal to
Report Question
0%
π
2
2
0%
π
2
0%
π
2
4
0%
2
π
2
The area bounding by
y
=
2
−
|
2
−
x
|
and y =
3
|
x
|
is :
Report Question
0%
4
+
3
ℓ
n
3
2
0%
4
−
3
ℓ
n
3
2
0%
3
2
+
ℓ
n
3
0%
1
2
+
ℓ
n
3
The area enclosed between the curves
y
=
a
x
2
and
x
=
a
y
2
(
a
>
0
)
is
1
sq.unit, then the value of
a
is
Report Question
0%
1
/
√
3
0%
1
/
2
0%
1
0%
1
/
3
The area inside the parabola
5
x
2
−
y
=
0
but outside the parabola
2
x
2
−
y
+
9
=
0
, is
Report Question
0%
12
√
3
0%
6
√
3
0%
8
√
3
0%
4
√
3
Explanation
From given, we have,
5
x
2
−
y
=
0
2
x
2
−
y
+
9
=
0
we have boundaries,
5
x
2
=
2
x
2
+
9
3
x
2
=
9
x
=
±
√
3
A
=
∫
√
3
−
√
3
5
x
2
−
(
2
x
2
+
9
)
∫
√
3
−
√
3
5
x
2
−
(
2
x
2
+
9
)
=
∫
√
3
−
√
3
3
x
2
d
x
−
∫
√
3
−
√
3
9
d
x
=
6
√
3
−
18
√
3
=
−
12
√
3
∴
A
r
e
a
=
|
−
12
√
3
|
=
12
√
3
The area of the figure bounded by the curves
y
=
ln
n
x
&
(
ln
n
x
)
2
is
Report Question
0%
e
+
1
0%
e
−
1
0%
3
−
e
0%
1
The area of the region bounded by x = 0, y = 0, x = 2, y = 2,
y
≤
e
x
and
y
≥
ℓ
n
x, is
Report Question
0%
6 - 4
ℓ
n
2
0%
4
ℓ
n
2 - 2
0%
2
ℓ
n
2 - 4
0%
6 - 2
ℓ
n
2
The area is bounded by
x
+
x
1
,
y
=
y
1
and
y
=
−
(
x
+
1
)
2
. Where
x
1
,
y
1
are the values of
x
,
y
satisfying the equation
s
i
n
−
1
x
+
s
i
n
−
1
y
=
−
π
will be (nearer to origin)
Report Question
0%
1
/
3
0%
3
/
2
0%
1
0%
2
/
3
Explanation
( Wrong question,
x
=
x
, instead of
x
+
x
1
)
Given that:
x
=
x
1
y
=
y
1
y
=
−
(
x
+
1
)
2
&
x
1
,
y
1
satisfy
sin
−
1
x
+
sin
−
1
y
=
z
We know:
−
z
2
≤
sin
−
1
x
≤
z
2
−
z
2
≤
sin
−
1
y
≤
z
2
∴
−
z
≤
sin
−
1
x
+
sin
−
1
y
≤
z
So, for
sin
−
1
x
+
sin
−
1
y
=
−
z
⇒
sin
−
1
x
=
−
π
2
&
sin
−
1
y
=
−
z
/
2
x
=
−
1
and
y
=
−
1
area required
=
1
−
|
∫
0
1
−
(
x
+
1
)
2
d
x
∣
=
1
−
|
∫
0
1
(
x
2
+
1
+
2
x
)
d
x
|
=
1
−
|
−
(
x
3
+
x
+
x
2
)
|
0
−
1
3
=
1
−
|
(
−
1
3
−
1
+
1
)
|
=
1
−
1
3
=
2
/
3
s
q
. Units
The area bounded by curves
y
=
|
x
|
−
1
and
y
=
−
|
x
|
+
1
is
Report Question
0%
1
0%
2
0%
3
0%
4
Explanation
First we form a table for different values of
x
and
y
for both equations as shown .
Here length of
A
C
=
2
units and distance(height) between vertices
D
and line
A
C
=
1
units and between vertices
B
and line
A
C
=
1
unit
∵
Area of triangle
=
1
2
×
b
×
h
∴
Area of
A
B
C
D
=
2
×
A
r
e
a
o
f
△
A
B
C
=
2
×
1
2
×
2
×
1
=
2
sq.units.
