12th Maths Monthly Test - 1 ( Ordinary Differential Equations )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Ordinary Differential Equations )-Aug 2020

Time : 2.40 Hours

Part - A

Multiple Choice Question - Answer All Question

The order and degree of y'+(y")² =(x+y")² are _________.

Integrating factor of the differential equation \(\frac{dy}{dx}=\frac{x+y+1}{x+1}\) is

\(\frac{1}{x+1}\)
x+1
\(\frac{1}{\sqrt{x+1}}\)
\({\sqrt{x+1}}\)

If sin x is the integrating factor of the linear differential equation \(\frac{dy}{dx}\) + Py = Q, Then P is

log sin x
cos x
tan x
cot x

The integrating factor of the differential equation \(\frac{dy}{dx}+y=\frac{1+y}{λ}\) is

\(\frac{x}{e^λ}\)
\(\frac{e^λ}{x}\)
λe^x
e^x

The solution of \(\frac{dy}{dx}\)+y cot x=sin 2x is

y sin x=\(\frac{2}{3}\) sin³ x+c
y sec x=\(\frac{x^2}{2}\)+c
y sin x =c+x
2y sin x=sin x-\(\frac{sin \space 3x}{3}\)+c

The differential equation corresponding to xy=c² where c is an arbitrary constant is ________.

xy"+x=0
y"=0
xy'+y=0
xy"-x=0

The integrating factor of the differential equation \(\frac{dy}{dx}\)+P(x)y=Q(x)is x, then P(x)

x
\(\frac{x^2}{2}\)
\(\frac{1}{x}\)
\(\frac{1}{x^2}\)

The solution of the differential equation \(\frac{dy}{dx}=2xy\) is
\(y=Ce^{x^2}\)

y= 2x² +C

\(y=Ce^{-x^2}\) + C

y = x² +C
The order and degree of the differential equation \(\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{1/3}+ x^{\frac{1}{4}}\) = 0 are respectively
2,3

3,3

2,6

2,4

The solution of the differential equation \(\frac{dy}{dx}+\frac{1}{\sqrt{1-x^2}}=0\)

y + sin^-1 x = c
x + sin^-1 y = 0
y²+2 sin^-1 x = c
x²+ 2 sin^-1y= 0

The general solution of the differential equation log \(\left(\frac{dy}{dx}\right)\) = x + y is

e^x+e^y =C
e^x+e^-y =C
e^-x+e^y =C
e^-x+e^-y =C

The solution of sec²x tan y dx+sec²y tan x dy=0 is

tan x+tan y =c
sec x+sec y=c
tan x tan y=c
sec x-sec y =c

The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}\)= 3x² and it passes through (-1,1). Then the equation of the curve is

y=x³+2
y=3x²+4
y=3x⁴+4
y=x³+5

If the solution of the differential equation \(\frac{dy}{dx}=\frac{ax+3}{2y+f}\) represents a circle, then the value of a is

2
-2
1
-1

The number of arbitrary constants in the general solutions of order n and n +1are respectively

n-1,n
n,n+1
n+1,n+2
n+1,n

The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is

Part - B

Generic 2 - Answer All Question

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes
are parallel to the x-axis.
Solve: \(\frac{dy}{dx}=\frac{2x}{x^2+1}\)

A differential equation, determine its order, degree (if exists)

\(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=xsin\left(\frac{d^2y}{dx^2}\right)\)

Determine the order and degree (if exists) of the following differential equations:

\(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

A differential equation, determine its order, degree (if exists)

\(x=e^{xy}\left(\frac{dy}{dx}\right)\)

Find value of m so that the function y = e^mx is a solution of the given differential equation.
y''− 5y' + 6y = 0

solve:xdy+ydx=xydx

If cos x is an I.F of \(\frac{dy}{dx}+py=Q,\)then p

1) tan x
2) -tan x

3) \(p=\frac{f ^1(x)}{f (x)}\) where f(x) =cos x

4) \(\frac{\frac{d}{dx}(cos \space x)}{cos \space x}\)

A differential equation, determine its order, degree (if exists)

\(\sqrt{\frac{dy}{dx}}-4\frac{dy}{dx}-7x=0\)

Show that each of the following expressions is a solution of the corresponding given differential
equation.y = ae^x + be^-x ; y − y = 0

If \(\frac{dy}{dx}=\frac{x-y}{x+y}\) then

1) xdy+y dx=x dx+y dy

2) ∫ d(xy) = ∫ xdx + ∫ ydy

3) x² -y² +2xy=c
4) x² -y² -2xy=c

Form the D.E corresponding to y=e^mx by eliminating 'm'.
Form the differential equation satisfied by are the straight lines in my-plane.

Find the order and degree of \(y+\frac{dy}{dx}=\frac{1}{4}\)∫ ydx

Solve the differential equation:

\(\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}\)

Solve:\(\frac{dy}{dx}=1+e^{x-y}\)

A differential equation, determine its order, degree (if exists)

\(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)

Part - C

Generic 3 - Answer All Question

Solve: \(\frac{dy}{dx}+y=cos x\)

Form the D.E to y² =a(b-x)(b+x) by eliminating a and b as its parameters.

