CBSE Questions for Class 12 Commerce Maths Differential Equations Quiz 1 - MCQExams.com

For $$x\epsilon R, x\neq 0$$, if $$y(x)$$ is a differentiable function such that $$x\int_{1}^{x}y (t) dt = (x + 1) \int_{1}^{x} t y (t) dt$$, then $$y (x)$$ equals:
(Where C is a constant)
  • $$Cx^{3} e^{\dfrac {1}{x}}$$
  • $$\dfrac {C}{x^{2}} e^{-\dfrac {1}{x}}$$
  • $$\dfrac {C}{x} e^{-\dfrac {1}{x}}$$
  • $$\dfrac {C}{x^{3}} e^{-\dfrac {1}{x}}$$
The differential equation $$\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}=\frac{\sqrt{1-\mathrm{y}^{2}}}{\mathrm{y}}$$ determines a family of circles with:
  • variable radii and a fixed centre at $$(0,1)$$
  • variable radii and a fixed centre at $$(0, -1)$$
  • fixed radius 1 and variable centres along the x-axis
  • fixed radius 1 and variable centres along the y-axis
Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=x^2-8x^3$$.
  • Not homogenous
  • Homogenous with degree 4
  • Homogenous with degree 2
  • None of these
Solution of differential equation $$\displaystyle \frac{dy}{dx} = sin  x  + 2x$$, is
  • $$y = x^2 - cos x + c$$
  • $$y = cos x + x^2 + c$$
  • $$y = cos x + 2$$
  • $$y = cos x + 2 + C$$
Which of the following is/are correct regarding homogenous functions?
  • A function is defined as a homogeneous function if its argument or input is multiplied by some factor, then the result is multiplied by some exponent of this factor.
  • Denoted by: $$f(kx)= k^n f(v)$$, where $$n$$ is the degree
  • Denoted by: $$f(kx)= nk f(v)$$, where $$n$$ is the degree
  • None of these
The solution of $$\dfrac{dy}{dx}=e^{logx}$$ is:
  • $$2y=x^{2}+c$$
  • $$y=x^{2}+c$$
  • $$y^{2}=x+c$$
  • $$xy=x^{2}+c$$
The solution of $$\dfrac{dy}{dx}=\dfrac{2x}{3y^{2}}$$ is:
  • $$y^{3}+x^{2}=c$$
  • $$y^{2}+x^{3}=c$$
  • $$y^{3}-x^{2}=c$$
  • $$y^{3}-x=c$$
The solution of $$\dfrac{dy}{dx}-\dfrac{2xy}{1+x^{2}}=0$$ is
  • $$y=c\left ( 1+x^{2} \right )$$
  • $$y=c\sqrt{1+x^{2}}$$
  • $$y=\dfrac{c}{1+x^{2}}$$
  • $$y=\dfrac{c}{\sqrt{1+x^{2}}}$$
The solution of $$x^{2} \cfrac{dy}{dx}=2$$ is
  • $$y=-\cfrac{2}{x}+c$$
  • $$y=\cfrac{x}{2}+c$$
  • $$y=\cfrac{2}{x}+c$$
  • $$y = 2x + c$$
Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=8x^4$$.
  • Homogenous with degree 4
  • Homogenous with degree 1
  • Not homogenous
  • None of these
If $$y = e^{-x}\cos 2x$$ then which of the following differential equations is satisfied?
  • $$\dfrac {d^{2}y}{dx^{2}} + 2 \dfrac {dy}{dx} + 5y = 0$$
  • $$\dfrac {d^{2}y}{dx^{2}} + 5 \dfrac {dy}{dx} + 2y = 0$$
  • $$\dfrac {d^{2}y}{dx^{2}} - 5 \dfrac {dy}{dx} - 2y = 0$$
  • $$\dfrac {d^{2}y}{dx^{2}} + 2 \dfrac {dy}{dx} - 5y = 0$$
What is the solution of the differential equation $$\dfrac {ydx - xdy}{y^{2}} = 0$$?
where $$c$$ is an arbitrary constant.
