Class 12 Maths MCQ Questions of Differential Equations with Answers?

1.	Class 12 Maths MCQ Questions of Differential Equations with Answers?
Answer» Important Class 12 Maths MCQ Questions of Differential Equations with Answers allows in getting ready for CBSE Board exams. Students can go through the important MCQ Questions for Class 12 Maths to revise all of the chapters and get extra marks withinside the CBSE final examinations. This chapter consists of many formulations and strategies for solving given differential equations. Apart from reading and practicing MCQ Questions of Differential equations from the NCERT book, college students shall additionally practice these vital MCQ Questions. The objective types of questions are taken in keeping with the syllabus of the CBSE board. These important MCQ Questions for class 12 help the students to gain the right marks withinside the class 12 board exam. Practice MCQ Question for Class 12 Maths chapter-wise 1. The radius of a circle is increasing at the rate of 0.7 cm/ s. The rate of increasing of its circumference is (a) 1.4 π cm/s (b) 0.8 π cm/s (c) 1.8 cm/s (d) None of these 2. Solution of differential equation xdy – ydx = Q represents (a) a rectangular hyperbola (b) parabola whose vertex is at origin (c) straight line passing through origin (d) a circle whose centre is at origin 3. Integrating factor of the differential equation \(cosx\frac{dy}{dx}+ysin x=1\) is (a) cos x (b) tan x (c) sec x (d) sin x 4. Family r = Ax + A³ of curves is represented by the differential equation of degree (a) 1 (b) 2 (c) 3 (d) 4 5. Which of the following is a second order differential equation? (a) (y’)² + x = y² (b) y’y” + y = sin x (c) y” + (y”)² + y = 0 (d) y’ = y² 6. The differential equation \(\frac{dy}{dx}+x=c\) represents (a) Family of hyperbolas (b) Family of parabolas (c) Family of ellipses (d) Family of circles 7. The general solution of e^xcosydx − e^xsinydy = 0 is (a) e^xcosy = k (b) e^xsiny = k (c) e^x= k cos y (d) e^x= k sin y 8. The solution of \(x\frac{dy}{dx}+y=e^x\) is (a) \(y=\frac{e^x}{x}+\frac{k}{x}\) (b) \(y=xe^x+cx\) (c) \(xe^x+k\) (d) \(x=\frac{e^y}{y}+\frac{k}{y}\) 9. The general solution of \(\frac{dy}{dx}=2x e^{x^{2-y}}\) is (a) \(e^{x^2-y}=c\) (b) \(e^{-y}+e^{x^2}=c\) (c) \(e^y=e^{x^2}+c\) (d) \(e^{x^2+y}=c\) 10. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is (a) an ellipse (b) parabola (c) circle (d) rectangular hyperbola 11. The number of arbitrary constants in the general solution of differential equation of fourth order is (a) 4 (b) 3 (c) 2 (d) 0 12. The order of differential equation of all circles of given radius ′a′ is (a) 4 (b) 2 (c) 1 (d) 3 13. The general solution of differential equation is y = ae^bx+c where a,b,c are arbitrary constant. The order of differential equation is: (a) 4 (b) 2 (c) 1 (d) 3 14. The degree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively (a) 2,1 (b) 1,2 (c) 3,2 (d) 2,3 15. The order and degree of the differential equation of the family having the same foci, are respectively (a) 2,1 (b) 2,1 (c) 2,2 (d) 1,2 16. The order and degree of the differential equation of the family of circles touching the x-axis at the origin, are respectively (a) 1,1 (b) 1,2 (c) 2,1 (d) 2,2 17. The solution of (x+ logy)dy +ydx =0 where y(0) =1 is (a) y(x−(A)) + ylogy = 0 (b) y(x−1+logy) + 1 = 0 (c) xy+ ylogy +1 = 0 (d) None of these 18. The number of arbitrary constants in the particular solution of a differential equation of third order is: (a) 3 (b) 2 (c) 1 (d) 0 19. What is the order of differential equation y’’ + 5y’ + 6 = 0? (a) 0 (b) 1 (c) 2 (d) 0 20. What is the degree of differential equation (y’’’)² + (y’’)³+ (y’)⁴ + y⁵= 0? (a) 2 (b) 3 (c) 4 (d) 5 Answer: 1. Answer: (a) 1.4 π cm/s Explanation: Let r be the radius and C the circumference of the circle. Then, C = 2πr dr/dt = 0.7 cm/sec. Now , C = 2πr ⇒ dC/dt = \(2\pi\frac{dr}{dt}\Rightarrow\) dC/dt \( =2\pi\times0.7cm/sec\) = 1.4 π cm/s 2. Answer: (c) straight line passing through origin Explanation: xdy − ydx = 0 ⇒ dy/y = dx/x On integration, we get logy = logx + logc y = xc which is equation of a straight line passing through origin. 