The area of the triangle formed by the lines joining the vertex of the parabola
x
2
=
12
y
to the ends of its latus rectum is-
Report Question
0%
16
sq. units
0%
12
sq. units
0%
18
sq. units
0%
24
sq. units
Explanation
Step 1: Make an appropriate diagram
Above figure shows a parabola of the form
x
2
=
4
a
y
Having vertex at
V
(
0
,
0
)
Focus at
S
(
0
,
a
)
Latus rectum is a line
A
B
passing through focus
S
and perpendicular to the axis of the parabola
Thus coordinates of end points of latus rectum is
A
(
−
2
a
,
a
)
and
B
(
2
a
,
a
)
Step 2: Find coordinates of end points of the triangle for given parabola
Given parabola is
x
2
=
12
y
⇒
4
a
=
12
∴
a
=
3
Therefore the coordinates of triangle are,
A
(
−
6
,
3
)
,
V
(
0
,
0
)
,
B
(
6
,
3
)
Area of
△
V
A
B
=
1
2
|
x
1
(
y
2
−
y
3
)
+
x
2
(
y
3
−
y
1
)
+
x
3
(
y
1
−
y
2
)
|
=
1
2
|
0
(
3
−
3
)
+
(
−
6
)
(
3
−
0
)
+
6
(
0
−
3
)
|
=
1
2
|
−
36
|
=
18
s
q
.
u
n
i
t
s
\
Hence, Option 'C' is correct
The area bounded by the curve
y
=
x
X
−
axis and the lines
x
=
−
1
and
x
=
1
is
Report Question
0%
0
0%
1
3
0%
2
3
0%
4
3
The area enclosed between the curves
y
=
l
o
g
e
(
x
+
e
)
,
x
=
l
o
g
e
(
1
y
)
and the x-axis is?
Report Question
0%
2
e
0%
e
0%
4
e
0%
None of these
The area (in sq. units) bounded by the parabola
y
=
x
2
−
1
, the tangent at the point (2,3) to it and the y-axis is:
Report Question
0%
14
3
0%
56
3
0%
8
3
0%
32
3
Explanation
Equation of tangent at
(
2
,
3
)
on
y
=
x
2
−
1
, is
y
=
(
4
x
−
5
)
.
.
.
.
.
.
.
(
i
)
∴
Required shaded area
=
a
r
(
△
A
B
C
)
−
3
∫
−
1
√
y
+
1
d
y
=
1
2
.
(
8
)
.
(
2
)
−
2
3
(
(
y
+
1
)
3
/
2
)
3
−
1
=
8
−
16
3
=
8
3
The area bounded by the curve
y
≤
x
2
+
3
x
,
0
≤
y
≤
4
,
0
≤
x
≤
3
, is
Report Question
0%
59
6
0%
57
4
0%
59
3
0%
57
6
Explanation
y
=
x
2
+
3
x
=
4
⇒
x
2
+
3
x
−
4
=
0
(
x
+
4
)
(
x
−
1
)
=
0
x
=
1
area
=
∫
1
0
(
x
2
+
3
x
)
d
x
+
2
(
4
)
=
1
3
+
3
2
+
8
=
11
6
+
8
=
59
6
The area of the region bounded by the curve
y
=
ϕ
(
x
)
,
y
=
0
and
x
=
10
is
Report Question
0%
81
4
0%
79
4
0%
73
4
0%
19
Let
f
(
x
,
y
)
=
{
(
x
,
y
)
:
y
2
≤
4
x
,
0
≤
x
≤
λ
}
and
s
(
λ
)
is area such that
S
(
λ
)
S
(
4
)
=
2
5
. Find the value of
λ
.
Report Question
0%
4
(
4
25
)
1
/
3
0%
4
(
2
25
)
1
/
3
0%
2
(
4
25
)
1
/
3
0%
2
(
2
25
)
1
/
3
Explanation
y
2
=
4
x
S
(
I
)
=
2
∫
λ
0
2
√
x
d
x
=
4
x
3
/
2
3
/
2
]
λ
0
=
8
3
λ
3
/
2
S
(
λ
)
S
(
4
)
=
2
5
⇒
λ
3
/
2
4
3
/
2
=
2
5
λ
=
4
(
2
5
)
2
/
3
=
4
(
4
25
)
1
/
3
.
