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JEE Questions for Maths Applications Of Derivatives Quiz 2 - MCQExams.com
JEE
Maths
Applications Of Derivatives
Quiz 2
The point on the curve y =
x
3
at which the tangent to the curve is parallel to the X-axis is
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0%
(2, 2)
0%
(3, 3)
0%
(4, 4)
0%
(0, 0)
The equation of normal to the curve
x
2
y =
x
2
- 3
x
+ 6 at the point with abscissa
x
= 3 is
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3x + 27y = 79
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27x - 3y = 79
0%
27x + 3y = 79
0%
3x - 27y = 79
The maximum area of the rectangle that can be inscribed in a circle of radius r is
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0%
πr2
0%
r2
0%
πr2/4
0%
2r2
The length of the subtangent to the curve
x
2
y
2
= a
4
at (-a, a) is
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0%
a/2
0%
2a
0%
a
0%
a/3
If the curves
x
2
= 9A(9 - y) and
x
2
= A(y +intersect; orthogonally, then the value of A is
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0%
3
0%
4
0%
5
0%
7
0%
9
The greatest value of sin
3
x
+ cos
3
x
is
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0%
1
0%
2
0%
√2
0%
√3
The equation of the normal line to the curve y =
x
log
e
x
parallel to 2
x
- 2y + 3 = 0 is
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x + y = 3e-2
0%
x - y = 6e-2
0%
x - y = 3e-2
0%
x - y = 6e2
A population p(t) of 1000 bacteria introduced into nutrient medium grows according to the relation p(t) = 1000 + (1000t/100 + t
2
). The maximum size of this bacterial population is
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0%
1100
0%
1250
0%
1050
0%
5250
The points on the curve y = 2
x
2
- 6
x
- 4 at which the tangent is parallel to the X - axis is
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0%
0%
2)
0%
0%
(0, -4)
0%
If
x
= t
2
and y = 2t, then equation of the normal at t = 1 is
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x + y - 3 = 0
0%
x + y - 1 = 0
0%
x + y + 1 = 0
0%
x + y + 3 = 0
If tangent to the curve
x
= at
2
, y = 2a is perpendicular to X - axis, then its point of contact is
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0%
(a, a)
0%
(0, a)
0%
(0, 0)
0%
(a, 0)
The perimeter of a sector is a constant. If its area is to be maximum, then the sectorial angle is
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0%
0%
2)
0%
0%
If ST and SN are the lengths of the subtangent and the subnormal at the point θ = π/2 on the curve
x
= a(θ + sin θ), y = a(1 - cos θ), a ≠ 1, then
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ST = SN
0%
ST = 2SN
0%
ST2 = aSN3
0%
ST3 = aSN
The length of the subtangent at any point (
x
1
, y
1
) on the curve y = a
x
(a >is
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0%
2 log a
0%
1/log a
0%
log a
0%
The absolute maximum of
x
40
-
x
20
on the interval [0, 1] is
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-1/4
0%
0
0%
1/4
0%
1/2
The point on the curve √
x
+ √y = √a, the normal at which is parallel to the X - axis is
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0%
(0, 0)
0%
(0, a)
0%
(a, 0)
0%
(a, a)
If
f
(
x
) = 1 + 2
x
2
+ 2
2
x
4
+ ...+2
10
x
20
. Then,
f
(
x
) has
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0%
more than one minimum
0%
exactly one minimum
0%
at least one maximum
0%
None of the above
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0%
-1, 1
0%
-2, 2
0%
-3, 3
0%
-4, 4
The equation of the tangent to the curve
x
= 2 cos
3
θ and y = sin
3
θ at the point, θ = π/4 is
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2x + 3y = 3√2
0%
2x - 3y = 3√2
0%
3x + 2y = 3√2
0%
3x + 2y = 3√2
The minimum value of 4e
2
x
+ 9e
-2
x
is
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0%
11
0%
12
0%
10
0%
14
Let
f '
(
x
) be differentiable for all
x
. If
f
(= - 2 and
f '
(
x
) ≥ 2, ∀
x
ϵ [1, 6], then
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f(< 8
0%
f(≥ 8
0%
f(≥ 5
0%
f(≤ 5
The value of c from the Lagrange's mean value theorem for which
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0%
5
0%
1
0%
√15
0%
None of these
What are the value of c for while Rolle's theorem for the functions
f
(
x
) =
x
3
- 3
x
2
+ 2
x
in the interval [0, 2] is verified?
