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CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 3 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Tangents And Its Equations
Quiz 3
Equation of the tangent to the parabola
y
2
=
4
x
+
5
which is parallel to the line
y
=
2
x
+
7
is
Report Question
0%
y
=
2
x
+
3
0%
y
=
2
x
−
3
0%
y
=
2
x
+
5
0%
y
=
2
x
−
5
Explanation
Differentiating the equation given,we obtain
2
y
′
y
=
2
y
′
=
2
2
y
=
2
y
=
1
x
=
−
1
(
y
−
1
)
=
2
(
x
+
1
)
y
=
2
x
+
3
For the parabola
y
2
=
8
x
, tangent and normal are drawn at
P
(
2
,
4
)
which meet the axis of the parabola in
A
and
B
, then the length of the diameter of the circle through
A
,
P
,
B
is
Report Question
0%
2
0%
4
0%
8
0%
6
Explanation
Considering the equation of the parabola
y
2
=
8
x
The axis of symmetry is the positive
x
−
axis.
By, differentiating with respect to
x
, we get
2
y
.
y
′
=
8
y
′
=
4
y
Hence,
y
′
(
2
,
4
)
=
1
Thus slope of the tangent at the point of contact
P
(
2
,
4
)
is
1
while that of normal is
−
1
.
Hence, equation of the tangent will be
y
−
4
x
−
2
=
1
−
x
+
y
=
2
...(i)
Hence, the tangent meets the x axis at
(
−
2
,
0
)
Therefore,
A
=
(
−
2
,
0
)
.
Equation of normal will be
y
−
4
x
−
2
=
−
1
x
+
y
=
6
...(ii)
Hence the normal meets the x axis at
(
6
,
0
)
.
Thus,
B
=
(
6
,
0
)
Therefore the three points through which the circle passes are
(
−
2
,
0
)
,
(
6
,
0
)
,
(
2
,
4
)
Now let the equation of the circle be
(
x
−
h
)
2
+
(
y
−
k
)
2
=
r
2
Therefore
(
−
2
−
h
)
2
+
k
2
=
r
2
→
(
2
+
h
)
2
+
k
2
=
r
2
(
6
−
h
)
2
+
k
2
=
r
2
Subtracting {ii} from {i}, we get
2
h
−
4
=
0
h
=
2
Therefore the equation of the circle reduces to
(
x
−
2
)
2
+
(
y
−
k
)
2
=
r
2
Now
(
2
−
2
)
2
+
(
4
−
k
)
2
=
r
2
→
(
4
−
k
)
2
=
r
2
(
6
−
2
)
2
+
k
2
=
r
2
→
16
+
k
2
=
r
2
Subtracting i from ii, we get
16
+
k
2
−
(
k
−
4
)
2
=
0
16
+
(
2
k
−
4
)
(
4
)
=
0
4
+
2
k
−
4
=
0
k
=
0
Thus the centre of the circle lies at
C
=
(
2
,
0
)
.
Hence the radius of the circle is
C
A
=
C
P
=
C
B
Now
C
A
=
√
(
2
−
(
−
2
)
)
2
+
0
2
=
2
−
(
−
2
)
=
4
=
r
Hence, the diameter of the circle is
8
units.
If the curves
y
=
x
2
−
1
,
y
=
8
x
−
x
2
−
9
touch each other at (2, 3) then equation of the common tangent is
Report Question
0%
4
x
−
y
=
5
0%
4
x
+
y
=
5
0%
x
−
4
y
=
5
0%
x
+
4
y
=
14
Explanation
y
1
1
=
2
x
y
1
2
=
8
−
2
x
at the tangent point slope will be
some.
2
x
=
8
−
2
x
4
x
=
8
x
=
2
y
=
3
y
1
=
4
(
y
−
3
)
=
4
(
x
−
2
)
y
=
4
x
−
5
The point on the hyperbola
y
=
x
−
1
x
+
1
at which the tangents are parallel to
y
=
2
x
+
1
are
Report Question
0%
(
0
,
−
1
)
only
0%
(
−
2
,
3
)
only
0%
(
0
,
−
1
)
,
(
−
2
,
3
)
0%
(
−
2
,
3
)
,
(
5
,
4
)
Explanation
y
′
=
(
x
+
1
)
−
(
x
−
1
)
(
x
+
1
)
2
y
′
=
2
(
∴
parallel to
y
=
2
x
+
1
)
2
(
x
+
1
)
2
=
2
(
x
+
1
)
2
=
1
x
=
0
or
x
=
−
2
Using the equation of hyperbola, when
x
=
0
,
y
=
−
1
.
when
x
=
−
2
,
y
=
3
Assertion(A): The tangent to the curve
y
=
x
3
−
x
2
−
x
+
2
at (1, 1) is parallel to the x axis.
Reason(R): The slope of the tangent to the above curve at (1, 1) is zero.
Report Question
0%
Both A and R are true R is the correct explanation of A
0%
Both A and R are true but R is not correct explanation of A
0%
A is true but R is false
0%
A is false but R is true
Explanation
y
1
=
3
x
2
−
2
x
−
1
y
1
|
x
=
1
=
0
=
m
m
=
0
∴
the curve is parallel to x-axis.
Assertion A: The curves
x
2
=
y
,
x
2
=
−
y
touch each other at (0, 0).
Reason R: The slopes of the tangents at (0, 0) for both the curves are equal.
