Free Objective Test 02 Practice Test - 11th and 12th
Question 1
The integrating factor of the differential equation dydx=y tan x−y2 sec x is
[MP PET 1995; Pb. CET 2002]
tan x
secx
-sec x
cot x
SOLUTION
Solution : B
The differential equation
is dydx−y tan x=−y2 sec xI.F.=e−∫ tan x dx
This is Bernoulli's equation i.e. reducible to
linear equation.
Dividing the equation by y2, we get
1y2dydx−1y tan x=−sec x............(i)
Put 1y=y⇒−1y2dydx=dYdx
Equation (i) reduces to −dydx=−y tan x=−sec x⇒dYdx+Y tan x=sec x, Which is a linear equation
Hence I.F.=e−∫ tan x dx=sec x.
Question 2
The orthogonal trajectories of the family of curves an−1y=xn are given by
SOLUTION
Solution : B
Differentiating, we have an−1dydx=nxn−1⇒an−1=nxn−1dxdy
Putting this value in the given equation, we havenxn−1dxdyy=xn
Replacing dydx by −dxdy we have ny=−xdxdy
⇒nydy+xdx=0⇒ny2+x2= constant. Which is the required family of orthogonal trajectories.
Question 3
The order of the differential equation whose general solution is given by y=C1e2x+C2+C3ex+C4sin(x+C5) is
SOLUTION
Solution : B
y=C1e2x+C2+C3ex+C4sin(x+C5)=C1.eC2e2x+C3ex+C4(sin x cos C5+cos x sin C5)=Ae2x+C3ex+B sin x+D cos x
Here, A=C1eC2,B=C4cosC5,D=C4sinC5
(Since equation consists of four arbitrary constants)
∴ order of differential equation = 4.
Question 4
The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively
SOLUTION
Solution : 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.
Question 5
If y1(x) is a solution of the differential equation dydx−f(x)y=0, then a solution of the differential equation dydx+f(x)y=r(x)
SOLUTION
Solution : A
dydx−f(x).y=0dyy=f(x)dx
ln y=∫f(x)dx
y1(x)=e∫f(x)dx Then for given equation I.F = e∫f(x)dx
Hence Solution y.y1(x)=∫r(x).y1(x)dx
y=1y1(x)∫r(x).y1(x)dx
Question 6
If xdydx=y(log y−log x+1), then the solution of the equation is
SOLUTION
Solution : C
dydx=yx(logyx+1)
Put y=vx⇒dydx=v+xdvdx
∴v+xdvdx=v log v+v⇒1vlogvdv=1xdx⇒∫1v log vdv=∫1xdx⇒log(log v)=logx+log c
⇒logyx=cx
Question 7
The solution of dydx+xy=x2 is
SOLUTION
Solution : B
dxdy+xy=x2⇒x−2dxdy+x−1y=1
Put x−1=t.Then−x−2dxdy=dtdy⇒x−2dxdy=−dtdy∴−dtdy+ty=1⇒dtdy−1yt=−1
It is linear in t. Here P=−1y.Q=−1
I.F=e∫(1y)dy=e−logy=1y
∴ Solution is t(1y)=∫(−1)1ydy=−logy+c⇒t=−y logy+cy⇒x−1cy−y logy.
Question 8
The solution of the differential equation (x2sin3y−y2cosx)dx+(x3cosysin2y−2ysinx)dy=0 is
SOLUTION
Solution : A
(x2sin3y−y2cos x)dx+(x3 cos y sin2 y−2y sin x)dy=0dydx=y2cosx−x2sin3yx3 cos y sin2−2y sinx(x3 cos y sin2y−2y sin x)dy=(y2 cos x−x2 sin3y)dx=0(x33dsin3 y−sin dy2)−sin3yd(x33)+y2d sin x=0
x33d sin2y+sin3yd(x33)−(sin dy2+y2 d sin x)
d(x33sin3y)−d(y2sinx)=0x33sin3y−y2sinx=c
Question 9
The solution of dydx=yx+tanyx is
SOLUTION
Solution : D
Put y=vx.Thendydx=v+xdvdx
Given equation is dydx=vx+tanyx⇒v+x tan v⇒cot v dv=dxx⇒log sin v=log x+logc
⇒sin v=cx⇒sin(yx=cx)
Question 10
The curves satisfying the differential equation (1−x2)y1+xy=ax are
SOLUTION
Solution : A
The given equation is linear DE and can be written as
dydx+x1−x2y=ax1−x2
Its integrating factor is e∫x1−x2dx=e−(12)ln(1−x2)=1√1−x2 [−1<x<1] and if x2>1 then I.F.=1√x2−1
ddx(y1√1−x2)=ax(1−x2)32=−12−2ax(1−x2)32
⇒y1√1−x2=a√1−x2+C⇒y=a+C√1−x2
⇒(y−a)2=C2(1−x2)⇒(y−a)2+C2x2=C2
Thus, if −1<x<1 the given equation represents an ellipse.
If x2>1 then the solution is of the form −(y−a)2+C2x2=C2 which represents a hyperbola.
Question 11
The solution of dydx+1=ex+y is
SOLUTION
Solution : A
x+y=z⇒1+dydx=dzdxdydx+1=ex+y⇒dydx=e−zdz=dx⇒∫e−zdz=∫dx⇒−e−z=x+c⇒e−(x+y)+x+c=0.
Question 12
Solution to the differential equation x+x33!+x55!+.....1+x22!+x44!+.....=dx−dydx+dy is
SOLUTION
Solution : B
Applying C and D, we get
dydx=e−xex=e−2x⇒2y=−e−2x+C
or 2ye2x=C.e2x−1.
Question 13
A curve is such that the mid point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets y-axis, lies on the line y = x. If the curve passes through (1, 0), then the curve is
SOLUTION
Solution : C
The point on y-axis is (0,y−xdydx)
According to given condition,
x2=y−x2dydx⇒dydx=2yx−1
Putting yx=v we get
xdvdx=v−1⇒ln∣∣yx−1∣∣=ln|x|+c⇒1−yx=x (as f(1)=0).
Question 14
The family of curves passing through (0,0) and satisfying the differential equation y′′y′=1 (where y′=dydx and y′′=d2ydx2) is
SOLUTION
Solution : D
dpdx=p(where p=dydx)
lnp=x+c⇒p=ex+c
dydx=kex⇒y=kex+λ
Satisfying (0,0), So λ=−k
y=k(ex−1)
Question 15
S1: The differential equation of parabolas having their vertices at the origin and foci on the x-axis is an equation whose variables are separable
S2: The differential equation of the straight lines which are at a fixed distance p from the origin is an equation of degree 2
S3: The differential equation of all conics whose both axes coincide with the axes of coordinates is an equation of order 2
SOLUTION
Solution : A
S1 -Equation of parabola is y2=±4ax
2ydydx=±ra
D.E of parabola ⇒y2=2yxdydx
2dyy=dxx
Which is variable seperable
S2 -Equation of line which is fixed distance. P from origin can be equation of tangent to circle
s2+y2=p2
Line is y=mx+p√1+m2(m=dydx)
(y−xdydx)2=P(1+(dydx)2)
So, degree is 2
S3 -Equation of conic whose both axis co-incide with co-ordinate axis is ax2+by2=1
As there are two constants, so order of D.E is 2