3. Answer: (c) sec x Explanation: \(cosx \frac{dy}{dx}+ysinx=1\) ⇒ dy/dx + sinx/cosx⋅y = 1/cosx ⇒ dy/dx+tanx⋅y = secx I.F = \(e^{\int\;tan\;xdx}\) = \(e^{log(sec x)}\) = sec x 4. Answer: (a) 1 Explanation: y = ax + a³.....(i) y' = a Linear differential equation of order 1 and degree 1. 5. Answer: (b) y’y” + y = sin x Explanation: The second order differential equation is y’y” + y = sin x 6. Answer: (d) Family of circles Explanation: Given differential equation is \(y\frac{dy}{dx}+x=c\) ⇒ ydy = (c−x)dx On integrating both sides, we get y²/2 = cx−x²/2 + d ⇒y²+x²−2cx −2d = 0 Hence, it represents a family of circles whose centres are on the x -axis. 7. Answer: (a) e^xcosy = k Explanation: e^xcosy dx−e^xsiny dy=0 ⇒e^xcosydx = e^xsinydy ⇒dx = tanydy On integrating, we get x = log(secy) + logk ⇒x = log[(secy)k] ⇒e^x= ksecy ⇒e^x/secy = k ⇒e^xcosy = k 8. Answer: (a) Explanation: \(x\frac{dy}{dx}+y=e^x\) \(\frac{dy}{dx}+\frac{1}{x}y=\frac{e^x}{x}\) It is a linear differential equation with I.F = \(e^{\int\;\frac{1}{x}dx}\) \(=e^{log x}=x\) Now, solution is y⋅x \(\int\frac{e^z}{x}.xdx+k\) ⇒y^x = ex + k \(y=\frac{e^x}{x}+\frac{k}{x}\) 9. Answer: (c) Explanation: \(\frac{dy}{dx}=2xe^{x^2-y}\) \(e^ydy=2xe ^{x^3}dx\) On integrating, we get \(e^y = e^{x^2} + c\) 10. Answer: (d) rectangular hyperbola Explanation: The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is rectangular hyperbola. 11. Answer: (a) 4 Explanation: The number of arbitrary constants in a solution of a differential equation of order n is equal to its order. So, here it is 4. 12. Answer: (b) 2 Explanation: Let the centre of circle be (h,k) and radius a, then its equation will be (x−h)²+(y−k)²=a² Since, there are two parameters h and k. So, the differential equation is of order 2 13. Answer: (b) 2 Explanation: ae^bx. e^c a,b,c are three constants in order to solve the above given equation we have to differentiate it twice so the given equation is d²y/dx² = ab²e^bx+c Thus, order is 2 14. Answer: (b) 1,2 Explanation: Parabola whose axis is x-axis is y²= 4ax 2y = dy/dx = 4a \(y\frac{dy}{dx}=2a\) \(\Rightarrow y \frac{d^2y}{dx^2}+(\frac{dy}{dx})^2=0\) degree =1, order =2 15. Answer: (b) 2,1 Explanation: Equation of family of parabola with x-axis as axis is y² = 4a(x + α) where α are two arbitrary constant. So, differential equation is of order 2 and degree 1. 16. Answer: (a) 1,1 Explanation: The system of circles touching x-axis at origin will have centres on y-axis. Let (0,a) be the centre of a circle. Then the radius of the circle should be a units, since the circle should touch x-axis at origin. Equation of a circle with centre at (0,a) and radius a is \((x-0)^2+(y-a)^2=a^2\) \(x^2+y^2--2ay=0\) The above equation represents the family of circles touching x-axis at origin. Here 'a' is an arbitrary constant. In order to find the differential equation of system of circles touching x-axis at origin, eliminate the the arbitrary constant from equation. Differentiating equation with respect to y, \(2x\frac{dx}{dy}+2y-2a=0\) \(\frac{dx}{dy}=2(y-a)\) Order is 1 and degree is also 1. 17. Answer: (b) Explanation: (x +logy)dy + ydx = 0 ∴d(xy) =xdy + ydx (xdy + ydx)+(logydy)=0 xy + (logy . -y/y dy) = 0 xy + ylogy − y + c=0 y(x−1) +ylogy + c = 0 y(x−1+logy) + c= 0 1(0−1+log1)+c=0 c=1 given y(0) = 1 i.e. (0, 1) is the solution of the differential equation So, overall solutions is (x +logy)dy + ydx = 0 18. Answer: (d) 0 Explanation: In the particular solution of a differential equation of third order, there is no arbitrary constant because in the particular solution of any differential equation, we remove all the arbitrary constant by substituting some particular values. 19. Answer: (c) 2 Explanation: The highest order derivative present in the differential equation is y’’. Hence, the order is 2. 20.Answer: (a) 2 Explanation: The degree is the power raised to the highest order derivative. Therefore, in the given differential equation, (y’’’)² + (y’’)³+ (y’)⁴ + y⁵= 0, the degree will be power raised to y’’’. Click here to practice more MCQ Question for Differential Equations Class 12