Report Question
0%
Both A and R are true R is the correct explanation of A
0%
Both A and R are true but R is not correct explanation of A
0%
A is true but R is false
0%
A is false but R is true
Explanation
y
′
=
2
x
y
′
|
x
=
0
=
0
=
m
1
y
′
=
−
2
x
y
′
|
x
=
0
=
0
=
m
2
m
1
=
m
2
∴
both the curve touches each other
Observe the following statements for the curve
y
=
2.
e
−
x
3
I : The slope of the tangent to the curve where it meets y-axis is
−
2
3
II:The equation of normal to the curve where it meets y-axis is
3
x
+
2
y
+
4
=
0
.
Which of the above statement is correct
Report Question
0%
only I
0%
only II
0%
both I and II
0%
neither I nor II
Explanation
y
′
=
−
2
3
e
−
x
y
′
|
x
=
0
=
−
2
3
y
=
2
,
x
=
0
(
y
−
2
)
=
3
2
(
x
−
0
)
2
y
−
3
x
−
4
=
0
only 2 is true
Match the points on the curve
2
y
2
=
x
+
1
with the slope of normals at those points and choose
the correct answer.
Point
Slope of normal
I :
(
7
,
2
)
a
)
−
4
√
2
II:
(
0
,
1
√
2
)
b
)
−
8
III :
(
1
,
1
)
c
)
−
4
IV:
(
3
,
√
2
)
d
)
−
2
√
2
Report Question
0%
i
−
b
,
i
i
−
d
,
i
i
i
−
c
,
i
v
−
a
0%
i
−
b
,
i
i
−
a
,
i
i
i
−
d
,
i
v
−
c
0%
i
−
b
,
i
i
−
c
,
i
i
i
−
d
,
i
v
−
a
0%
i
−
b
,
i
i
−
d
,
i
i
i
−
a
,
i
v
−
c
Explanation
y
′
=
1
4
y
=
m
Slope of normal
m
′
=
−
4
y
For given points slope is
(1)
m
′
1
=
−
4
×
2
==
−
8
(2)
m
′
2
=
−
4
×
1
√
2
=
−
2
√
2
(3)
m
′
3
=
−
4
×
1
=
−
4
(4)
m
′
4
=
−
4
×
√
2
=
−
4
√
2
Option A is correct answer
The equation of the tangent to the curve
y
=
e
−
|
x
|
at the point where the curve cuts the line
x
=
1
is
Report Question
0%
x
+
y
=
e
0%
e
(
x
+
y
)
=
1
0%
y
+
e
x
=
1
0%
x
+
e
y
=
2
Explanation
y
=
e
−
x
y
′
=
−
1
e
−
x
e
−
x
y
′
=
−
e
−
1
a
t
x
=
1
y
=
e
−
1
a
t
x
=
1
(
y
−
e
−
1
)
=
−
e
−
1
(
x
−
1
)
x
+
e
y
=
2
The equation of the normal at
x
=
2
a
for the curve
y
=
8
a
3
4
a
2
+
x
2
is
Report Question
0%
2
x
−
y
=
3
a
0%
2
x
+
y
=
2
a
0%
x
+
2
y
=
6
a
0%
x
+
y
=
a
Explanation
Equation of normal at
x
=
2
a
for
y
=
8
a
3
4
a
2
+
x
2
⇒
x
=
2
a
⇒
y
=
8
a
3
4
a
2
+
(
2
a
)
2
=
a
y
′
=
d
y
d
x
=
−
8
a
3
(
2
x
)
(
4
a
2
+
x
2
)
2
,
x
=
2
a
⇒
y
′
=
−
8
a
3
×
2
×
2
a
(
4
a
2
+
4
a
2
)
2
=
−
8
a
3
×
4
a
64
a
4
=
−
1
2
Slope of normal
=
−
1
y
′
=
2
⇒
y
−
a
=
2
(
x
−
2
a
)
2
x
−
4
a
=
y
−
a
∴
2
x
−
y
=
3
a
is a equation of normal.
The distance of the origin from the normal to the curve
y
=
e
2
x
+
x
2
at
x
=
0
is
Report Question
0%
2
5
0%
2
√
5
0%
2
√
5
0%
5
√
2
Explanation
y
′
=
2
e
2
x
+
2
x
y
=
1
at
x
=
0
y
′
=
2
m
of normal
=
−
1
2
(
y
−
1
)
=
−
1
2
(
x
−
0
)
x
+
2
y
−
2
=
0
distance from origin
d
=
|
x
1
+
2
y
1
−
2
√
1
2
+
2
2
|
(
x
1
y
1
)
=
(
0
,
0
)
d
=
2
√
5
Equation of the tangent line to
y
=
b
e
−
x
a
where it crosses y-axis is
Report Question
0%
a
x
+
b
y
=
1
0%
x
a
+
y
b
=
1
0%
x
b
+
y
a
=
1
0%
a
x
−
b
y
=
1
Explanation
y
=
b
e
−
x
a
a
t
x
=
0
y
=
b
y
′
=
−
b
e
−
x
a
y
′
=
−
b
a
(
a
t
x
=
0
)
(
y
−
b
)
=
−
b
a
(
x
−
0
)
x
a
+
y
b
=
1
The portion of the tangent to xy
=
a
2
at any point on it between the axes is
Report Question
0%
Trisected at that point
0%
bisected at that point
0%
constant
0%
with ratio
1
:
4
at the point
Explanation
Let xy tangent at
(
a
,
a
)
x
y
′
+
y
=
0
y
1
=
−
y
x
y
′
=
−
1
(
y
−
a
)
=
−
1
(
x
−
a
)
Bisected by that point.
y
′
a
t
B
=
−
y
x
=
−
a
y
2
t
y
2
(
y
−
a
2
t
)
=
−
a
2
t
2
(
x
−
t
)
x
=
0
,
y
=
2
a
2
t
y
=
0
,
x
=
2
t
(
t
,
a
2
t
)
to the mid point of (2+10) and
(
0
,
2
a
2
t
)
.