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0%
c = ± 1
0%
c = 1 ± 1/ √3
0%
c = ± 2
0%
None of these
The function
x
x
is increasing, when
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x > 1/e
0%
x < 1/e
0%
x < 0
0%
∀ x
If
f
(
x
) =
x
3
and g(
x
) =
x
3
- 4
x
in -2 ≤
x
≤ 2, then consider the statements I
f
(
x
) and g(
x
) satisfy mean value theorem. II
f
(
x
) and g(
x
) both satisfy rolle's theorem. of these Statements
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0%
I and II are correct
0%
Only I is correct
0%
None is correct
0%
I and II are correct
The function
f
(
x
) = 2
x
3
- 15
x
2
+ 36
x
+ 6 is strictly decreasing in the interval
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0%
(2, 3)
0%
(-∞, 2)
0%
(3, 4)
0%
(-∞,∪(4, ∞)
0%
(-∞,∪ (3, ∞)
If
f
(
x
) be a differentiable function in [2, 7]. If
f
(= 3 and
f '
(
x
) ≤ 5 for all
x
in (2, 7), then the maximum possible value of
f
(
x
) at
x
= 7 is
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0%
7
0%
15
0%
28
0%
14
If
f
(
x
) is differentiable and strictly increasing function, then the value of
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0%
-1
0%
0
0%
1
0%
2
The function
x
- cot
x
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0%
never decreases
0%
always increases
0%
always decreases
0%
sometimes increases and sometimes decreases
If
f
is a real valued differentiable function, such that
f
(
x
)
f '
(
x
) < 0 for all real
x
, then
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f(x) must be an increasing function
0%
f(x) must be decreasinf function
0%
|f(x)| mus be increasing function
0%
|f(x)| must be decreasing function
Rolle's theorem is applicable in the interval [-2, 2] for the function
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f(x) = x2
0%
f(x) = 4x4
0%
f(x) = 2x3 + 3
0%
f(x) = π|x|
Which of the following function is decreasing on (0, π/?
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sin 2x
0%
cos 3x
0%
tan x
0%
cos 2x
The value of c in mean value theorem for the function
f
(
x
) = 2
x
2
+ 3
x
+ 4 in the interval [1, 2] is
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0%
1/2
0%
1/3
0%
3/2
0%
2/3
The function
f
(
x
) = [
x
(
x
- 2)]
2
is increasing in the set
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0%
(-∞,∪ (2, ∞)
0%
(-∞, 1)
0%
(0,∪ (2, ∞)
0%
(1, 2)
0%
(0, 2)
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increasing when x > 0
0%
strictly increasing when x > 0
0%
strictly increasing at x = 0
0%
not continuous at x = 0 and so it is not increasing when x > 0
The function
f
(
x
) = a
x
+ b is strictly increasing for all real
x
, if
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a > 0
0%
a < 0
0%
a = 0
0%
a ≤ 0
If
f
(
x
) = k
x
- cos
x
is monotonically increasing for all
x
ϵ R, then
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0%
K > - 1
0%
K < 1
0%
K > 1
0%
None of these
The value of c in (0,satisfying the mean value theorem for the function
f
(
x
) =
x
(
x
- 1)
2
,
x
ϵ [0, 2] is equal to
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0%
3/4
0%
4/3
0%
1/3
0%
2/3
0%
5/3
In which of the following functions Rolle's theorem is applicable ?
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0%
0%
2)
0%
0%
If the mean value theorem is
f
(b) - (b - a)
f '
(c). Then, for the function
x
2
- 2
x
+ 3 in [1, 3/2], the value of c is
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0%
6/5
0%
5/4
0%
4/3
0%
7/6
A function
f
is defined
f
(
x
) = e
x
sin
x
in [0, π]. Which of the following is not correct ?
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f is continuous in [0, π]
0%
f is differentiable in [0, π]
0%
f(= f(π)
0%
Rolle's theorem is not true in [0, π]
The function
f
(
x
) = (9 -
x
2
)
2
increases in
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(-3,∪(3, ∞)
0%
(-∞, -∪ (3, ∞)
0%
(-∞, -∪ (0, 3)
0%
(-3, 3)
0%
(3, ∞)
If a < 0, the function (e
a
x
+ e
-a
x
) is a decreasing function for all values of
x
, where
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0%
x < 0
0%
x > 0
0%
x < 1
0%
x > 1
The function
f
defined by
f
(
x
) =
x
3
- 6
x
2
- 36
x
+ 7 is increasing, if
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0%
x > 2 and also x > 6
0%
x > 2 and also x < 6
0%
x > -2 and also x < 6
0%
x < - 2 and also x > 6
The function
f
(
x
) =
x
2
e
-
x
increases in the interval
Report Question
0%
(0, 2)
0%
(2, 3)
0%
(3, 4)
0%
(4, 5)
If
f
(
x
) = k
x
- sin
x
in monotonically increasing, then
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0%
k > 1
0%
k > - 1
0%
k < 1
0%
k < - 1
If 2a + 3b + 6c = 0, then at least one root of the equation a
x
2
+ b
x
+ c = 0 lies in the interval
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0%
(0, 1)
0%
(1, 2)
0%
(2, 3)
0%
(1, 3)
In the interval [0, 1], the function
x
2
-
x
+ 1 is
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0%
increasing
0%
decreasing
0%
neither increasing nor decreasing
0%
do not say anything
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0%
(0, ∞)
0%
(-∞, 0)
0%
(-∞, ∞)
0%
None of these
The function
f
(
x
) = 1 -
x
3
Report Question
0%
increases everwhere
0%
decreases in (0, ∞ )
0%
increases in (0, ∞)
0%
None of the above
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