If the area (in sq. units) of the region
{
(
x
,
y
)
:
y
2
≤
4
x
,
x
+
y
≤
1
,
x
≥
0
,
y
≥
0
}
is
a
√
2
+
b
, then
a
−
b
is equal to?
Report Question
0%
8
3
0%
10
3
0%
6
0%
−
2
3
Explanation
{
(
x
,
y
)
:
y
2
≤
4
x
,
x
+
y
≤
1
,
x
≥
0
,
y
≥
0
}
A
∫
3
−
2
√
2
0
2
√
x
d
x
+
1
2
(
1
−
(
3
−
2
√
2
)
)
(
1
−
(
3
−
2
√
2
)
)
=
2
[
x
3
/
2
]
3
−
2
√
2
0
3
/
2
+
1
2
(
2
√
2
−
2
)
(
2
√
2
−
2
)
=
8
√
2
3
+
(
−
10
3
)
a
=
8
3
,
b
=
−
10
3
a
−
b
=
6
.
Region formed by
|
x
−
y
|
≤
2
and
|
x
+
y
|
≤
2
is
Report Question
0%
Rhombus of side is
2
0%
Square of area is
6
0%
Rhombus of area is
8
√
2
0%
Square of side is
2
√
2
Explanation
ABCD is a square
A
r
e
a
=
4
×
1
2
×
2
×
2
=
8
side
=
2
√
2
The region represented by
|
x
−
y
|
≤
2
and
|
x
+
y
|
≤
2
is bounded by a:
Report Question
0%
Square of side length
2
√
2
units
0%
Rhombus of side length
2
units
0%
Square of area
16
s
q
units
0%
Rhombus of area
8
√
2
s
q
.
units
Explanation
|
x
−
y
|
≤
2
and
|
x
+
y
|
≤
2
⇒
x
−
y
≤
2
or
x
−
y
≥
2
and
x
+
y
≤
2
and
x
+
y
≥
−
2
After plotting them on the graph the common region is
Square whose side is
2
√
2
Area of the region bounded by the curve
y
=
cos
x
between
x
=
0
and
x
=
π
is
Report Question
0%
1
sq. units
0%
4
sq. units
0%
2
sq. units
0%
3
sq. units
Explanation
Required area enclosed by the curve
y
=
cos
x
,
x
=
0
and
x
=
π
A
=
∫
π
/
2
0
cos
x
d
x
+
|
∫
π
π
/
2
cos
x
d
x
|
=
[
s
i
n
x
]
π
/
2
0
+
[
s
i
n
x
]
π
π
/
2
=
[
sin
π
2
−
sin
0
]
+
|
sin
π
2
−
sin
π
|
=
1
+
1
=
2
s
q
u
n
i
t
s
.
Let
S
(
α
)
=
{
(
x
,
y
)
:
y
2
≤
x
,
0
≤
x
≤
α
}
and
A
(
α
)
is area of the region
S
(
α
)
. If for a
λ
,
0
<
λ
<
4
,
A
(
λ
)
:
A
(
4
)
=
2
:
5
, then
λ
equals
Report Question
0%
2
(
4
25
)
1
3
0%
4
(
4
25
)
1
3
0%
2
(
2
5
)
1
3
0%
4
(
2
5
)
1
3
Explanation
S
(
α
)
=
{
(
x
,
y
)
:
y
2
≤
x
,
0
≤
x
≤
α
}
A
(
α
)
=
2
∫
α
0
√
x
d
x
=
2
α
3
2
A
(
4
)
=
2
×
4
3
/
2
=
16
A
(
λ
)
=
2
×
λ
3
/
2
A
(
λ
)
A
(
4
)
=
2
5
⇒
λ
=
4.
(
4
25
)
1
/
3
The area (in sq. units) of the region
A
=
{
(
x
,
y
)
:
x
2
≤
y
≤
x
+
2
}
is?