H.P.
Area of the triangle formed by the normal to the curve
x
=
e
sin
y
at (1, 0) with the coordinate axes is
Report Question
0%
1
4
0%
1
2
0%
3
4
0%
1
The arrangment of the slopes of the normals to the curve
y
=
e
log
(
c
o
s
x
)
in the ascending order at the points given below.
A
)
x
=
π
6
,
B
)
x
=
7
π
4
,
C
)
x
=
11
π
6
,
D
)
x
=
π
3
Report Question
0%
C
,
B
,
D
,
A
0%
B
,
C
,
A
,
D
0%
A
,
D
,
C
,
B
0%
D
,
A
,
C
,
B
Explanation
We have,
y
=
e
log
cos
x
=
(
cos
x
)
log
e
[
∵
a
log
b
c
=
c
log
b
a
]
=
cos
x
⇒
d
y
d
x
=
−
sin
x
=
m
(say)
Thus slope of normal to the given curve at any point is,
m
′
=
−
1
m
=
1
sin
x
Now,
m
′
π
6
=
2
,
m
′
7
π
4
=
−
√
2
,
m
′
π
6
=
−
2
,
m
′
π
6
=
√
3
2
Clearly ascending order is,
C
,
B
,
D
,
A
Note: Given function is not defined where
cos
x
<
0
lf the normal at the point
p
(
θ
)
of the curve
x
2
3
+
y
2
3
=
a
2
3
passes through the origin then
Report Question
0%
θ
=
π
3
0%
θ
=
π
6
0%
θ
=
π
4
0%
θ
=
π
2
Explanation
x
=
a
cos
3
θ
y
=
a
sin
3
θ
y
1
=
a
3
sin
2
θ
cos
θ
a
3
cos
2
θ
−
cos
θ
−
sin
θ
cos
θ
(
y
−
a
sin
3
θ
)
=
cos
θ
sin
θ
(
x
−
a
cos
3
θ
)
(
0
,
0
)
satisfies the eq
n
sin
4
θ
=
cos
4
θ
θ
=
π
4
The equations of the tangents at the origin to the curve
y
2
=
x
2
(
1
+
x
)
are
Report Question
0%
y
=
±
x
0%
y
=
±
2
x
0%
y
=
±
3
x
0%
x
=
±
2
y
Explanation
y
=
+
−
√
x
2
+
x
3
y
′
=
+
−
2
x
+
3
x
2
2
√
x
2
+
x
3
y
′
=
+
−
2
+
3
x
2
√
1
+
x
y
′
|
x
=
0
=
+
−
1
y
=
+
−
x
The equation of the common normal at the point of contact of the curves
x
2
=
y
and
x
2
+
y
2
−
8
y
=
0
Report Question
0%
x
=
y
0%
x
=
0
0%
y
=
0
0%
x
+
y
=
0
Explanation
y
2
+
y
−
8
y
=
0
y
=
0.
y
=
7
x
=
0
,
x
=
√
7
y
1
=
2
x
=
m
1
y
y
1
−
4
y
1
=
−
x
y
1
=
x
4
−
y
=
m
2
m
1
,
m
2
a
t
(
0
,
0
)
m
1
=
m
2
=
0
Slope of the normal
=
α
∴
x
=
0
is the required line
cl. y-axis
lf the chord joining the points where
x
=
p
,
x
=
q
on the curve
y
=
a
x
2
+
b
x
+
c
is parallel to the tangent drawn to the curve at
(
α
,
β
)
then
α
=
Report Question
0%
2
p
q
0%
√
p
q
0%
p
+
q
2
0%
p
−
q
2
Explanation
y
′
=
2
a
x
+
b
y
′
|
x
=
b
=
(
2
a
α
+
b
)
2
a
α
+
b
=
(
y
2
−
y
1
x
2
−
x
1
)
2
a
α
+
b
=
a
p
2
+
b
p
−
a
q
2
−
b
q
(
p
−
q
)
=
a
(
p
−
q
)
(
p
+
q
)
+
b
(
p
−
a
)
(
p
−
q
)
2
a
α
+
b
=
a
(
p
+
q
)
+
b
α
=
(
p
+
q
2
)
The arrangement of the following curves in the ascending order of slopes of their tangents at the given points.
