Eamcet - Maths - Applications Of Derivatives

An aeroplane flying with uniform speed horizontally one kilometer above the ground is observed at an elevation of 600. After 10 seconds if the elevation is observed to be 300, then the speed of the plane (in km/hr) is:

A.  240 / √3

B.  200√3

C.  240√3

D.  120 / √3

Observe the following statements :A : Integrating factor of (dy/dx) + y = x2 is exR :Integrating factor of (dy/dx) + P(x) y = Q(x) is e∫p(x)dx. Then the true statement among the following is

A.  A is true, R is false

B.  A is false, R is true

C.  A is true, R is true, R => A

D.  A is false, R is false

The circumference of a circle is measured as 56 cm with an error 0,02 cm. The percentage error in its area is

A.  1/7

B.  1/28

C.  1/14

D.  1/56

The angle between the curves xy=4 and x2-y2=15 at the point (-4, -1) is

A.  600

B.  900

C.   tan-(1/2)

D.   tan-(5/7)

If the rate of change in y=2x3+3x2-30x+7 is 6 times the rate of change in x, then x=

A.  1, 5

B.  1, -5

C.  2, 3

D.  2, -3

If x is real, the maximum value of 3x2+9x+17/3x2-9x+17 is

A.  1

B.   17/7

C.  ¼

D.  41

If there is a possible error of 0.02 cm in the measurement of the diameter of a spare then the possible percentage error in its volume when the radius 10 cm is

A.  0.1

B.  0.2

C.  0.3

D.  0.4

The maximum value of x4+3x3-2x2-9x+6 is

A.  11

B.  3/8

C.  3

D.  12

The points on the curve y=x2+√1-x2 at which the tangent is perpendicular to x-axis are

A.  (1, 1) only

B.  (±1, 1)

C.  (1, ±1)

D.  (-1, 1) only

If an error of 0.02cm is made while   measuring the radius 10cm of a sphere, then the error in the volume is

A.  8 π cubic.cms

B.  80 π cubic.cms

C.  0.06 π cubic.cms

D.  16 π cubic.cms

The stationary points of 2x3-9x2-24x+16 are

A.  (-1, 29), (4, -96)

B.  (1, 29)

C.   (1, 1)

D.  None

If y= ax+b/(x-1)(x-4) has a maximum value at the point (2, -1) then

A.  A=10, b=20

B.  a=1, b=0

C.   a=5, b=5

D.  None

The approximate percentage reduction in the volume of a cube of ice, if each side of the ice cube to reduced by 0.7% due to melting to:

A.  2.1%

B.  2.5%

C.  3.2%

D.  3.3%

The equation of the normal to the curve 3y2=4x+1 at (1, 2) is

A.  3x+y+5=0

B.  3x+y-5=0

C.  3x-y+5=0

D.  3x-y-5=0

The angle between the curves y=x2 and y=4-x2 is

A.   tan-1(6/13)

B.  tan-1(3/4)

C.  tan-1(5/√3)

D.  tan-1(4√2/7)

If the normal to the curve x3-y2 =0 at (m2, -m3) is y=mx-2m3, then the value of m2  is

A.  1/3

B.  1/6

C.  2/3

D.  -2/3

The curves y=x2-1, y=8x-x2-9 touch each other at the point (2, 3). The equation of the common normal is

A.  4x+y+5=0

B.  4x-y-5=0

C.  x+4y-14=0

D.  x-4y+14=0

If the subnormal of the curve xyn=an+1 is constant, then the value of n is

A.  1

B.  -1

C.  5

D.   -2

The equation of tangent to the curve y=x+9/x+5 so that is passes through the origin is

A.  x+y=0

B.   x-y=0

C.  x+y=1

D.  x-y=1

If θ is the angle between the curves y=sin x, y =cosx at x=π/4 then tan θ=

A.  2

B.  √2

C.  1/√2

D.  2√2

A curve passes through the point (2, 0) and the slope of the tangent at any point is x2-2x for all values of x. The point of maximum or donation the curve is

A.   (0, 2/3)

B.   (0, 4/3)

C.  (0, 1/3)

D.  (0, 5/3)

The function f(x) = tan x has

A.  No max points

B.  No min points

C.  Neither max nor min points

D.  None

f(x)= x-1/x is

A.  Increasing in R

B.  Decreasing in R+

C.  Not decreasing

D.   Not increasing

The function f(x)=cot-1 x+x increases in the interval

A.  (1, ∞)

B.   (-1, ∞)

C.  (-∞,∞)

D.  (0, ∞)

The point P in the first quadrant of the ellipse x2/8+y2/18=1 so that the area of the triangle formed by the tangent at P and the coordinate axes is least

A.   (2, 3)

B.  (√8, 0)

C.   (√18, 0)

D.  None

A rectangular sheet of dimensions 30 cm * 80 cm four equal squares of side x cm are removed at the corners, and the sides are then turned up so as to form an open rectangle box. The value of x, so that the value of the box is the greatest is

A.  20/3

B.   10/3

C.  15/2

D.  5

The displacement of a body of mass 100kg in a rectilinear motion is given by the formula s=2t2+3t+1. The K.E of the body 5 sec after the start is

A.  56000

B.  26450

C.  20000

D.  none

If the slope of the tangent to the curve xy+ax=by at (1, 1) is 2, then (a, b)

A.  (0, 1)

B.  (1, 2)

C.  (-1, 2)

D.  (1, -2)

The function f(x)=a sin x+1/3x has maximum value at x =π/3. The value of a is

A.   3

B.   1/3

C.  2

D.  1/2

The portion of the tangent drawn at any point on x2/3+y2/3=a2/3 (a>0), except the points on the coordinate axes, included between the coordinate axes is

A.  a

B.  2a

C.  a2/3

D.  a2