Class 12 Mathematics Chapter 4 – Differential Equations MCQs| FBISE | Grade 12 Math - MCQs

Practice 50 chapter-wise MCQs for FBISE HSSC-II Mathematics.

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1. A differential equation is an equation involving

A) a dependent variable and its derivativesB) only constantsC) only algebraic numbersD) only independent variables

2. The order of a differential equation is the order of the

A) highest power of the dependent variableB) highest derivative presentC) lowest derivative presentD) independent variable

3. The degree of a differential equation is defined when the equation is

A) free from derivativesB) linear in the independent variableC) a polynomial in derivativesD) homogeneous only

4. The order of $\frac{d^2y}{dx^2}+3\frac{dy}{dx}+y=0$ is

A) 1B) 0C) 3D) 2

5. The degree of $\left(\frac{d^2y}{dx^2}\right)^3+\frac{dy}{dx}=0$ is

A) 3B) 2C) 1D) 4

6. A first-order differential equation contains derivatives of order

A) exactly 2B) at most 1C) at least 2D) only 0

7. A differential equation obtained from a practical situation is called a

A) numerical equationB) constant equationC) modelD) matrix

8. The general solution of a first-order differential equation normally contains

A) no arbitrary constantB) three arbitrary constantsC) two arbitrary constantsD) one arbitrary constant

9. The equation $\frac{dy}{dx}=3x^2$ is

A) separableB) second orderC) algebraicD) not a differential equation

10. A separable differential equation can be written in the form

A) $f(x)+g(y)=0$ onlyB) $f(y)dy=g(x)dx$ C) $x+y=0$ onlyD) $dy/dx=x+y$ only

11. To solve $\frac{dy}{dx}=xy$ , the variables can be separated as

A) $dy=x/y\,dx$ B) $y\,dy=x\,dx$ C) $dy/y=x\,dx$ D) $dx/x=y\,dy$

12. Integrating $\frac{dy}{y}=x\,dx$ gives

A) $1/y=x^2+C$ B) $y=x^2+C$ C) $\ln y=x+C$ D) $\ln|y|=x^2+C$

13. The general solution of $\frac{dy}{dx}=2x$ is

A) $y=x^2+C$ B) $y=2x+C$ C) $y=x+C$ D) $y=2x^2+C$

14. The general solution of $\frac{dy}{dx}=6x^2$ is

A) $y=6x^3+C$ B) $y=2x^3+C$ C) $y=3x^2+C$ D) $y=2x^2+C$

15. The solution of $\frac{dy}{dx}=y$ is

A) $y=x+C$ B) $y=Cx$ C) $y=Ce^x$ D) $y=e^x+C$

16. For $\frac{dy}{dx}=ky$ , the general solution is

A) $y=e^{x+k}$ B) $y=kx+C$ C) $y=Cx^k$ D) $y=Ce^{kx}$

17. The equation $\frac{dy}{dx}=\frac{x}{y}$ can be separated as

A) $y\,dy=x\,dx$ B) $dy/y=x\,dx$ C) $dx/x=y\,dy$ D) $y/x=dy/dx$

18. Integrating $y\,dy=x\,dx$ gives

A) $y^2=x^2+C$ B) $y^2/2=x+C$ C) $y=x+C$ D) $y=x^2+C$

19. The solution of $\frac{dy}{dx}=\frac{2x}{y}$ satisfies

A) $y^2=x^2+C$ B) $y=2x^2+C$ C) $y^2=2x^2+C$ D) $y=\sqrt{x}+C$

20. An equation of the form $\frac{dy}{dx}=f(x)g(y)$ is generally

A) not differentialB) always second orderC) always linearD) separable

21. The differential equation $\frac{dy}{dx}=x+y$ is

A) not separable in its given formB) separable in its given formC) algebraicD) second order

22. The differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ can be recognized as

A) second orderB) homogeneousC) constantD) algebraic

23. A first-order homogeneous differential equation often has the form

A) $dy/dx=F(x)+F(y)$ B) $dy/dx=x+y+1$ C) $dy/dx=F(y/x)$ D) $dy/dx=C$ only

24. For a homogeneous equation involving $y/x$ , a useful substitution is

A) $v=x+y$ B) $x=vy$ onlyC) $y=x+v$ D) $y=vx$

25. If $y=vx$ , then $dy/dx$ equals

A) $v+x\,dv/dx$ B) $x\,dv/dx$ C) $v$ D) $x/v$

26. The substitution $y=vx$ is useful because it changes $y/x$ into

A) $x/v$ B) $v$ C) $x+y$ D) $1/v$

27. For $\frac{dy}{dx}=F(y/x)$ , the equation is typically classified as

A) algebraicB) second-order linearC) homogeneous first-orderD) constant

28. A particular solution is obtained from the general solution by

A) removing all variablesB) setting the independent variable to zeroC) differentiating twiceD) assigning a value to the arbitrary constant