A
)
y
=
1
1
+
x
2
at
x
=
0
B
)
y
=
2
e
−
x
4
,
where it cuts the y-axis
C
)
y
=
c
o
s
(
x
)
at
x
=
−
π
4
D
)
y
=
4
x
2
at
x
=
−
1
Report Question
0%
DCBA
0%
ACBD
0%
ABCD
0%
DBAC
Explanation
(A)
y
1
=
−
2
x
(
1
+
x
2
)
y
1
|
x
=
0
=
0
(B)
y
1
=
−
2
x
(
4
y
1
|
x
=
0
=
−
1
2
(C)
y
1
=
−
s
i
n
x
y
1
|
x
=
−
p
i
/
4
=
1
√
2
(D)
y
1
=
8
x
y
1
|
x
=
−
1
=
−
8
(
C
)
>
(
A
)
>
(
B
)
>
(
D
)
Observe the following lists for the curve
y
=
6
+
x
−
x
2
with the slopes of tangents at the given points; I, II, III, IV
Point
Tangent slope
I:
(
1
,
6
)
a)
3
II:
(
2
,
4
)
b)
5
III:
(
−
1
,
4
)
c)
−
1
IV:
(
−
2
,
0
)
d)
−
3
Report Question
0%
a
,
b
,
c
,
d
0%
b
,
c
,
d
,
a
0%
c
,
d
,
b
,
a
0%
c
,
d
,
a
,
b
Explanation
y
′
=
1
−
2
x
(A)
m
1
=
−
1
(B)
m
2
=
−
3
(C)
m
3
=
3
(D)
m
4
=
5
lf the tangent to the curve
2
y
3
=
a
x
2
+
x
3
at the point
(
a
,
a
)
cuts off intercepts
α
and
β
on the coordinate axes such that
α
2
+
β
2
=
61
, then
a
is equal to
Report Question
0%
±
30
0%
±
5
0%
±
6
0%
±
61
Explanation
2
×
3
×
y
2
y
′
=
2
a
x
+
3
x
2
y
′
=
2
a
x
+
3
x
2
6
y
2
m
=
y
′
|
(
a
,
a
)
=
5
6
Equation of the tangent at
(
a
,
a
)
with slope,
m
=
5
6
(
y
−
a
)
=
5
6
(
x
−
a
)
Intersection points with coordinate axes are,
(
0
,
a
6
)
and
(
−
a
5
,
0
)
So,
α
=
a
6
and
β
=
−
a
5
Given that,
α
2
+
β
2
=
61
⇒
a
2
36
+
a
2
25
=
61
⇒
a
2
=
30
2
⇒
a
=
±
30
Assertion(A): If the tangent at any point
P
on the curve
x
y
=
a
2
meets the axes at
A
and
B
then
A
P
:
P
B
=
1
:
1
Reason(R): The tangent at
P
(
x
,
y
)
on the curve
X
m
.
Y
n
=
a
m
+
n
meets the axes at
A
and
B
. Then the ratio of
P
divides
¯
A
B
is
n
:
m
.
Report Question
0%
Both A and R are true R is the correct explanation of A
0%
Both A and R are true but R is not correct explanation of A
0%
A is true but R is false
0%
A is false but R is true
Explanation
y
=
a
2
x
y
′
=
−
a
2
x
2
y
′
=
−
a
2
t
2
(
y
−
a
2
t
)
=
−
a
2
t
2
(
n
−
t
)
B
=
(
2
+
10
)
A
=
(
0
,
2
a
2
t
)
A
P
B
P
=
1
The point on the curve
√
x
+
√
y
=
2
a
2
, where the tangent is equally inclined to the axes, is
Report Question
0%
(
a
4
,
a
4
)
0%
(
0
,
4
a
4
)
0%
(
4
a
4
,
0
)
0%
none of these
Explanation
y
=
x
is the equation of line as it is equally inclined to axes
So let point
(
k
,
k
)
is the point of tangency
Therefore,
√
k
+
√
k
=
2
a
2
k
=
a
4
Hence point is
(
a
4
,
a
4
)
Match List-I with List-II and select the correct answer using the code given below. A B C D
List-I
List-II
a) Equation of tangent to the curve
y
=
b
e
−
x
/
a
at
x
=
0
1)
x
−
2
y
=
2
b) Equation of tangent to the curve
y
=
x
2
+
1
at
(
1
,
2
)
2)
y
=
2
x
c) Equation of normal to the curve
y
=
2
x
−
x
2
at
(
2
,
0
)
3)
x
−
y
=
π
d) Equation of normal to the curve
y
=
sin
x
at
x
=
π
4)
x
a
+
y
b
=
1
Report Question
0%
4 1 2 3
0%
4 2 1 3
0%
1 2 3 4
0%
1 4 3 2
Explanation
(A)
y
′
=
−
b
a
e
−
x
/
a
m
=
−
b
a
(
y
−
b
)
=
−
b
a
(
x
−
0
)
x
a
+
y
b
=
1
(B)
y
′
=
2
x
y
′
|
x
=
1
=
2
(
y
−
2
)
=
2
(
x
−
1
)
y
=
2
x
(C)
y
′
=
2
−
2
x
y
′
|
x
=
2
=
−
2
Slope of normal
=
1
2
(
y
−
0
)
=
1
2
(
x
−
2
)
2
y
=
x
−
2
(D)
y
′
=
cos
x
y
′
|
x
=
π
=
−
1
Slope of normal =1
(
y
−
0
)
=
1
(
x
−
π
)
x
−
y
=
π
Area of the triangle formed by the tangent, normal at
(
1
,
1
)
on the curve
√
x
+
√
y
=
2
and the y axis is (in sq. units)
Report Question
0%
1
0%
2
0%
1
2
0%
4
Explanation
1
√
x
+
y
1
√
y
=
0
y
′
=
−
√
y
x
y
′
=
−
1
a
t
(
n
=
1
,
y
=
1
)
(
y
−
1
)
=
−
1
(
x
−
1
)
y
=
2
(
y
−
1
)
=
1
(
x
−
1
)
y
=
0
A
r
e
a
=
1
2
×
2
×
1
=
1
A curve with equation of the form
y
=
a
x
4
+
b
x
3
+
c
+
c
x
+
d
has zero gradient at the point (0,1) and also touches the x-axis at the point (-1, 0) then the values of the x for which the curve has a negative gradient are :
Report Question
0%
x
>
−
1
0%
x
<
1
0%
x
<
−
1
0%
−
1
≤
x
≤
1
Explanation
d
y
d
x
|
x
=
0
=
4
a
x
3
+
3
b
x
2
+
c
=
0
∴
c
=
0
d
y
d
x
<
0
4
a
x
3
+
3
b
x
2
<
0
x
2
(
4
a
x
+
3
b
)
<
0
x
<
−
3
b
4
a
Assertion (A): The points on the curve
y
=
x
3
−
3
x
at which the tangent is parallel to
x
-axis are
(
1
,
−
2
)
and
(
−
1
,
2
)
.
