CLASS : XII | BATCH: 2 YR JEE 2023

TOPIC: Differential Equations DPP NO. - 6

1. The order of the differential equation whose general solution is given by y = (C1 + C2) cos (x + C3) – C4 e x  C5 ,

where C1, C2,C3 , C4, C5 are arbitrary constants, is-

[A] 5 [B] 4 [C] 3 [D] 2
2. The differential equation of all circles in the first quadrant which touch the coordinate axes is of order-
[A] 1 [B] 2 [C] 3 [D] None of these
3. The differential equations of all circles passing through origin and having their centres on the x-axis is

dy y2  x2 dy y2  x2 d2 y y2  x2 dy y2  x2
[A] = [B] = [C] = [D] =
dx 2xy dx 2x dx 2 2 xy dx 2 xy

dy
4. The solution of the equation + y tan x = xm cos x is-
dx
[A] (m + 1) y = xm+1 cos x + c (m+1) cos x [B] my = (xm + c) cos x
[C] y = (xm+1 + c) cos x [D] None of these
5. The solution of the differential equation (sin x + cos x) dy + (cos x – sin x)dx = 0 is-
[A] ex (sin x + cos x) + c = 0 [B] ey (sin x + cos x) = c
[C] ey (cos x – sin x) = c [D] ey (sin x – cos x) = c

dy 3x2 sin 2 x
6. The solution of the differential equation + y = is -
dx 1  x3 1  x3
1 1
[A] y (1 + x3) = x + sin 2x + c [B] y (1 + x3) = cx + sin 2x
2 2

1 x 1
[C] y (1 + x3) = cx – sin 2x [D] y (1 + x3) = – sin 2x + c
2 2 4

y y
7. The slope of the tangent at (x, y) to a curve passing through 1,   is given by – cos2   , then the equation
x x
 4
of the curve is-

  e    x 
[A] y = tan–1  log    [B] y = x tan–1  log   
  x    e 

  e 
[C] y = x tan–1  log    [D] None of these
  x 
8. The Solution of the equation (1– x2) dy + xy dx = xy2 dx is-
[A] (y – 1)2 (1 – x2) = 0 [B] (y – 1)2 (1 – x2) = c2 y2
[C] (y – 1)2 (1 + x2) = c2 y2 [D] None of these

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 dy e x (sin2 x  sin 2x )
9. The solution of = is-
dx y (2 log y  1)
[A] y2 (log y) – ex sin2 x + c = 0 [B] y2 (log y) – ex cos2 x + c = 0

[C] y2 (log y) + ex cos2 x + c = 0 [D] None of these

10. The solution of (cosec x log y)dy + (x2y) dx = 0 is -
2
logy  logy 
[A] + (2 – x2) cos x + 2 sin x = c [B]   + (2 – x2) cos x + 2x sin x = c
2  2 

(log y) 2
[C] + (2 – x2) cos x + 2x sin x = c [D] None of these
2

dy (1  y 2 )(1  x  x2 )
11. The solution of the differential equation xy = is -
dx (1  x2 )

1 1
[A] log (1 + y2) = log x – tan–1 x + c [B] log (1 + y2) = log x + tan–1x + c
2 2
[C] log (1 + y2) = log x – tan–1x + c [D] log (1 + y2) = log x + tan–1 x + c

12. The solution of (x 1  y 2 )dx + (y 1  x2 )dy = 0 is -

[A] 1  x2  1  y 2 = c [B] 1  x2  1  y 2 = c
[C] (1 + x2)3/2 + (1 + y2)3/2 = c [D] None of these
13. The solution of the differential equation cos y log (sec x + tan x) dx = cos x log (sec y + tan y ) dy is
[A] sec2 x + sec2 y =c [B] sec x + sec y = c [C] sec x – sec y = c [D] None of these
14. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the
point (4, 3). The equation of the curve is-
[A] x2 = y + 5 [B] y2 = x – 5 [C] y2 = x + 5 [D] x2 = y – 5

dy
15. The solution of the differential equation = ex – y + x2e–y is
dx

1 1 3 1 3
[A] y = ex – y + x2e–y + c [B] ey – ex = x3 + c [C] ex + ey = x +c [D] ex – ey = x +c
3 3 3

dy y  log y  1
16. The solution of the equation =   is-
dx x  x 

y y
[A] log   = cx [B] = log y + c [C] y = log y + 1 [D] y = xy + c
x x

dy xy
17. The solution of the differential equation = 2 is-
dx x  y2

2
/ y2 2 2
[A] ay2 = e x [C] y = e x + e y + c
2
[B] ay = ex/y [D] y = e x + y2 + c

dy
18. Solution of the differential equation sin = a with y (0) = 1 is-
dx

y  1 y  1 1  y  y
[A] sin–1 = a [B] sin = a [C] sin
x x 1  x  = a [D] sin
x  1 = a

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19. Which of the following differential equations has the same order and degree-

6 4 2
d4 y  dy   d3 y   dy 
[A]   + 5y = ex    1   + 5y = x8
4 + 8 [B] 5  3  + 8
dx  dx   dx   dx 
2/3
  dy  3  d3 y dy  dy 
2

[C] 1     = 4 [D] y = x2 + 1  
  dx   dx 3 dx  dx 

20. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point
(4, 3). The equation of the curve is -
[A] x2 = y + 5 [B] y2 = x – 5 [C] y2 = x + 5 [D] x2 = y – 5

ANSWERS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
C A D A B D C B A C B A D C B A A B C C

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