Lakshya Education MCQs

Question: The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively
Options:
A.1,2
B.3,2
C.2,3
D.2,1
Answer: Option A
: A

Equation of family of parabolas with x-axis as axis is y2=4a(x+α) where a,α are two arbitrary constants. So differential equation is of order 2 and degree 1.

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Question 1. The solution of dydx=yx+tanyx is
  1.    y sin(yx)=cx
  2.    y sin(yx)=cy
  3.    sin(xy)=cx
  4.    sin(yx)=cx
Answer: Option D
: D

Put y=vx.Thendydx=v+xdvdx
Given equation is dydx=vx+tanyxv+xtanvcotvdv=dxxlogsinv=logx+logc
sinv=cxsin(yx=cx)
Question 2. If y1(x) is a solution of the differential equation dydxf(x)y=0, then a solution of the differential equation dydx+f(x)y=r(x)
  1.    1y1(x)∫r(x)y1(x)dx
  2.    y1(x)∫r(x)y1(x)dx
  3.    ∫r(x)y1(x)dx
  4.    None of these
Answer: Option A
: A

dydxf(x).y=0dyy=f(x)dx
ln y=f(x)dx
y1(x)=ef(x)dxThen for given equation I.F = ef(x)dx
Hence Solution y.y1(x)=r(x).y1(x)dx
y=1y1(x)r(x).y1(x)dx
Question 3. If xdydx=y(log ylog x+1), then the solution of the equation is
  1.    y log(xy)=cx
  2.    x log(yx)=cy
  3.    log(yx)=cx
  4.    log(xy)=cx
Answer: Option C
: C

dydx=yx(logyx+1)
Put y=vxdydx=v+xdvdx
v+xdvdx=vlogv+v1vlogvdv=1xdx1vlogvdv=1xdxlog(logv)=logx+logc
logyx=cx
Question 4. The solution of the differential equation (x2sin3yy2cosx)dx+(x3cosysin2y2ysinx)dy=0 is 
  1.    x3sin3y=3y2sinx+C
  2.    x3sin3y+3y2sinx=C  
  3.    x2sin3y+y3sinx=C  
  4.    2x2siny+y2sinx=C
Answer: Option A
: A

(x2sin3yy2cosx)dx+(x3cosysin2y2ysinx)dy=0dydx=y2cosxx2sin3yx3cosysin22ysinx(x3cosysin2y2ysinx)dy=(y2cosxx2sin3y)dx=0(x33dsin3ysindy2)sin3yd(x33)+y2dsinx=0
x33dsin2y+sin3yd(x33)(sindy2+y2dsinx)
d(x33sin3y)d(y2sinx)=0x33sin3yy2sinx=c
Question 5. The order of the differential equation whose general solution is given by y=C1e2x+C2+C3ex+C4sin(x+C5) is 
  1.    5
  2.    4
  3.    3
  4.    2
Answer: Option B
: B

y=C1e2x+C2+C3ex+C4sin(x+C5)=C1.eC2e2x+C3ex+C4(sinxcosC5+cosxsinC5)=Ae2x+C3ex+Bsinx+Dcosx
Here, A=C1eC2,B=C4cosC5,D=C4sinC5
(Since equation consists of four arbitrary constants)
order of differential equation = 4.

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