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Important Class 12 Maths MCQ Questions of Differential Equations with Answers allows in getting ready for CBSE Board exams. Students can go through the important MCQ Questions for Class 12 Maths to revise all of the chapters and get extra marks withinside the CBSE final examinations. This chapter consists of many formulations and strategies for solving given differential equations.

Apart from reading and practicing MCQ Questions of Differential equations from the NCERT book, college students shall additionally practice these vital MCQ Questions. The objective types of questions are taken in keeping with the syllabus of the CBSE board. These important MCQ Questions for class 12 help the students to gain the right marks withinside the class 12 board exam.

Practice MCQ Question for Class 12 Maths chapter-wise

1. The radius of a circle is increasing at the rate of 0.7 cm/ s. The rate of increasing of its circumference is

(a) 1.4 π cm/s
(b) 0.8 π cm/s
(c) 1.8 cm/s
(d) None of these

2. Solution of differential equation xdy – ydx = Q represents

(a) a rectangular hyperbola
(b) parabola whose vertex is at origin
(c) straight line passing through origin
(d) a circle whose centre is at origin

3. Integrating factor of the differential equation \(cosx\frac{dy}{dx}+ysin x=1\) is

(a) cos x
(b) tan x
(c) sec x
(d) sin x

4. Family r = Ax + A³ of curves is represented by the differential equation of degree

(a) 1
(b) 2
(c) 3
(d) 4

5. Which of the following is a second order differential equation?

(a) (y’)² + x = y²
(b) y’y” + y = sin x
(c) y” + (y”)² + y = 0
(d) y’ = y²

6. The differential equation \(\frac{dy}{dx}+x=c\) represents

(a) Family of hyperbolas
(b) Family of parabolas
(c) Family of ellipses
(d) Family of circles