Find the differential equation of the family of circles passing through the origin and having their
centres on the x -axis.
Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x² y"+ xy'+y= 0.

Solve the Linear differential equation:
cos x\(\frac{dy}{dx}\)+y sin x=1

Show that the differential equation representing the family of curves y² = 2a\(\left(x+a\frac{2}{3}\right)\)where a is a
positive parameter, is \(\left(y^2-2xy\frac{2}{3}\right)^3\) =8 \(\left(y\frac{dy}{dx}\right)^5\).

Solve : ydx+(x-y²)dy=0
Solve: \(X\frac{dy}{dx}+2y=x^2\)

Solve : \(\frac{dy}{dx}+2y^2=0\),y=(1) and find the corresponding solution of the curve.

Express the physical statement in the form of the differential equation.
Radium decays at a rate proportional to the amount Q present.

Solve :(1+e²x)dy+(1+y²)e dx=0

Find the differential equation of the curve represented by xy = ae^x + be^-x + x² .

Part - D

Generic 5 - Answer All Question

Solve : (1+y²)(1+logx)dx+xdy=0, given that x=1,y=1.

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the
detective also measured the body temperature and found it to be 70^o F. Two hours later, the detective
measured the body temperature again and found it to be 60^o F. If the room temperature is 50^o F, and
assuming that the body temperature of the person before death was 98.6^o F, at what time did the
murder occur?
[log(2.43)=0.88789; log(0.5)=-0.69315]
Solve :\(\frac{dy}{dx}=-\frac{x+ycos}{1+sinx}\) .Also find the domain of the function.

Solve \(\frac{dy}{dx}\)=\(\frac{x − y + 5}{2 ( x − y ) + 7}\).

Solve: \(\frac{dv}{dx}\)+ 2y cot x = 3x²cosec²x

Solve the Linear differential equation: \(\left(y−e^{sin − 1}x\right)\)\(\frac{dx}{dy}+\sqrt{1-x^2}=0\)

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12th Maths Monthly Test - 1 ( Ordinary Differential Equations )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Ordinary Differential Equations )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

The order and degree of y'+(y")2 =(x+y")2 are _________.

Integrating factor of the differential equation \(\frac{dy}{dx}=\frac{x+y+1}{x+1}\) is

If sin x is the integrating factor of the linear differential equation \(\frac{dy}{dx}\) + Py = Q, Then P is

The integrating factor of the differential equation \(\frac{dy}{dx}+y=\frac{1+y}{λ}\) is

The solution of \(\frac{dy}{dx}\)+y cot x=sin 2x is

The differential equation corresponding to xy=c2 where c is an arbitrary constant is ________.

The integrating factor of the differential equation \(\frac{dy}{dx}\)+P(x)y=Q(x)is x, then P(x)

The solution of the differential equation \(\frac{dy}{dx}=2xy\) is

The order and degree of the differential equation \(\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{1/3}+ x^{\frac{1}{4}}\) = 0 are respectively

The solution of the differential equation \(\frac{dy}{dx}+\frac{1}{\sqrt{1-x^2}}=0\)

The general solution of the differential equation log \(\left(\frac{dy}{dx}\right)\) = x + y is

The solution of sec2x tan y dx+sec2y tan x dy=0 is

The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}\)= 3x2 and it passes through (-1,1). Then the equation of the curve is

If the solution of the differential equation \(\frac{dy}{dx}=\frac{ax+3}{2y+f}\) represents a circle, then the value of a is

The number of arbitrary constants in the general solutions of order n and n +1are respectively

The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is

Part - B

Generic 2 - Answer All Question

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis.

Solve: \(\frac{dy}{dx}=\frac{2x}{x^2+1}\)

A differential equation, determine its order, degree (if exists) \(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=xsin\left(\frac{d^2y}{dx^2}\right)\)

Determine the order and degree (if exists) of the following differential equations: \(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

A differential equation, determine its order, degree (if exists) \(x=e^{xy}\left(\frac{dy}{dx}\right)\)

Find value of m so that the function y = emx is a solution of the given differential equation. y''− 5y' + 6y = 0

solve:xdy+ydx=xydx

If cos x is an I.F of \(\frac{dy}{dx}+py=Q,\)then p 1) tan x 2) -tan x 3) \(p=\frac{f ^1(x)}{f (x)}\) where f(x) =cos x 4) \(\frac{\frac{d}{dx}(cos \space x)}{cos \space x}\)

A differential equation, determine its order, degree (if exists) \(\sqrt{\frac{dy}{dx}}-4\frac{dy}{dx}-7x=0\)

Show that each of the following expressions is a solution of the corresponding given differential equation.y = aex + be-x ; y − y = 0

If \(\frac{dy}{dx}=\frac{x-y}{x+y}\) then 1) xdy+y dx=x dx+y dy 2) ∫ d(xy) = ∫ xdx + ∫ ydy 3) x2 -y2 +2xy=c 4) x2 -y2 -2xy=c

Form the D.E corresponding to y=emx by eliminating 'm'.