Report Question
0%
10
3
0%
9
2
0%
31
6
0%
13
6
Explanation
x
2
≤
y
≤
x
+
2
x
2
=
y
;
y
=
x
+
2
x
2
=
x
+
2
x
2
−
x
−
2
=
0
(
x
−
2
)
(
x
−
1
)
=
0
x
=
2
,
−
1
Area
=
∫
2
−
1
(
x
+
2
)
−
x
2
d
x
=
[
x
2
2
+
2
x
−
x
3
3
]
2
−
1
=
2
+
4
−
8
3
−
1
2
+
2
−
1
3
=
9
2
.
If the area enclosed by the curves
y
2
=
4
λ
x
and
y
=
λ
x
is
1
9
square units then value of
λ
is
Report Question
0%
24
0%
37
0%
48
0%
38
Explanation
y
2
=
4
λ
x
;
y
=
λ
x
If
λ
>
0
then
Hence
∫
4
/
λ
0
(
2
√
λ
√
x
−
λ
x
)
d
x
=
1
9
(
2
√
λ
x
3
/
2
3
/
2
−
λ
x
2
2
)
4
λ
=
1
9
⇒
4
3
√
λ
8
λ
3
/
2
−
λ
8
λ
2
=
1
9
⇒
32
3
λ
=
1
9
⇒
λ
=
24
If the area (in sq. units) bounded by the parabola
y
2
=
4
λ
x
and the line
y
=
λ
x
,
λ
>
0
, is
1
9
, then
λ
is equal to
Report Question
0%
24
0%
48
0%
4
√
3
0%
2
√
6
Explanation
A
r
e
a
=
1
9
=
∫
4
λ
0
(
√
4
λ
x
−
λ
x
)
d
x
⇒
λ
=
24
.
The area (in sq. units) of the region bounded by the curves
y
=
2
x
and
y
=
|
x
+
1
|
, in the first quadrant is:
Report Question
0%
3
2
−
1
log
e
2
0%
1
2
0%
log
e
2
+
3
2
0%
3
2
Explanation
Required Area
∫
1
0
(
(
x
+
1
)
−
2
x
)
d
x
=
(
x
2
2
+
x
−
2
x
ln
2
)
1
0
=
(
1
2
+
1
−
2
ln
2
)
−
(
0
+
0
−
1
ln
2
)
=
3
2
−
1
ln
2
Area of the region bounded by
y
2
≤
4
x
,
x
+
y
≤
1
,
x
≥
0
,
y
≥
0
is
a
√
2
+
b
, then value of
a
−
b
is?
Report Question
0%
4
0%
6
0%
8
0%
12
Explanation
Let P be the point common to
x
+
y
=
1
&
y
2
=
4
x
So
y
2
=
4
(
1
−
y
⇒
y
2
+
4
y
−
4
=
0
⇒
y
=
−
4
±
√
16
+
16
2
⇒=
−
2
+
2
√
2
Hence
P
(
3
,
−
2
√
2
,
−
2
+
2
√
2
)
Hence started area
=
Area of region (OPN)
+
Area of (
Δ
OPQ)
=
∫
3
−
2
√
2
0
2
√
x
d
x
+
1
2
[
−
1
−
(
3
−
2
√
2
)
]
2
=
2
3
⋅
2
(
√
2
−
1
)
(
3
−
2
√
2
)
+
1
2
[
2
(
√
2
−
1
)
]
2
=
4
3
{
−
7
+
5
√
2
}
+
2
(
3
−
2
√
2
)
=
(
20
3
−
4
)
√
2
+
6
−
28
3
=
8
3
√
2
−
10
3
Hence
a
=
8
3
,
b
=
−
10
3
So
a
−
b
=
6
.
The area bounded by the line
y
=
x
, x-axis and ordinates
x
=
−
1
and
x
=
2
is?
Report Question
0%
3
2
0%
5
2
0%
2
0%
3
The area of the region
{
(
x
,
y
)
:
x
y
≤
8
,
1
≤
y
≤
x
2
}
is
Report Question
0%
16
log
6
2
−
6
0%
8
log
6
2
−
7
3
0%
16
log
6
2
−
14
3
0%
8
log
6
2
−
14
3
Explanation
x
y
≤
8
1
≤
y
≤
x
2
x
2
.
x
=
8
x
=
2
Require Area
=
∫
4
1
(
8
y
−
√
y
)
d
y
=
[
8
l
n
y
−
y
3
/
2
3
/
2
]
4
1
=
8
l
n
4
−
2
3
.