  • $$xy = c$$
  • $$y = cx$$
  • $$x + y = c$$
  • $$x - y = c$$
What is the solution of the differential equation $$\sin \left (\dfrac {dy}{dx}\right ) - a = 0$$?
where $$c$$ is an arbitrary constant.
  • $$y = x \sin^{-1} a + c$$
  • $$x = y \sin^{-1} a + c$$
  • $$y = x + x \sin^{-1} a + c$$
  • $$y = \sin^{-1} a + c$$
Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=4-x^2$$.
  • Homogenous with degree 2
  • Not homogenous
  • Homogenous with degree 4
  • None of these
Which of the following is true regarding the function $$f(x, y)= x^4 \sin \dfrac{x}{y}$$ ?
  • Not homogenous
  • Homogenous with degree 2
  • Homogenous with degree 3
  • None of the above
Find the value of $$k$$ for the function: $$2x^2y+3xyz+z^k$$ to be homogenous.
  • $$6$$
  • $$3$$
  • $$2$$
  • None of these
If $$\dfrac{dy}{dx}=x^{-3}$$ then $$y$$
  • $$-\dfrac{1}{2x^2}+c$$
  • $$\dfrac{1}{2x^2}+c$$
  • $$-\dfrac{1}{3x^3}+c$$
  • $$-\dfrac{1}{4x^4}+c$$
The number of arbitrary constants in the particular solution of the differential equation of order $$3$$ is ______.
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
The number of arbitrary constant in the general solution of differential equation of order $$3$$ is _________.
  • $$0$$
  • $$2$$
  • $$3$$
  • $$4$$
What is the solution of the differential equation $$\dfrac {dx}{dy} + \dfrac {x}{y} - y^{2} = 0$$?
where $$c$$ is an arbitrary constant.
  • $$xy = x^{4} + c$$
  • $$xy = y^{4} + c$$
  • $$4xy = y^{4} + c$$
  • $$3xy = y^{3} + c$$
What is the general solution of the differential equation $$x\, dy - y\, dx \,y^2$$ ?
  • $$x = cy $$
  • $$y^2=cx$$
  • $$x + xy - cy = 0$$
  • None of the above
The solution of the differential equation $$x^4\dfrac{dy}{dx}+x^3y+cosec(xy)=0$$ is equal to 
  • $$2 \cos (xy)+x^{-2}=C$$
  • $$2\cos(xy)+y^{-2}=C$$
  • $$2\sin(xy)+x^{-2}=C$$
  • $$2\sin (xy)+y^{-2}+C$$
The differential equation $$y\cfrac { dy }{ dx } +x=a$$ where '$$a$$' is any constant represents:
  • A set of straight lines
  • A set of ellipses
  • A set of circles
  • None of the above
What is the solution of the differential equation $$\dfrac{dy}{dx} +\dfrac{y}{x} = 0 $$ ?
Where c is a constant.
