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12th Maths Weekly Test -1 (Applications of Integration)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Applications of Integration)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The order and degree of the differential equation\(\left[(\frac{d^2y}{dx^2})+(\frac{dy}{dx})\right]^\frac{1}{2}=\frac{d^3y}{dx^3 }\) are

1,2
2,1
3,2
2,3

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 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 I.F of y log y\(\frac{dx}{dy}\)+ x - log y = 0 is
log(log y)

log y

\(\frac{1}{log y}\)

\(\frac{1}{log (log\space y)}\)
The transformation y=vx reduces\(\frac{dy}{dx}\) = \(\frac{x+y}{3x}\)
\(\frac{3av}{4v+1}=\frac{dx}{x}\)

\(\frac{3dv}{v+1}=\frac{dx}{x}\)

\(2x\frac{dv}{dx}=v\)

\(\frac{3dv}{1-2v}==\frac{dx}{x}\)

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

1,1
1.2
2,1
2,2

On finding the differential equation corresponding to y=e^mx where m is the arbitrary constant, then m is ________.

\(\frac{y}{y^1}\)
\(\frac{y^1}{y}\)
y'
y

The solution of log \(\left(\frac{dy}{dx}\right)\)=ax+by is______.

\(\frac{e^{ax}}{a}+\frac{e^{-by}}{b}+c = 0\)
ae^ax -be^-by +c=0
ae^x +be^y =k
be^ax +ae^-by =k

The differential equation of x² y = k is _________.

\(x^2\frac{dy}{dx}=0\)
\(x^2\frac{dy}{dx}+y=0\)
\(x\frac{dy}{dx}+2y=0\)
\(y\frac{dy}{dx}+2x=0\)

The I.F. of (1+y²)dx=(tan^-1 y-x)dy is ________.

e^tan-1 y
e^tan-1 x
tan^-1 y
tan^-1 x

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

Part - B

Generic 2 - Answer All Question

The differential equation obtained by eliminating a and b from y=ae^3x+be^-3x is

1)\(\frac{d^2y}{dx^2}-9y\)

2)\(\frac{d^2y}{dx^2}+9y\)

3)y''-9y=0

4) y' =3ae^3x -3be^-3x

y=cx=c² is the general solution of the differential equation.
1) y'=c
2) y=y'x-(y')²
3) y"=0
4) (y')² -xy'+y=0

A curve passing through the origin has its slope e^x, Find the equation of the curve.

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

Determine the order and degree of\(\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]}{\frac{d^2y}{dx^2}}\)=k
Solution of \(\frac{dy}{dx}\)+mx=0 where m< 0 is

1) \(\frac{dy}{dx}\)=-m dy
2) y=ce^mx
3) log x=-my+log x
4) x=ce^-my

Part - C

Generic 3 - Answer All Question

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

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

Form the D.E of family of curves represented by y=c(x-c)².where c is the parameter.

Find the D.E of all circles touching x-axis at the origin.

Obtain the D.E of all circles of radius ‘r’
Solve: \(\frac{dy}{dx}\)=(4x+y+1)²

Part - D

Generic 5 - Answer All Question

Solve: (1 + e^2x) dy + (1 + y²)e^x dx = 0 when y(0) = 1

Solve:\(\frac{dy}{dx}\) = (3x+2y+1)²

Solve: \(\frac{dy}{dx}\)=(sin²x cos²x + xe^x)dx
The surface area of a balloon being inflated changes at a constant rate. If initially, its radius 3 units
and after 2 seconds it is 5 units, find the radius after t seconds.

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