8
−
0
+
2
3
=
16
l
n
2
−
14
3
.
The area bounded by curve
y
=
sin
2
x
(
x
=
0
t
o
x
=
π
)
and X-axis is ______
Report Question
0%
4
0%
2
0%
1
0%
3
2
Explanation
As the crest and trugh are same we can write
A
=
2
∫
π
/
2
0
sin
2
x
d
x
=
2
[
−
cos
2
x
2
]
π
/
2
0
=
(
1
+
1
)
A
=
2
The area (in sq. units) of the region
{
(
x
,
y
)
∈
R
2
:
x
2
≤
y
≤
3
−
2
x
}
, is:
Report Question
0%
29
3
0%
34
3
0%
31
3
0%
32
3
Explanation
From given data in question,
Point of intersection of given two curve
y
=
x
2
and
y
=
−
2
x
+
3
x
2
=
−
2
x
+
3
x
=
−
3
,
1
Area of shaded region
=
∫
1
−
3
(
(
−
2
x
+
3
)
−
x
2
)
d
x
=
[
−
x
2
+
3
x
−
x
3
3
]
1
−
3
=
−
(
1
2
−
3
2
)
+
3
×
(
1
−
(
−
3
)
)
−
1
3
+
3
3
3
=
12
+
8
−
28
3
=
32
3
.
The area bounded by
y
=
sin
2
x
,
x
=
π
2
and
x
=
π
is
Report Question
0%
π
2
0%
p
i
4
0%
π
8
0%
π
16
0%
2
π
Explanation
Required area
=
∫
π
π
2
sin
2
x
d
x
=
∫
π
π
2
[
1
−
cos
2
x
2
]
d
x
=
1
2
∫
π
π
2
(
1
−
cos
2
x
)
d
x
=
1
2
[
x
−
sin
2
x
2
]
π
π
2
=
1
2
[
(
π
−
0
)
−
(
π
2
−
0
)
]
=
1
2
[
π
2
]
=
π
4
The area of the region bounded by the curve
y
=
2
x
−
x
2
and the line
y
=
x
is ________ square units.
Report Question
0%
1
6
0%
1
2
0%
1
3
0%
7
6
Explanation
We note that the region bounded by these curves is in the region
x
∈
[
0
,
1
]
. In this region, the curve
y
=
x
lies below the curve
y
=
2
x
−
x
2
So, to calculate the area of said region, we evaluatie the following integral:
∫
1
0
2
x
−
x
2
−
x
d
x
=
∫
1
0
x
−
x
2
d
x
=
[
x
2
2
−
x
3
3
]
x
=
1
x
=
0
=
1
2
−
1
3
=
1
6
.
Given
f
(
x
)
=
{
x
,
0
≤
x
<
1
2
1
2
,
x
=
1
2
1
−
x
,
1
2
<
x
≤
1
and
g
(
x
)
=
(
x
−
1
2
)
2
,
x
ϵ
R
, Then the area (in sq.units) of the region bounded by the curves
y
=
f
(
x
)
and
y
=
g
(
x
)
between the lines
2
x
=
1
and
2
x
=
√
3
, is:
Report Question
0%
1
3
+
√
3
4
0%
1
2
−
√
3
4
0%
1
2
+
√
3
4
0%
√
3
4
−
1
3
Explanation
{
x
,
0
<
n
<
1
2
1
2
,
n
=
1
2
1
−
n
,
1
2
<
n
<
1
and
g
(
x
)
=
(
x
−
1
2
)
2
graph of the current is as
refer above image.