Reason (R): The tangent at
(
x
1
,
y
1
)
on the curve
y
=
f
(
x
)
is vertical then
d
y
d
x
at
(
x
1
,
y
1
)
is not defined.
Report Question
0%
Both A and R are true and R is the correct explanation for A
0%
Both A and R are true but R is not the correct explanation for A
0%
A is true but R is false
0%
A is false but R is true
Explanation
y
′
=
3
x
2
−
3
y
′
=
0
∴
tangent is parallel to x-axis
x
2
=
1
x
=
+
−
1
y
=
−
2
y
=
2
y tangent is vertical then
d
y
d
x
is not defined.
lf the tangent at any point on the curve
x
4
+
y
4
=
c
4
cuts off intercepts
a
and
b
on the coordinate axes, the value of
a
−
4
/
3
+
b
−
4
/
3
is
Report Question
0%
c
−
4
/
3
0%
c
−
1
/
2
0%
c
1
/
2
0%
c
4
/
3
Explanation
Given equation of curve is
x
4
+
y
4
=
c
4
d
y
d
x
=
−
x
3
y
3
Slope of tangent at point
(
x
1
,
y
1
)
on the curve is
−
x
3
1
y
3
1
Let the equation of tangent to curve be
x
a
+
y
b
=
1
Since, this line intersects X-axis at
(
a
,
0
)
and
(
0
,
b
)
x
c
4
/
x
3
1
+
y
c
4
/
y
3
1
=
1
⇒
a
=
c
4
x
3
1
,
b
=
c
4
y
3
1
⇒
a
−
4
/
3
=
c
−
16
/
3
x
−
4
1
,
b
−
4
/
3
=
c
−
16
/
3
y
−
4
1
⇒
a
−
4
/
3
+
b
−
4
/
3
=
c
−
16
/
3
(
x
4
1
+
y
4
1
)
⇒
a
−
4
/
3
+
b
−
4
/
3
=
c
−
4
/
3
I. lf the curve
y
=
x
2
+
b
x
+
c
touches the straight line
y
=
x
at the point
(
1
,
1
)
then
b
and
c
are given by
1
,
1.
II. lf the line
P
x
+
m
y
+
n
=
0
is a normal to the curve
x
y
=
1
, then
P
>
0
,
m
<
0
.
Which of the above statements is correct
Report Question
0%
only I
0%
only II
0%
both I and II
0%
Neither I nor II
Explanation
(I)
1
+
b
+
c
=
1
b
+
c
=
0
y
′
=
2
x
+
b
y
′
=
1
2
+
b
=
1
(
b
=
−
1
)
c
=
1
(II)
y
′
=
−
1
x
2
m
=
−
1
x
2
−
P
m
=
x
2
x
−
P
=
x
2
m
p and m will be of opposite num.
At origin the curve
y
2
=
x
3
+
x
Report Question
0%
Touches the x-axis
0%
Touches the y-axis
0%
Bisects the angle between the axes
0%
touches both the axes
Explanation
d
y
d
x
=
3
x
2
+
1
2
y
At origin
x
=
0
,
y
=
0
tan
θ
=
∞
θ
=
π
2
∴
Curve touches the y-axis
Observe the following statements
I: If
p
and
q
are the lengths of perpendiculars from the origin on the tangent and normal at any point on the curve
x
2
3
+
y
2
3
=
1
then
4
p
2
+
q
2
=
1
.
II: If the tangent at any point
P
on the curve
x
3
.
y
2
=
a
5
cuts the coordinate axes at
A
and
B
then
A
P
:
P
B
=
3
:
2
Report Question
0%
only I
0%
only II
0%
both I and II
0%
neither I nor II
Explanation
x
=
a
cos
3
θ
y
=
a
sin
3
θ
y
′
=
−
tan
θ
(
y
−
s
i
n
3
θ
)
=
−
tan
θ
(
x
−
cos
3
θ
)
y
+
n
tan
θ
−
tan
θ
cos
3
θ
−
sin
3
θ
=
0
Perpendicular from origin
|
tan
θ
cos
3
θ
+
sin
3
θ
√
1
+
tan
2
θ
|
=
p
p
2
sin
θ
cos
θ
p
=
sin
2
θ
2
(
y
−
sin
3
θ
)
=
1
tan
θ
(
x
−
cos
3
θ
)
y
tan
θ
−
sin
3
θ
tan
θ
−
x
+
cos
3
θ
=
0
Perpendicular frm the origin
|
cos
3
θ
−
sin
3
θ
tan
θ
√
1
+
tan
2
θ
|
=
q
q
=
cos
2
θ
4
p
2
+
q
2
=
1
Observe the following statements for the curve
x
=
a
t
3
,
y
=
a
t
4
at
t
=
1
.
