Solution of the differential equationcos x dy = y (sin x - y) d

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 Multiple Choice QuestionsMultiple Choice Questions

11.

The area (in square units) bounded by the curves y = straight y space equals space square root of straight x comma 2y-x+3 =0, X -axis and lying in the first quadrant is 

  • 9

  • 36

  • 18

  • 18

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12.

The intercepts on X- axis made by tangents to the curve, straight y space equals space integral subscript 0 superscript straight x vertical line straight t vertical line space dt comma space straight x space straight epsilon space straight R which are parallel to the line y =2x, are equal to

  • ±1

  • ±2

  • ±3

  • ±3

160 Views

13.

Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equationd V(t)/dt = - k(T - t), where k > 0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is

  • straight T squared minus 1 over straight k
  • straight I space minus KT squared over 2
  • straight I space minus space kT squared over 2
  • straight I space minus space kT squared over 2
290 Views

14.

For x ∈, (0, 5π/2) define f(x). Then f (x) = integral subscript 0 superscript straight x space square root of straight t space sin space straight t space dt has

  • local maximum at π and 2π.

  • local minimum at π and 2π

  • local minimum at π and the local maximum at 2π.

  • local minimum at π and the local maximum at 2π.

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15.

The area of the region enclosed by the curves y = x, x = e, y =1/x and the positive x-axis is

  • 1/2 square units

  • 1 square units

  • 3/2 square units

  • 3/2 square units

139 Views

16.

The area bounded by the curve y = cos x and y = sin x between the ordinates x = 0 and x = 3π/2 is 

  • left parenthesis 4 minus square root of 2 minus 2 right parenthesis space sq space units
  • left parenthesis 4 minus square root of 2 plus 2 right parenthesis space sq space units
  • left parenthesis 4 minus square root of 2 minus 1 right parenthesis space sq space units
  • left parenthesis 4 minus square root of 2 minus 1 right parenthesis space sq space units
444 Views

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17.

Solution of the differential equation
cos x dy = y (sin x - y) dx, 0 < x < π/2, is 

  • sec x = (tan x + C ) y 

  •  y sec x = tan x + C

  •  y tan x = sec x + C 

  •  y tan x = sec x + C 


A.

sec x = (tan x + C ) y 

since cos xdy = y sin x dx - ydx
rightwards double arrow space 1 over straight y squared space dy over dx minus 1 over straight y space tan space straight x space equals space minus space sec space straight x
Put space minus 1 over straight y space equals space straight z
rightwards double arrow 1 over straight y squared dy over dx space equals space dz over dx
rightwards double arrow space dz over dx space plus space left parenthesis tan space straight x right parenthesis straight z space equals space minus space sec space straight x
This is linear differential equation,
Therefore,
IF space equals space straight e to the power of integral tan space straight x space dx end exponent equals space straight e to the power of log space sec space straight x space end exponent space equals space sec space straight x
Hence comma space the space solution space is
straight z. space left parenthesis sec space straight x right parenthesis space equals space integral negative sec space straight x. space sec space straight x space dx space plus space straight C subscript 1
rightwards double arrow space minus 1 over straight y space sec space straight x space equals space minus space tan space straight x space plus space straight C subscript 1
rightwards double arrow space sec space straight x space equals space straight y space left parenthesis space tan space straight x space plus space straight C right parenthesis

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18.

The area (in sq. units) of the region
{(x, y} : x  ≥ 0, x + y ≤ 3, x2 ≥ 4y and y ≤ 1 +√x}

  • 5/2

  • 59/12

  • 3/2

  • 3/2

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19.

The area of the region bounded by the parabola (y – 2)2 = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is 

  • 3

  • 6

  • 9

  • 9

212 Views

20.

The area of the plane region bounded by the curves x + 2y2= 0 and x + 3y2= 1 is equal to 

  • 5/3

  • 1/3

  • 2/3

  • 2/3

163 Views

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