29. If $y=Ce^x$ and $y(0)=5$ , then $C$ is

A) 5B) 1C) 0D) e

30. If $y=x^2+C$ and $y(1)=4$ , then $C$ is

A) 2B) 3C) 4D) 5

31. The initial condition $y(x_0)=y_0$ is used to

A) increase the orderB) remove the derivativeC) determine the arbitrary constantD) change the independent variable

32. The solution of $dy/dx=3y$ , with $y(0)=2$ , is

A) $y=3x+2$ B) $y=3e^{2x}$ C) $y=2e^x+3$ D) $y=2e^{3x}$

33. The solution of $dy/dx=2x$ , with $y(0)=3$ , is

A) $y=x^2+3$ B) $y=2x+3$ C) $y=2x^2+3$ D) $y=x^2$

34. For $dy/dx=x/y$ with $y(0)=2$ , the relation is

A) $y=x^2+2$ B) $y^2=x^2+4$ C) $y^2=2x^2+4$ D) $y=x+2$

35. A differential equation representing exponential growth is commonly

A) $dy/dt=t^2$ B) $dy/dt=k+y$ C) $dy/dt=ky$ D) $dy/dt=1/y$ only

36. For $dy/dt=ky$ , if $k>0$ , the solution represents

A) zero motionB) constant motionC) linear decay onlyD) exponential growth

37. For $dy/dt=ky$ , if $k<0$ , the solution represents

A) exponential decayB) exponential growthC) quadratic growthD) periodic motion

38. A population model $dP/dt=kP$ assumes the growth rate is proportional to

A) timeB) populationC) square of timeD) a constant independent of population

39. If $dP/dt=kP$ , then $\frac{1}{P}dP$ equals

A) $t\,dP$ B) $P\,dt$ C) $k\,dt$ D) $dt/k$

40. Integrating $dP/P=k\,dt$ gives

A) $1/P=kt+C$ B) $P=kt+C$ C) $\ln P=t/k+C$ D) $\ln|P|=kt+C$

41. The solution of $dP/dt=kP$ can be written as

A) $P=Ce^{kt}$ B) $P=Ckt$ C) $P=C+t^k$ D) $P=kP_0+t$

42. In the model $P=P_0e^{kt}$ , $P_0$ represents

A) growth rateB) initial populationC) time constant onlyD) final population

43. In $P=P_0e^{kt}$ , the parameter $k$ controls the

A) time unit onlyB) initial population onlyC) growth or decay rateD) arbitrary constant only

44. The differential equation $\frac{dy}{dx}=\frac{2y}{x}$ is separable and leads to

A) $y=C/x^2$ B) $y=2x+C$ C) $y=Ce^{2x}$ D) $y=Cx^2$

45. The equation $\frac{dy}{dx}=\frac{y}{x}$ has general solution

A) $y=Cx$ B) $y=Ce^x$ C) $y=C/x$ D) $y=x+C$

46. The differential equation $\frac{dy}{dx}=\frac{x^2}{y^2}$ can be separated as

A) $dy/y^2=x^2dx$ B) $y^2dy=x^2dx$ C) $y^2dx=x^2dy$ D) $ydy=x^2dx$

47. Integrating $y^2dy=x^2dx$ gives

A) $y^3=x^2+C$ B) $y^2=x^2+C$ C) $y^3/3=x^3/3+C$ D) $y=x^3+C$

48. The differential equation $\frac{dy}{dx}=\frac{1+y}{x}$ can be separated as

A) $dy=x(1+y)dx$ B) $dy/(1+y)=x\,dx$ C) $(1+y)dy=x\,dx$ D) $dy/(1+y)=dx/x$

49. A practical differential-equation model is useful because it

A) relates changing quantitiesB) eliminates all measurementsC) always gives a numerical answer immediatelyD) contains no assumptions

50. When solving a separable differential equation, integration constants are generally

A) always set equal to zeroB) combined into one arbitrary constantC) never requiredD) replaced by the independent variable

Explore More Class 12 Mathematics MCQs

Chapter 1 – Functions and Graphs
Chapter 2 – Limit, Continuity and Derivative
Chapter 3 – Integration
Chapter 4 – Differential Equations
Chapter 5 – Kinematics of Motion in a Straight Line
Chapter 6 – Analytical Geometry
Chapter 7 – Conic Section
Chapter 8 – Inverse Trigonometric Functions and Their Graphs
Chapter 9 – Solution of Trigonometric Equations
Chapter 10 – Numerical Methods

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This set covers the Class 12 Mathematics Differential Equations content represented in the uploaded chapter file, including differential equations, order and degree, first-order equations, general solutions, and related applications.

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