  • $$xy = c$$
  • $$x = cy$$
  • $$y = cx$$
  • None of the above
The differential equation of $$y=c{ x }^{ 3 },c$$ is a arbitrary constant is _______
  • $$x.\cfrac { dy }{ dx } =3y$$
  • $${ x }^{ 3 }.\cfrac { dy }{ dx } =3y$$
  • $$3x\cfrac { dy }{ dx } =y$$
  • $$x.\cfrac { dy }{ dx } =3{ y }^{ 2 }$$
The equation of the curve passing through $$(0,1)$$ which is a solution of the differential equation $$\left( 1+{ y }^{ 2 } \right) dx+\left( 1+{ x }^{ 2 } \right) dy=0$$ is given by
  • $$\tan ^{ -1 }{ \left( x \right) } +\tan ^{ -1 }{ \left( y \right) =0 }$$
  • $$\tan ^{ -1 }{ \left( x \right) } +\tan ^{ -1 }{ \left( y \right) =\dfrac {\pi}{4}}$$
  • $$\sin h ^{ -1 }{ \left( x \right) } +\sin h ^{ -1 }{ \left( y \right) =0 }$$
  • $$\sinh ^{ -1 }{ \left( x \right) } +\sinh ^{ -1 }{ \left( y \right) } \log { \left( 1+\sqrt { 2 } \right) =0 }$$
The solution of $$\dfrac{ydx-xdy}{y^{2}}=0$$ represents a family of
  • Straight lines passing through the origin
  • Circles
  • Parabola
  • Hyperbola
The $$D.E$$ $$y\dfrac { dy }{ dx } +x=a$$ represents
  • a circle whose centra is on $$X-$$axis
  • a circle whose centre is on $$Y-$$axis
  • a circle whose centre is origin
  • a parabola
The solution of $$\left( { x }^{ 2 }{ y }^{ 3 }+{ x }^{ 2 } \right) dx+\left( { y }^{ 2 }{ x }^{ 3 }+{ y }^{ 2 } \right) dy=0$$ is
  • $$\left( { x }^{ 3 }+1 \right) \left( { y }^{ 3 }+1 \right) =c$$
  • $$\left( { x }^{ 3 }-1 \right) \left( { y }^{ 3 }-1 \right) =c$$
  • $$\left( { x }^{ 3 }-1 \right) \left( { y }^{ 3 }+1 \right) =c$$
  • $$\left( { x }^{ 3 }+1 \right) \left( { y }^{ 3 }-1 \right) =c$$
Which of the following transformation reduce the differential equation $$\dfrac{{dz}}{{dx}} + \dfrac{z}{x}\log z = \dfrac{z}{{{x^2}}}{\left( {\log z} \right)^2}$$ into the form $$\dfrac{{dv}}{{dx}} + P\left( x \right)v = Q\left( x \right)$$
  • $$v=logz$$
  • $$v=e^2$$
  • $$ v=\dfrac{1}{logz}$$
  • $$v=(logz)^2$$
The differential equation for all the straight lines which are at a unit distance from the origin is  
  • $${\left( {y - x\frac{{dy}}{{dx}}} \right)^2} = 1 - {\left( {\dfrac{{dy}}{{dx}}} \right)^2}$$
  • $${\left( {y + x\frac{{dy}}{{dx}}} \right)^2} = 1 + {\left( {\dfrac{{dy}}{{dx}}} \right)^2}$$
  • $${\left( {y - x\frac{{dy}}{{dx}}} \right)^2} = 1+{\left( {\dfrac{{dy}}{{dx}}} \right)^2}$$
  • $${\left( {y + x\frac{{dy}}{{dx}}} \right)^2} = 1 - {\left( {\dfrac{{dy}}{{dx}}} \right)^2}$$
Which of the following differential equation is linear ?
  • $$\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+2y=0$$
  • $$\dfrac{d^2y}{dx^2}+y\dfrac{dy}{dx}+x=0$$
  • $$\dfrac{d^{2}y}{dx^2}+\dfrac{y}{x}+\sin y=x^2$$
  • $$(1+x)\dfrac{dy}{dx}-xy=1$$
Solution of the different equation, $$ydx - xdy + x{y^2}dx = 0$$ can be.
  • $$2x + {x^2}y = \lambda y$$
  • $$2y + {y^2}x = \lambda y$$
  • $$2y - {y^2}x = \lambda y$$
  • none of these
Solve $$(1+\cos x)dy=(1-\cos x)dx$$.
