Mathematics - Application of Derivatives
Exam Duration: 45 Mins Total Questions : 30
The time T of oscillation of a simple pendulum of length l is given by \(T = 2\pi\sqrt{1\over g}\). The percentage error in T corresponding to an error of 2% in the value of l is
- (a)
2%
- (b)
1%
- (c)
3%
- (d)
1.2%
The number of solution of the equation \(f(x)=-2x^3+21x^2-60x+41,\) then
- (a)
f(x) is decreasing in \((-\infty, 1)\)
- (b)
f(x) is decreasing in \((-\infty, 2)\)
- (c)
f(x) is increasing in \((-\infty, 1)\)
- (d)
f(x) is increasing in \((-\infty, 2)\)
The function \(f(x)=x^2+{\lambda\over x}\) has a
- (a)
minimum at x = 2, if \(\lambda = 16\)
- (b)
minimum at x = 2, if \(\lambda = 15\)
- (c)
maximum for all real value of \(\lambda\)
- (d)
maximum at x = 2, if \(\lambda=4\)
If \(f(x)=\int^x_0{(e^t-1)}(t-1)^2(t+1)^3\ dt\), then f(x) has
- (a)
maximum at x = 5
- (b)
minimum at x = 0
- (c)
minimum at x = -1
- (d)
maximum at x = -2
If m is the slope of a tangent to the curve \(e^y=1+x^2, then\)
- (a)
\(|m| \le 1\)
- (b)
m > -1
- (c)
m > 1
- (d)
|m| > 1
The equation of the tangent to the curve \(y=e^{ -|x| }\) at the point where the curve cuts the line x = 1 is
- (a)
x + y = e
- (b)
e(x + y) = 1
- (c)
y + q; = 1
- (d)
none of these
If the tangent at (1, 1) on y2 = x(2 - X)2 meets the curve again at.P, then P is
- (a)
(4,4)
- (b)
(-1,2)
- (c)
(9/4,3/8)
- (d)
none of these
The curve y - exy + x = 0 has a vertical tangent at the point
- (a)
(1, 1)
- (b)
at no point
- (c)
(0,1)
- (d)
(1, 0)
The slope of the tangent to the curve y \(\int _{ x }^{ x^{ 2 } }{ cos } ^{ -1 }t^{ 2 }dt\quad x=\frac { 1 }{ \sqrt { 2 } } \)
- (a)
\(\left( \frac { \sqrt [ 4 ]{ 8 } }{ 2 } -\frac { 3 }{ 4 } \right) \pi \)
- (b)
\(\left( \frac { \sqrt [ 4 ]{ 8 } }{ 2 } -\frac { 1 }{ 4 } \right) \pi \)
- (c)
\(\left( \frac { \sqrt [ 5 ]{ 8 } }{ 4 } -\frac { 1 }{ 3 } \right) \pi \)
- (d)
none of these
Apoint on the ellipse \(4x^{ 2 }+9y^{ 2 }=36\) where the tangent is equally inclined to the axes is
- (a)
\(\left( \frac { 9 }{ \sqrt { 13 } } ,\frac { 4 }{ \sqrt { 13 } } \right) \)
- (b)
\(\left( -\frac { 9 }{ \sqrt { 13 } } ,\frac { 4 }{ \sqrt { 13 } } \right) \)
- (c)
\(\left( \frac { 9 }{ \sqrt { 13 } } -\frac { 4 }{ \sqrt { 13 } } \right) \)
- (d)
none of these
The rate of change of the surface area of a sphere of radius r, when the radius is increasing at the rate of 2cm/s is proportional
- (a)
1/r
- (b)
1/r2
- (c)
r
- (d)
r2
the function f(x)=x2-2x is strict decreasing in the interval
- (a)
(-∞, 1]
- (b)
[1, ∞)
- (c)
R
- (d)
None of these
The equation of the normal to the curve y=sin x at (0,0) is
- (a)
x=0
- (b)
x+y=0
- (c)
y=0
- (d)
x-y=0
Find a point on the curve y=(x-2)2 at which the tangent is parallel to the chord joining the points (2,0) and (4,4)
- (a)
(3,1)
- (b)
(4,1)
- (c)
(6,1)
- (d)
(5,1)
If an error k% is made in measuring the radius of a sphere, then percentage error in its volume is
- (a)
k%
- (b)
3k%
- (c)
2k%
- (d)
k/3 %
Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%
- (a)
1.06x3m3
- (b)
1.26x3m3
- (c)
2.50x3m3
- (d)
0.06x3m3
Find the approximate value of f(3.02), where f(x)=3x^2+5x+3
- (a)
45.46
- (b)
45.47
- (c)
44.76
- (d)
44.46
Find the maximum value of the following functions. f(x)=sin(sinx) for all x∈R
- (a)
-sin1
- (b)
sin6
- (c)
sin1
- (d)
-sin3
Find the local minimum value of the function
\(f(x)=sin^4x+cos^4x, 0<x<{\pi\over2}\)
- (a)
1/√2
- (b)
1/2
- (c)
√3/2
- (d)
0
Find the maximum profit that a company can make if the profit function is given by P(x)=41+24x-18x2
- (a)
25
- (b)
43
- (c)
62
- (d)
49
The function f(x)=x5-5x4+5x3-1 has
- (a)
one minima and two maxima
- (b)
two minima and one maxima
- (c)
two minima and two maxima
- (d)
one minima and one maxima
Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are
- (a)
(10,10)
- (b)
(12,8)
- (c)
(15,5)
- (d)
none of these
Amongst all pairs of positive numbers with product 256, find those whose sum is the least
- (a)
16,14
- (b)
16,16
- (c)
64,4
- (d)
32,8
The curve y = x1/5 at (0, 0) has
- (a)
a vertical tangent (parallel to y-axis)
- (b)
a horizontal tangent (parallel to x-axis)
- (c)
an oblique tangent
- (d)
no tangent
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is
- (a)
1
- (b)
0
- (c)
-6
- (d)
6
Let the f: R ⟶ R be defined by f(x) = 2x + cosx, then f
- (a)
has a minimum at x = π
- (b)
has a maximum, at x = 0
- (c)
is a decreasing function
- (d)
is an increasing function
Statement-I: The ordinate of a point describing the circle x2 + y2 = 25 decreases at the rate of 1.5 cm/s. The rate of change of the abscissa of the point when
ordinate equals 4 cm is 2 cm/s.
Statement-II: xdx + ydy = 0
- (a)
If both Statement-I and Statement-If are true and Staternent-Il is the correct explanation of Statement-1.
- (b)
If both Statement-I and Staternent-Il are true but Statement-Il is not the correct
explanation of Statement-1. - (c)
If Staternent-I is true but Staternent-Il is false
- (d)
If Statement-I is false and Staternent-Il is true.