Required image =are of trapezium
A
B
C
D
−
∫
√
3
2
1
2
(
x
−
1
2
)
2
d
x
=
1
2
(
√
3
−
1
2
)
(
1
2
+
1
−
√
3
2
)
−
1
3
(
(
x
−
1
2
)
3
)
√
3
2
−
1
2
=
√
3
4
−
1
3
If the curve
y
=
a
x
1
2
+
b
x
passes through the point
(
1
,
2
)
and lies above the
x
−
a
x
i
s
for
0
≤
x
≤
9
and the area enclosed by the curve, the
x
−
a
x
i
s
and the line
x
=
4
is
8
sq.units. Then
Report Question
0%
a
=
1
0%
b
=
1
0%
a
=
3
0%
b
=
−
1
Explanation
Since the curve
y
=
a
x
1
2
+
b
x
passes through the point
(
1
,
2
)
∴
2
=
a
+
b
(
1
)
By observation the curve also passes through
(
0
,
0
)
.
Therefore, the area enclosed by the curve,
x
−
a
x
i
s
and
x
=
4
is given by
A
=
∫
4
0
(
a
x
1
2
+
b
x
)
d
x
=
8
⇒
2
a
3
×
8
+
b
12
×
16
=
8
⇒
2
a
3
+
b
=
1
(
2
)
Solving
(
1
)
,
(
2
)
we get
a
=
3
,
b
=
−
1
.
The area of the region, enclosed by the circle
x
2
+
y
2
=
2
which is not common to the region bounded by the parabola
y
2
=
x
and the straight line
y
=
x
, is:
Report Question
0%
1
3
(
15
π
−
1
)
0%
1
6
(
24
π
−
1
)
0%
1
6
(
12
π
−
1
)
0%
1
3
(
6
π
−
1
)
Explanation
Step-1: Find the radius of the circle
The given equation of the circle is
x
2
+
y
2
=
2
Comparing with the general form of equation of a circle,
x
2
+
y
2
=
r
2
,
We get,
r
2
=
2
r
=
√
2
Step-2: Find the area of required region
Here, Required Area= Area of circle-Area of shaded region
=
π
(
√
2
)
2
−
∫
1
0
(
√
x
−
x
)
d
x
=
2
π
−
(
2
3
−
1
2
)
=
2
π
−
1
6
=
12
π
−
1
6
Hence, Area of the required region is
12
π
−
1
6
The area (in sq. units) of the region
{
(
x
,
y
)
∈
R
2
|
4
x
2
≤
y
≤
8
x
+
12
}
is :
Report Question
0%
128
3
0%
127
3
0%
125
3
0%
124
3
Explanation
Shaded are is the area between curve
Y
=
4
x
2
__(I)
and
Y
=
8
x
+
12
__(II)
on solving equation (I) & (II) (to find intersection point of the curves).
4
x
2
−
8
x
−
12
=
0
4
(
x
+
1
)
(
x
−
3
)
=
0
x
=
−
1
or
x
=
3
∴
shaded Area
=
∫
3
−
1
[
(
8
x
+
12
)
−
(
4
x
2
)
]
d
x
=
∫
3
−
1
8
x
d
x
+
∫
3
−
1
12
d
x
−
4
∫
3
−
1
x
2
d
x
=
[
4
x
2
+
12
x
−
4
3
x
3
]
3
−
1
=
4
⋅
9
+
12
⋅
3
−
4
3
⋅
27
−
4
+
12
−
4
3
=
128
3
The area bounded by the curve
y
=
x
2
+
2
x
+
1
and tangent at
(
1
,
4
)
and y -axis and
Report Question
0%
2
3
sq units
0%
1
3
sq units
0%
2 sq units
0%
None of these
Area of the region bounded by the curve
y
=
e
x
,
y
=
e
−
x
and the straight line x= 1 given by
Report Question
0%
e
−
e
−
1
+
2
0%
e
−
e
−
1
−
2
0%
e
+
e
−
1
−
2
0%
None of these
The area bounded by the curve
y
=
(
x
)
the x-axis and the ordinate
x
=
1
and
x
=
b
is
(
b
−
1
)
c
o
s
(
3
b
+
4
)
, then
f
(
x
)
is given by
Report Question
0%
(
x
−
1
)
s
i
n
(
3
x
+
4
)
0%
(
x
−
1
)
s
i
n
(
3
x
−
4
)
0%
−
3
(
x
−
1
)
s
i
n
(
3
x
+
4
)
+
c
o
s
(
3
x
+
4
)
0%
None of these
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers
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