  • $$y = \displaystyle \cot \cfrac{x}{2} - x +C$$
  • $$y = \displaystyle \tan \cfrac{x}{2} - x +C$$
  • $$y = \displaystyle \sin \cfrac{x}{2} - x +C$$
  • None of these
The solution of the differential equation $$\dfrac{dy}{dx} = \dfrac{3y - 7x - 3}{3x - 7y + 7}$$ is 
  • $$(y - x - 2)^5 (y + x - 5)^7 = c$$
  • $$(y - x - 5)^2 (y + x - 1)^7 = c$$
  • $$(y - x - 7)^2 (y + x - 5) = c$$
  • $$(y - x - 1)^2 (y + x - 1)^5 = c$$
If $$\dfrac{dy}{dx}+y\tan x=\sin 2x$$ and $$y(0)=1$$, then $$y(\pi)=?$$
  • $$1$$
  • $$-1$$
  • $$-5$$
  • $$5$$
The solution of the differential equaton $$3{e^x}\tan ydx + \left( {1 - {e^x}} \right){\sec ^2}ydy = 0$$ is 
  • $$\tan y = c{\left( {1 - {e^x}} \right)^3}$$
  • $${\left( {1 - {e^x}} \right)^3}\tan y = c$$
  • $$\tan y = c\left( {1 - {e^x}} \right)$$
  • $$\left( {1 - {e^x}} \right)\tan y = c$$
Solve:
$$ \dfrac {dy}{dx} = y \cot x$$
  • $$y=k\tan x$$
  • $$y=k\csc x$$
  • $$y=k\sin x$$
  • $$y=k\cos x$$
The D.E. obtained from $$(y - a)^2 = 4 (x - b)$$ is given by
  • $$2y_2 + y_3 = 0$$
  • $$2y_2 = y_3$$
  • $$2y_1 + y^3_2 = 0$$
  • $$2y_2 + y^3_1 = 0$$
The differential equation $$\dfrac{dy}{dx} = 2$$ represents
  • A straight line of slope $$2$$ units
  • A circle with radius $$2$$ units
  • A parabola with foci $$2$$ units
  • None of these
The solution of the differential equation  $$\left( 1+y^{ { 2 } } \right) +\left( x-e^{ { { \tan   }^{ -1 }y } } \right) \dfrac { dy }{ dx } =0,$$  is
  • $$( x - 2 ) = k e ^ { - \tan ^ { - 1 } y }$$
  • $$2 x e ^ { 2 \tan ^ { - 1 } y } = e ^ { 2 \tan ^ { - 1 } y } = k$$
  • $$x e ^ { \tan ^ { - 1 } y } = \tan ^ { - 1 } y + k$$
  • $$xe^{ { 2\tan ^{ { -1 } } y } }=e^{ { \tan ^{ { -1 } } y } }+k$$
The order of the differential equation of the parabola whose axis is parallel to y-axis is:
  • 1
  • 2
  • 3
  • 4
The order of the differential equation of the curve $$y^{2}=4ax$$ is:
  • 2
  • 1
  • 3
  • Cant be determined
The differential equation of all conics with the co-ordinate axes as axes is of the order
  • 1
  • 2
  • 3
  • 4
The solution of $$\dfrac{d^{2}y}{dx^{2}}=0$$ represents
  • a straight line
  • a circle
  • a parabola
  • a point
The order of the differenital equation of $$(x-a)^{2}+(y-b)^{2}=a^{2}$$ is:
  • 4
  • 3
  • 2
  • 1
The differential equation $$y\dfrac{dy}{dx}+x=A$$ where A is a constant represents a set of:
  • circles centred on y axis
  • circles centred on x axis
  • parabolas
  • ellipses
The differential equation of all conics with centre at origin is of order
  • 2
  • 3
  • 4
  • 1
The differential equation of the family of circles whose center lies on $$x-$$axis and passing through origin is
  • $$x^{2}+y^{2}+\dfrac{dy}{dx}=0$$
  • $$(y^{2}-x^2)dx-2xydy=0$$
  • $$y^{2}dx+(x^{2}+2xy)dy=0$$
  • $$xdy+ydx+x^{2}dx+y^{2}dy=0$$
The solution of $$\displaystyle \frac{dy}{dx}+\displaystyle \frac{\sqrt{1+y^{2}}}{\sqrt{1+x^{2}}}=0$$ is:
  • $$\sin^{-1}x+\sin^{-1}y=c$$
  • $$\tan^{-1}x+\tan^{-1}y=c$$
  • $$\sinh^{-1}x+\sinh^{-1}y=c$$
  • $$\cot^{-1}x+\cot^{-1}y=c$$
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