Form the differential equation satisfied by are the straight lines in my-plane.

Find the order and degree of \(y+\frac{dy}{dx}=\frac{1}{4}\)∫ ydx

Solve the differential equation: \(\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}\)

Solve:\(\frac{dy}{dx}=1+e^{x-y}\)

A differential equation, determine its order, degree (if exists) \(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)

Part - C

Generic 3 - Answer All Question

Solve: \(\frac{dy}{dx}+y=cos x\)

Form the D.E to y2 =a(b-x)(b+x) by eliminating a and b as its parameters.

Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.

Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x2 y"+ xy'+y= 0.

Solve the Linear differential equation: cos x\(\frac{dy}{dx}\)+y sin x=1

Show that the differential equation representing the family of curves y2 = 2a\(\left(x+a\frac{2}{3}\right)\)where a is a positive parameter, is \(\left(y^2-2xy\frac{2}{3}\right)^3\) =8 \(\left(y\frac{dy}{dx}\right)^5\).

Solve : ydx+(x-y2)dy=0

Solve: \(X\frac{dy}{dx}+2y=x^2\)

Solve : \(\frac{dy}{dx}+2y^2=0\),y=(1) and find the corresponding solution of the curve.

Express the physical statement in the form of the differential equation. Radium decays at a rate proportional to the amount Q present.

Solve :(1+e2x)dy+(1+y2)e dx=0

Find the differential equation of the curve represented by xy = aex + be-x + x2 .

Part - D

Generic 5 - Answer All Question

Solve : (1+y2)(1+logx)dx+xdy=0, given that x=1,y=1.

Solve :\(\frac{dy}{dx}=-\frac{x+ycos}{1+sinx}\) .Also find the domain of the function.

Solve \(\frac{dy}{dx}\)=\(\frac{x − y + 5}{2 ( x − y ) + 7}\).

Solve: \(\frac{dv}{dx}\)+ 2y cot x = 3x2cosec2x

Solve the Linear differential equation: \(\left(y−e^{sin − 1}x\right)\)\(\frac{dx}{dy}+\sqrt{1-x^2}=0\)

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12th Maths Monthly Test - 2 ( Discrete Mathematics

The order and degree of y'+(y")² =(x+y")² are _________.

The differential equation corresponding to xy=c² where c is an arbitrary constant is ________.

The solution of sec²x tan y dx+sec²y tan x dy=0 is

The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}\)= 3x² and it passes through (-1,1). Then the equation of the curve is

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes
are parallel to the x-axis.

A differential equation, determine its order, degree (if exists)

\(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=xsin\left(\frac{d^2y}{dx^2}\right)\)

Determine the order and degree (if exists) of the following differential equations:

\(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

A differential equation, determine its order, degree (if exists)

\(x=e^{xy}\left(\frac{dy}{dx}\right)\)

Find value of m so that the function y = e^mx is a solution of the given differential equation.
y''− 5y' + 6y = 0

If cos x is an I.F of \(\frac{dy}{dx}+py=Q,\)then p

1) tan x
2) -tan x

3) \(p=\frac{f ^1(x)}{f (x)}\) where f(x) =cos x

4) \(\frac{\frac{d}{dx}(cos \space x)}{cos \space x}\)

A differential equation, determine its order, degree (if exists)

\(\sqrt{\frac{dy}{dx}}-4\frac{dy}{dx}-7x=0\)

Show that each of the following expressions is a solution of the corresponding given differential
equation.y = ae^x + be^-x ; y − y = 0

If \(\frac{dy}{dx}=\frac{x-y}{x+y}\) then

1) xdy+y dx=x dx+y dy

2) ∫ d(xy) = ∫ xdx + ∫ ydy

3) x² -y² +2xy=c
4) x² -y² -2xy=c

Form the D.E corresponding to y=e^mx by eliminating 'm'.

Solve the differential equation:

\(\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}\)

A differential equation, determine its order, degree (if exists)

\(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)

Form the D.E to y² =a(b-x)(b+x) by eliminating a and b as its parameters.

Find the differential equation of the family of circles passing through the origin and having their
centres on the x -axis.

Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x² y"+ xy'+y= 0.

Solve the Linear differential equation:
cos x\(\frac{dy}{dx}\)+y sin x=1

Show that the differential equation representing the family of curves y² = 2a\(\left(x+a\frac{2}{3}\right)\)where a is a
positive parameter, is \(\left(y^2-2xy\frac{2}{3}\right)^3\) =8 \(\left(y\frac{dy}{dx}\right)^5\).

Solve : ydx+(x-y²)dy=0

Express the physical statement in the form of the differential equation.
Radium decays at a rate proportional to the amount Q present.

Solve :(1+e²x)dy+(1+y²)e dx=0

Find the differential equation of the curve represented by xy = ae^x + be^-x + x² .

Solve : (1+y²)(1+logx)dx+xdy=0, given that x=1,y=1.

Solve: \(\frac{dv}{dx}\)+ 2y cot x = 3x²cosec²x