I : The equation of the tangent to the curve is
4
x
−
3
y
−
a
=
0
II : The equation of the normal to the curve is
3
x
+
4
y
−
7
a
=
0
III: Angle between tangent and normal at any point on the curve is
π
2
Which of the above statements are correct.
Report Question
0%
I and II
0%
II and III
0%
I and III
0%
I, II, III
Explanation
d
y
d
x
=
4
a
t
3
3
a
t
2
d
y
d
x
=
4
t
3
d
y
d
x
|
t
=
1
=
4
3
(
y
−
a
)
=
4
3
(
x
−
a
)
(tangent)
4
x
−
3
y
−
a
=
0
(
y
−
a
)
=
−
3
4
(
x
−
a
)
(Normal)
4
y
+
3
x
−
7
a
=
0
Tangent and normal are always at an angle of
π
/
2
to each other
Assertion (A): The normal to the curve
a
y
2
=
x
3
(
a
≠
0
,
x
≠
0
)
at a point
(
x
,
y
)
on it makes equal intercepts on the axes, then
x
=
4
a
9
.
Reason (R): The normal at
(
x
1
,
y
1
)
on the curve
y
=
f
(
x
)
makes equal intercepts on the coordinate axes, then
d
y
d
x
|
(
x
1
,
y
1
)
=
1
Report Question
0%
Both (A) and (R) are true and (R) is the correct explanation for (A).
0%
Both (A) and (R) are true but (R) is not the correct explanation for (A).
0%
(A) is true but (R) is false.
0%
(A) is false but (R) is true.
Explanation
a
y
2
=
x
3
2
a
y
d
y
d
x
=
3
x
2
⇒
d
y
d
x
=
3
x
2
2
a
y
Now, slope of normal
=
−
d
x
d
y
=
−
2
a
y
3
x
2
=
−
2
a
×
x
3
2
3
x
2
√
a
=
−
2
√
a
3
√
x
For the normal to make equal intercepts on the axes, the slope has to be
±
1
.
Hence,
−
2
√
a
3
√
x
=
±
1
On squaring both sides,
4
a
9
x
=
1
⇒
x
=
4
a
9
.
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
The point of intersection of the tangents drawn to the curve
x
2
y
=
1
−
y
at the points where it is met by the curve
x
y
=
1
−
y
is given by
Report Question
0%
(
0
,
−
1
)
0%
(
1
,
1
)
0%
(
0
,
1
)
0%
(
1
,
∞
)
Explanation
x
2
y
=
x
y
y
=
1
,
x
=
0
y
=
1
/
2
,
x
=
1
2
x
d
x
d
y
=
−
−
1
y
2
d
y
d
x
=
−
2
x
y
2
(
x
=
0
)
d
y
d
x
=
0
(
y
−
1
)
=
x
−
0
y
=
1
------- (1)
(
y
−
1
2
)
=
−
1
2
(
x
−
1
)
-----(2)
x
=
0
(By (1) & (2))
(
0
,
1
)
The number of tangents to the curve
x
3
/
2
+
y
3
/
2
=
a
3
/
2
, where the tangents are equally inclined to the axes, is
Report Question
0%
2
0%
1
0%
0
0%
4
Explanation
d
y
d
x
=
−
1
(where x is equally inclined)
3
2
x
1
/
2
+
3
/
2
y
1
/
2
d
y
d
x
=
0
d
y
d
x
=
−
√
x
y
=
1
−
√
x
y
=
−
1
√
x
=
√
y
x
=
y
x
3
/
2
+
x
3
/
2
=
a
3
/
2
2
x
3
/
2
=
a
3
/
2
x
=
a
2
2
/
3
y
=
a
2
2
/
3
∴
only one tangent.
