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CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 13 - MCQExams.com
CBSE
Class 12 Commerce Maths
Application Of Derivatives
Quiz 13
Let
∅
(
x
)
=
(
f
(
x
)
)
3
−
3
(
f
(
x
)
)
2
+
4
f
(
x
)
+
5
x
+
3
s
i
n
x
+
4
c
o
s
x
∀
x
∈
R,where
f
(
x
)
is a differentiable function
∀
x
∈
R
, then
Report Question
0%
∅
is increasing whenever f is increasing
0%
∅
is increasing whenever f is decreasing
0%
∅
is decreasing whenever f is decreasing
0%
∅
is decreasing if f'(x)=-11
Let N be the set of positive integers. For all
n
∈
N
, let
f
n
=
(
n
+
1
)
1
/
3
−
n
1
/
3
and
A
=
{
n
∈
N
:
f
n
+
1
<
1
3
(
n
+
1
)
2
/
3
<
f
n
}
Then
Report Question
0%
A = N
0%
A is a finite set
0%
the complement of A in N is nonempty, but finite
0%
A and its complement in N are both infinite
A Stationary point of
f
(
x
)
=
√
16
−
x
2
is
Report Question
0%
(4,0)
0%
(-4,0)
0%
(0,4)
0%
(-4,4)
If
f
:
R
→
R
is the function defined by
f
(
x
)
=
e
x
2
−
e
−
x
2
e
x
2
+
e
−
x
2
,
then
Report Question
0%
f
(
x
)
is an increasing function
0%
f
(
x
)
is an decreasing function
0%
f
(
x
)
is onto (surjective)
0%
Into function
If
∫
2
x
2
(
x
2
+
1
)
2
d
x
=
f
(
x
)
+
c
where f (0) = 0, then
Report Question
0%
f(x) is an increasing function
0%
x = 0 is point of extremum
0%
f(x) is concave up for
x
∈
(
−
∞
,
−
1
)
∪
(
0
,
1
)
0%
the curve f(x) has 3 points of inflection
A
s
a
t
i
o
n
a
r
y
p
o
int
o
f
f
(
x
)
=
√
16
−
x
2
i
s
Report Question
0%
(
4
,
0
)
0%
(
−
4
,
0
)
0%
(
0
,
4
)
0%
(
−
4
,
4
)
The slope of the tangent to the curve at a point (x, y) on it is proportional to (x-2). If the slope of the tangent to the curve at (10,-9) on it -The equation of the curve is
Report Question
0%
y
=
k
(
x
−
2
)
2
0%
y
=
−
3
16
(
x
−
2
)
2
+
1
0%
y
=
−
3
16
(
x
−
2
)
2
+
3
0%
y
=
k
(
x
+
2
)
2
Explanation
Step -1: Find the value of slope
.
The slope of the tangent to the curve at a point
(
x
,
y
)
on it is proportional to
(
x
−
2
)
.
=
d
y
d
x
=
m
(
x
−
2
)
=
d
y
d
x
=
k
(
x
−
2
)
[
where
k
is a proportionality constant
]
Slope of the tangent to the curve at
(
10
,
−
9
)
is
−
3.
⇒
−
3
=
k
(
10
−
2
)
⇒
−
3
=
8
k
⇒
k
=
−
3
8
⇒
d
y
d
x
=
−
3
8
(
x
−
2
)
Step -2: Integrate the formed equation and solve it further
.
Integrating both sides,
⇒
y
=
−
3
8
(
x
−
2
)
2
2
+
c
.
.
.
.
.
.
.
(
i
)
[
Eqaution of curve
]
Also,
(
10
,
−
9
)
satisfy the equation of curve as it lies on the curve
.
∴
−
9
=
−
3
8
(
10
−
2
)
2
2
+
c
⇒
−
9
=
−
3
16
×
8
2
+
c
⇒
c
=
3
Hence, the equation of the curve is
y
=
−
3
16
(
x
−
2
)
2
+
3.