7. The general solution of e^xcosydx − e^xsinydy = 0 is

(a) e^xcosy = k
(b) e^xsiny = k
(c) e^x= k cos y
(d) e^x= k sin y

8. The solution of \(x\frac{dy}{dx}+y=e^x\) is

(a) \(y=\frac{e^x}{x}+\frac{k}{x}\)
(b) \(y=xe^x+cx\)
(c) \(xe^x+k\)
(d) \(x=\frac{e^y}{y}+\frac{k}{y}\)

9. The general solution of \(\frac{dy}{dx}=2x e^{x^{2-y}}\) is

(a) \(e^{x^2-y}=c\)
(b) \(e^{-y}+e^{x^2}=c\)
(c) \(e^y=e^{x^2}+c\)
(d) \(e^{x^2+y}=c\)

10. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

(a) an ellipse
(b) parabola
(c) circle
(d) rectangular hyperbola

11. The number of arbitrary constants in the general solution of differential equation of fourth order is

(a) 4
(b) 3
(c) 2
(d) 0

12. The order of differential equation of all circles of given radius ′a′ is

(a) 4
(b) 2
(c) 1
(d) 3

13. The general solution of differential equation is y = ae^bx+c where a,b,c are arbitrary constant. The order of differential equation is:

(a) 4
(b) 2
(c) 1
(d) 3

14. The degree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively

(a) 2,1
(b) 1,2
(c) 3,2
(d) 2,3

15. The order and degree of the differential equation of the family having the same foci, are respectively

(a) 2,1
(b) 2,1
(c) 2,2
(d) 1,2

16. The order and degree of the differential equation of the family of circles touching the x-axis at the origin, are respectively

(a) 1,1
(b) 1,2
(c) 2,1
(d) 2,2

17. The solution of (x+ logy)dy +ydx =0 where y(0) =1 is

(a) y(x−(A)) + ylogy = 0
(b) y(x−1+logy) + 1 = 0
(c) xy+ ylogy +1 = 0
(d) None of these

18. The number of arbitrary constants in the particular solution of a differential equation of third order is:

(a) 3
(b) 2
(c) 1
(d) 0

19. What is the order of differential equation y’’ + 5y’ + 6 = 0?

(a) 0
(b) 1
(c) 2
(d) 0

20. What is the degree of differential equation (y’’’)² + (y’’)³+ (y’)⁴ + y⁵= 0?

(a) 2
(b) 3
(c) 4
(d) 5

Answer:

1. Answer: (a) 1.4 π cm/s

Explanation: Let r be the radius and C the circumference of the circle. Then, C = 2πr

dr/dt = 0.7 cm/sec.

Now , C = 2πr ⇒ dC/dt = \(2\pi\frac{dr}{dt}\Rightarrow\) dC/dt

\( =2\pi\times0.7cm/sec\)

= 1.4 π cm/s

2. Answer: (c) straight line passing through origin

Explanation: xdy − ydx = 0

⇒ dy/y = dx/x

On integration, we get

logy = logx + logc

y = xc which is equation of a straight line passing through origin.

3. Answer: (c) sec x

Explanation: \(cosx \frac{dy}{dx}+ysinx=1\)

⇒ dy/dx + sinx/cosx⋅y = 1/cosx

⇒ dy/dx+tanx⋅y = secx

I.F = \(e^{\int\;tan\;xdx}\)

= \(e^{log(sec x)}\)

= sec x

4. Answer: (a) 1

Explanation: y = ax + a³.....(i)

y' = a

Linear differential equation of order 1 and degree 1.

5. Answer: (b) y’y” + y = sin x

Explanation: The second order differential equation is y’y” + y = sin x

6. Answer: (d) Family of circles

Explanation: Given differential equation is

\(y\frac{dy}{dx}+x=c\)

⇒ ydy = (c−x)dx

On integrating both sides, we get

y²/2 = cx−x²/2 + d

⇒y²+x²−2cx −2d = 0

Hence, it represents a family of circles whose centres are on the x -axis.