The points on the hyperbola
x
2
−
y
2
=
2
closest to the point (0, 1) are
Report Question
0%
(
±
3
2
,
1
2
)
0%
(
1
2
,
±
3
2
)
0%
(
1
2
,
1
2
)
0%
(
±
3
4
±
3
2
)
Explanation
Let point be
(
√
2
s
e
c
θ
,
√
2
t
a
n
θ
)
closest to (0, 1)
So distance
=
√
2
s
e
c
2
θ
+
(
√
2
t
a
n
θ
−
1
)
2
=
√
2
s
e
c
2
θ
+
2
t
a
n
2
θ
−
2
√
2
t
a
n
θ
+
1
=
√
4
t
a
n
2
θ
−
2
√
2
t
a
n
θ
+
3
=
2
√
t
a
n
2
θ
−
t
a
n
θ
√
2
+
3
4
=
2
√
(
t
a
n
θ
−
1
2
√
2
)
2
+
3
4
−
1
8
So distance is min when
t
a
n
θ
=
1
2
√
2
s
e
c
θ
=
±
3
2
√
2
lf the tangent to the curve
f
(
x
)
=
x
2
at any point
(
c
,
f
(
c
)
)
is parallel to the line joining points
(
a
,
f
(
a
)
)
and
(
b
,
f
(
b
)
)
on the curvel then
a
,
c
,
b
are in
Report Question
0%
AP
0%
GP
0%
HP
0%
AGP
Explanation
∂
′
M
=
2
s
at
(
c
,
∂
(
c
)
)
=
2
c
∂
(
a
)
−
∂
(
L
)
a
−
b
=
2
c
a
2
=
b
2
a
−
b
=
2
c
a
+
b
=
2
c
∴
a
,
c
,
b
are in AP
A
curve is given by the equations
x
=
sec
2
θ
,
y
=
cot
θ
. If the tangent at
P
where
θ
=
π
4
meets the curve again at
Q
, then length
P
Q
, is
Report Question
0%
√
5
2
0%
3
√
5
2
0%
10
0%
20
Explanation
y
=
cot
θ
⇒
d
y
d
θ
=
−
cosec
2
θ
x
=
sec
2
θ
⇒
d
x
d
θ
=
2
sec
θ
⋅
sec
θ
tan
θ
d
y
d
x
|
θ
=
π
4
=
−
cosec
2
θ
2
sec
2
θ
tan
θ
=
−
2
2
×
2
×
1
=
−
1
2
y
=
1
tan
θ
x
=
1
+
tan
2
θ
x
=
1
+
1
y
2
⇒
y
2
x
=
y
2
+
1
⇒
x
=
2
,
y
=
1
Now, at
(
x
,
y
)
=
(
2
,
1
)
, equation of a line with slope
−
1
2
, is
(
y
−
1
)
=
−
1
2
(
x
−
2
)
⇒
2
y
+
x
=
4
⇒
y
2
(
4
−
2
y
)
=
y
2
+
1
⇒
4
y
2
−
2
y
3
=
y
2
+
1
⇒
2
y
3
−
3
y
2
+
1
=
0
⇒
y
=
1
,
−
1
2
⇒
x
=
2
,
5
P
Q
=
√
(
2
−
5
)
2
+
(
1
+
1
2
)
2
=
(
3
2
+
3
2
4
)
0.5
=
3
√
5
2
Hence, option B.
lf the parametric equation of a curve given by
x
=
e
t
cos
t
,
y
=
e
t
sin
t
, then the tangent to the curve at the point
t
=
π
4
makes with axis of
x
the angle.
Report Question
0%
0
0%
π
4
0%
π
3
0%
π
2
Explanation
a
t
t
=
π
/
4
y
=
e
t
s
i
n
t
d
y
d
l
=
e
t
(
s
i
n
t
+
c
o
s
t
)
d
x
d
l
=
d
t
(
c
o
s
t
+
s
i
n
t
)
d
y
d
x
=
s
i
n
t
+
c
o
s
t
c
o
s
t
−
s
i
n
t
=
α
∴
t
a
n
−
1
(
∞
)
=
π
/
2
The normal to the curve
x
=
a
(
1
+
cos
θ
)
,
y
=
a
sin
θ
at any point
θ
always passes through the fixed point:
Report Question
0%
(
a
,
0
)
0%
(
0
,
a
)
0%
(
0
,
0
)
0%
(
a
,
a
)
Explanation
x
=
a
(
1
+
cos
θ
)
,
y
=
a
sin
θ
Differentiating w.r.t
θ
, we get
d
x
d
θ
=
a
(
−
sin
θ
)
,
d
y
d
θ
=
a
cos
θ
∴
d
y
d
x
=
−
cos
θ
sin
θ
Therefore equation of normal at
(
a
(
1
+
cos
θ
)
,
sin
θ
)
is
(
y
−
a
sin
θ
)
=
sin
θ
cos
θ
(
x
−
a
(
1
+
cos
θ
)
)
It is clear that in the given options normal passes through the point
(
a
,
0
)
lf the curve
y
=
p
x
2
+
q
x
+
r
passes through the point (1, 2) and the line
y
=
x
touches it at the origin, then the values of
p
,
q
and
r
are
Report Question
0%
p
=
1
,
q
=
−
1
,
r
=
0
0%
p
=
1
,
q
=
1
,
r
=
0
0%
p
=
−
1
,
q
=
1
,
r
=
0
0%
p
=
1
,
q
=
2
,
r
=
3
Explanation
2
=
p
+
q
+
r
r
=
0
d
x
d
y
|
x
=
0
=
1
d
y
d
x
=
2
p
x
+
q
q
=
1
2
=
p
+
1
+
0
p
=
1
For the curve
y
=
3
sin
θ
cos
θ
,
x
=
e
θ
sin
θ
,
0
≤
θ
≤
π
; the tangent is parallel to
x
-axis when
θ
is
Report Question
0%
0
0%
π
2
0%
π
4
0%
π
6
Explanation
d
y
d
x
=
0
d
y
d
θ
=
d
3
2
(
s
i
n
2
θ
)
d
y
d
θ
=
d
3
4
(
c
o
s
2
θ
)
---- (1)
d
y
d
θ
=
e
θ
s
i
n
θ
+
e
θ
c
o
s
θ
---- (2)
d
x
d
y
=
3
4
c
o
s
2
θ
e
θ
(
s
i
n
θ
+
c
o
s
θ
)
=
0
c
o
s
2
θ
=
0
2
θ
=
π
/
2
θ
=
π
/
4
If the circle
x
2
+
y
2
+
2
g
x
+
2
f
y
+
c
=
0
is touched by
y
=
x
at
P
such that
O
P
=
6
√
2
,
then the value of
c
is
Report Question
0%
36
0%
144
0%
72
0%
None of these
Explanation
Let the point of contact be
x
1
,
y
1
However it lies on the line
y
=
x
Hence
x
1
=
y
1
Applying distance formula, we get
√
x
2
1
+
x
2
1
=
6
√
2
2
x
2
1
=
72
x
1
=
6
...(positive, since it lies on y=x.)