At any point on the curve
2
x
2
y
2
−
x
4
=
c
, the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to
Report Question
0%
ordinate
0%
radius vector
0%
x-intercept of tangent
0%
sub-tangent
Given
g
(
x
)
=
x
+
2
x
−
1
and the line 3x + y -10 =0, then the line is
Report Question
0%
tangent to g(x)
0%
normal to g(x)
0%
chord of g(x)
0%
none of these
If the is an error of k% in measuring the edge of a cube, then the percent error in estimating its volume is
Report Question
0%
k
0%
3k
0%
k
3
0%
None of these
If the line joining the point (0, 3 ) and (5, -2) is a tangent to the curve
y
=
c
x
+
1
, then the value of c is
Report Question
0%
1
0%
-2
0%
4
0%
None of these
f
(
x
)
=
{
−
x
2
,
for
x
<
0
x
2
+
8
,
for
x
≥
0
Let . Then x-intercept of the line, thet is , the tangent to the graph of f(x) is
Report Question
0%
zero
0%
-1
0%
-2
0%
-4
N
characters of information are held on magnetic tape, in batches of
x
characters each, the batch processing time is
α
+
β
x
2
seconds.
α
and
β
are constants. The optical value of
x
for last processing is,
Report Question
0%
α
β
0%
β
α
0%
√
α
β
0%
√
β
α
The equation of the curve
y
=
b
e
−
x
/
a
at the point where it crosses the y-axis is
Report Question
0%
x
a
−
y
b
=
1
0%
a
x
=
b
y
=
1
0%
a
x
−
b
y
=
1
0%
x
a
+
y
b
=
1
A curve is represented by the equations
x
=
s
e
c
2
t
and
y
=
cot
t
,
where t is a parameter. If the tangent at the point P on the curve, where
t
=
π
/
4
, meets the curve again at the point Q, then
|
P
Q
|
is equal to
Report Question
0%
5
√
3
2
0%
5
√
5
2
0%
2
√
5
3
0%
3
√
5
2
Let f be a continuous, differetiable and bijective function. If the tangent to y= f (x) at x = a is also the normal to y = f (x) at x = b then there exists at least one
c
ϵ
(
a
,
b
)
such that
Report Question
0%
f'(c) = 0
0%
f
(
c
)
>
0
0%
f
(
c
)
<
0
0%
None of these
Given the curves
y
=
f
(
x
)
passing through the point
(
0
,
1
)
and
y
=
∫
x
−
∞
f
(
t
)
passing through the point
(
0
,
1
2
)
.
The tangents drawn to both the curves at the points with equal abscissae intersect on the
x
- axis. Then the curve
y
=
f
(
x
)
is
Report Question
0%
f
(
x
)
=
x
2
+
x
+
1
0%
f
(
x
)
=
x
2
e
x
0%
f
(
x
)
=
e
2
x
0%
f
(
x
)
=
x
−
e
x
If f(x) and g(x) are differentiable function for
0
≤
x
≤
1
such that f(0) = 10 , g(0) = 2, f(1) = 2, g(1) =4, then in the interval (0,1)
Report Question
0%
f (x) = 0 for all x
0%
f (x) + 4g' (x) =0for at least one x
0%
f (x) +2g' (x) for at most one x
0%
none of these
The angle formed bt the positive y-axis and the tangent to
y
=
x
2
+
4
x
−
17
at
(
5
/
2
,
−
3
/
4
)
is
Report Question
0%
tan
−
1
(
9
)
0%
π
2
tan
−
1
(
9
)
0%
π
2
tan
−
1
(
9
)
0%
None of these
The abscissa of a point on the curve
x
y
=
(
a
+
y
)
2
, the normal which cuts off numerically equal intercept from the coordinate axes, is
Report Question
0%
−
a
√
2
0%
√
2
a
0%
a
√
2
0%
−
√
2
a
The co-ordinates of the point (s) on the graph of the function
f
(
x
)
=
x
3
3
−
5
x
2
2
+
7
x
−
4
, where the tangent drawn cuts off intercept from the co-ordinate axes which
Report Question
0%
(2, 8/3)
0%
(3, 7/2)
0%
(1, 5/6)
0%
None of these
The triangle by the tangent to the curve
f
(
x
)
=
x
2
b
x
−
b