7. Answer: (a) e^xcosy = k

Explanation: e^xcosy dx−e^xsiny dy=0

⇒e^xcosydx = e^xsinydy

⇒dx = tanydy

On integrating, we get

x = log(secy) + logk

⇒x = log[(secy)k]

⇒e^x= ksecy

⇒e^x/secy = k

⇒e^xcosy = k

8. Answer: (a)

Explanation: \(x\frac{dy}{dx}+y=e^x\)

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

It is a linear differential equation with

I.F = \(e^{\int\;\frac{1}{x}dx}\)

\(=e^{log x}=x\)

Now, solution is y⋅x \(\int\frac{e^z}{x}.xdx+k\)

⇒y^x = ex + k

\(y=\frac{e^x}{x}+\frac{k}{x}\)

9. Answer: (c)

Explanation: \(\frac{dy}{dx}=2xe^{x^2-y}\)

\(e^ydy=2xe ^{x^3}dx\)

On integrating, we get \(e^y = e^{x^2} + c\)

10. Answer: (d) rectangular hyperbola

Explanation: The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is rectangular hyperbola.

11. Answer: (a) 4

Explanation: The number of arbitrary constants in a solution of a differential equation of order n is equal to its order. So, here it is 4.

12. Answer: (b) 2

Explanation: Let the centre of circle be (h,k) and radius a, then its equation will be

(x−h)²+(y−k)²=a²

Since, there are two parameters h and k. So, the differential equation is of order 2

13. Answer: (b) 2

Explanation: ae^bx. e^c a,b,c are three constants

in order to solve the above given equation we have to differentiate it twice so the given equation is

d²y/dx² = ab²e^bx+c

Thus, order is 2

14. Answer: (b) 1,2

Explanation: Parabola whose axis is x-axis is y²= 4ax

2y = dy/dx = 4a

\(y\frac{dy}{dx}=2a\)

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

degree =1, order =2

15. Answer: (b) 2,1

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

16. Answer: (a) 1,1

Explanation: The system of circles touching x-axis at origin will have centres on y-axis.
Let (0,a) be the centre of a circle. Then the radius of the circle should be a units, since the circle should touch x-axis at origin. Equation of a circle with centre at (0,a) and radius a is

\((x-0)^2+(y-a)^2=a^2\)

\(x^2+y^2--2ay=0\)

The above equation represents the family of circles touching x-axis at origin. Here 'a' is an arbitrary constant. In order to find the differential equation of system of circles touching x-axis at origin, eliminate the the arbitrary constant from equation. Differentiating equation with respect to y,

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

\(\frac{dx}{dy}=2(y-a)\)

Order is 1 and degree is also 1.

17. Answer: (b)

Explanation: (x +logy)dy + ydx = 0

∴d(xy) =xdy + ydx

(xdy + ydx)+(logydy)=0

xy + (logy . -y/y dy) = 0

xy + ylogy − y + c=0

y(x−1) +ylogy + c = 0

y(x−1+logy) + c= 0

1(0−1+log1)+c=0

c=1 given y(0) = 1 i.e. (0, 1) is the solution of the differential equation So, overall solutions is

(x +logy)dy + ydx = 0

18. Answer: (d) 0

Explanation: In the particular solution of a differential equation of third order, there is no arbitrary constant because in the particular solution of any differential equation, we remove all the arbitrary constant by substituting some particular values.

19. Answer: (c) 2

Explanation: The highest order derivative present in the differential equation is y’’. Hence, the order is 2.

20.Answer: (a) 2

Explanation: The degree is the power raised to the highest order derivative. Therefore, in the given differential equation, (y’’’)² + (y’’)³+ (y’)⁴ + y⁵= 0, the degree will be power raised to y’’’.

Click here to practice more MCQ Question for Differential Equations Class 12

Class 12 Maths MCQ Questions of Differential Equations with Answers?

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