Differentiating the equation of circle with respect to x.
2
x
+
2
y
y
′
+
2
g
+
2
f
y
′
=
0
Now
y
′
=
1
since
y
′
is the slope of the line
y
=
x
.
By substituting, we get
x
+
y
=
−
(
g
+
f
)
Now
x
1
=
y
1
=
6
Hence
12
=
−
(
g
+
f
)
...(i)
Substituting
x
=
y
=
6
in the equation of the circle, we get
36
+
36
+
2
(
6
)
(
g
+
f
)
+
c
=
0
72
+
12
(
−
12
)
+
c
=
0
c
=
144
−
72
c
=
72
If
x
+
4
y
=
14
is a normal to the curve
y
2
=
α
x
3
−
β
at
(
2
,
3
)
, then the value of
α
+
β
is
Report Question
0%
9
0%
−
5
0%
7
0%
−
7
Explanation
Given equation of curve is
y
2
=
α
x
3
−
β
Since, it passes through (2,3)
4
=
8
α
−
β
d
y
d
x
=
3
α
x
2
2
y
Slope of tangent at (2,3) is
2
α
Slope of normal at
(
2
,
3
)
=
−
1
2
α
Given equation of normal is
x
+
4
y
=
14
Slope of normal is
−
1
4
⇒
α
=
2
⇒
β
=
7
Hence
α
+
β
=
9
The number of tangents to the curve
x
3
/
2
+
y
3
/
2
=
2
a
3
/
2
,
a
>
0
, which are equally inclined to the axes, is
Report Question
0%
2
0%
1
0%
0
0%
4
Explanation
For a tangent to be equally inclined to the axes, the slope of the tangent should be 1.
Thus
y
′
=
1
.
Now
x
3
2
+
y
3
2
=
2
a
3
2
.
3
2
(
√
x
)
+
3
2
(
√
y
)
=
0
√
x
=
−
√
y
x
=
y
x
1
=
y
1
Substituting int he above equation, we get
2
x
3
2
=
2
a
3
2
x
=
y
=
a
.
Hence the tangent touches the curve at
(
a
,
a
)
.
Thus we get only one tangent, satisfying that criteria.
If
m
is the slope of a tangent to the curve
e
y
=
1
+
x
2
, then
Report Question
0%
|
m
|
>
1
0%
m
>
1
0%
m
>
−
1
0%
|
m
|
≤
1
Explanation
e
y
=
1
+
x
2
d
y
d
x
=
2
x
1
+
x
2
So
|
m
|
=
|
2
x
x
2
+
1
|
Now, since
x
2
≥
0
⇒
x
2
+
1
≥
1
⇒
1
x
2
+
1
≤
1
⇒
2
x
x
2
+
1
≤
1
|
m
|
≤
1
If the circle
x
2
+
y
2
+
2
g
x
+
2
f
y
+
c
=
0
is touched by y = x at P in the first quadrant, such that
O
P
=
6
√
2
, then the value of
c
is
Report Question
0%
36
0%
144
0%
72
0%
None of these
Explanation
Let the point of contact be
x
1
,
y
1
However it lies on the line
y
=
x
Hence
x
1
=
y
1
Applying distance formula, we get
√
x
2
1
+
x
2
1
=
6
√
2
2
x
2
1
=
72
x
1
=
6
...(positive, since it lies on y=x.)
Differentiating the equation of circle with respect to x.
2
x
+
2
y
y
′
+
2
g
+
2
f
y
′
=
0
Now
y
′
=
1
since
y
′
is the slope of the line
y
=
x
.
By substituting, we get
x
+
y
=
−
(
g
+
f
)
Now
x
1
=
y
1
=
6
Hence
12
=
−
(
g
+
f
)
...(i)
Substituting
x
=
y
=
6
in the equation of the circle, we get
36
+
36
+
2
(
6
)
(
g
+
f
)
+
c
=
0
72
+
12
(
−
12
)
+
c
=
0
c
=
144
−
72
c
=
72
The equations of the tangents to the curve
y
=
x
4
from the point (2, 0) not on the curve, are given by
Report Question
0%
y
=
0
0%
y
−
1
=
5
(
x
−
1
)
0%
y
−
4098
81
=
2048
27
(
x
−
8
3
)
0%
y
−
32
243
=
80
81
(
x
−
2
3
)
Explanation
y
=
4
(
8
3
)
3
(
x
−
2
)
y
−
0
=
m
(
x
−
2
)
y
=
m
(
x
−
2
)
m
(
x
1
−
2
)
=
x
4
1
4
x
4
1
−
8
x
3
1
=
x
4
1
(
m
=
4
x
3
1
)
3
x
4
1
=
8
x
3
1
x
1
=
8
3
∴
(
y
−
4098
81
)
=
2048
27
(
x
−
8
3
)
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
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