at the point (1, 1) and the co-ordinate axes lies in the first quadrant. If its area is 2, then the value of b is
Report Question
0%
-1
0%
3
0%
-3
0%
1
If the normal to the curve y = f(x) at the point (3, 4) makes an angle
3
π
4
with the positive x-axis, then f'(3) is equal to
Report Question
0%
-1
0%
−
3
4
0%
4
5
0%
1
Let
f
(
x
,
y
)
be a curve in the
x
−
y
plane having the property that distance from the origin of any tangent to the curve is equal to distance of point of contact from the
y
−
axis. Of
f
(
1
,
2
)
=
0
, then all such possible curves are
Report Question
0%
x
2
+
y
2
=
5
x
0%
x
2
−
y
2
=
5
x
0%
x
2
y
2
=
5
x
0%
A
l
l
o
f
t
h
e
s
e
The angle between the tangents at ant point P and the line joining P to the original, where P is a point on the curve in
(
x
2
+
y
2
)
=
c
tan
−
1
y
x
,
c
is a constnt, is
Report Question
0%
independent of x
0%
independent of y
0%
independent of x but dependent on y
0%
independent of y but dependent on x
Consider a curve
y
=
f
(
x
)
in
x
y
- plane. The curve passes through
(
0
,
0
)
and has the property that a segment of tangent drawn at any point
P
(
x
,
f
(
x
)
)
and the line
y
=
3
gets bisected by the line
x
+
y
=
1
, then the equation of the curve is
Report Question
0%
y
2
=
9
(
x
−
y
)
0%
(
y
−
3
)
2
=
9
(
1
−
x
−
y
)
0%
(
y
+
3
)
2
=
9
(
1
−
x
−
y
)
0%
(
y
−
3
)
2
−
9
(
1
+
x
+
y
)
The percentage error in the
11
t
h
root of the number
28
is approximately _______ times the percentage error in
28
.
Report Question
0%
11
0%
28
0%
1
28
0%
1
11
Explanation
Given: To find percentage error in
11
t
h
root
28
with respect to
28
∴
As we know that if number
=
x
n
then
%
error in
x
n
=n times the (
%
error in
x
)
∴
We can say that percentage error of nth root of any number is approximately
1
/
n
times percentage error in number.
∴
11
t
h
root of
28
=
28
1
/
11
here
x
=
28
,
n
=
1
/
11
hence ,
1
/
11
The curve possessing the property text the intercept made by the tangent at any point of the curve on the
y
−
axis is equal to square of the abscissa of the point of tangency, is given by
Report Question
0%
y
2
=
x
+
C
0%
y
=
2
x
2
+
C
0%
y
=
−
x
2
+
c
x
0%
N
o
n
e
o
f
t
h
e
s
e
The tangent at a point
P
of a curve meets the
y
−
axis at
A
and the line parallel to
y
−
axis at
A
, and the line parallel to
y
−
axis through
P
meets the
x
−
axis at
B
. If area of
Δ
O
A
B
is constant (
O
being the origin). Then the curve is
Report Question
0%
c
x
2
−
x
y
+
k
=
0
0%
x
2
+
y
2
=
c
x
0%
3
x
2
+
4
y
2
=
k
0%
x
y
−
x
2
y
2
+
k
x
=
0
Consider the curved mirror
y
=
f
(
x
)
passing through
(
0
,
6
)
having the property that all light rays emerging from origin, after getting reflected from the mirror becomes parallel to
x
−
axis, then the equation of curve is
Report Question
0%
y
2
=
4
(
x
−
y
)
or
y
2
=
36
(
9
+
x
)
0%
y
2
=
4
(
1
−
x
)
or
y
2
=
36
(
9
−
x
)
0%
y
2
=
4
(
1
+
x
)
or
y
2
=
36
(
9
−
x
)
0%
N
o
n
e
o
f
t
h
e
s
e
A normal
P
(
x
,
y
)
on a curve meets the
X
−
axis at
Q
and
N
is the ordinate at
P
.
If
N
Q
=
x
(
1
+
y
2
)
1
+
x
2
.
Then the equation of curves passing through
(
3
,
1
)
is
Report Question
0%
5
(
1
+
y
2
)
=
(
1
+
x
2
)
0%
5
(
1
+
y
2
)
=
5
(
1
+
x
2
)
0%
5
(
1
+
x
2
)
=
(
1
+
y
2
)
0%
None of these
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