This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 14 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 14 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The inverse differential operator D1 acting on a function f(x) is equivalent to which mathematical operation? A) Differentiation with respect to x. B) Taking the reciprocal of the function. C) Integration with respect to x. D) Squaring the function. Show Answer Correct Answer: C) Integration with respect to x. 2. What is the significance of the order of a differential equation? A) It indicates the complexity of the equation. B) It shows the highest derivative present. C) It defines the type of function involved. D) It determines the number of solutions. Show Answer Correct Answer: B) It shows the highest derivative present. 3. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? A) $\ln\left(\frac{27}{2}\right)$. B) $\ln\left(\frac{9}{2}\right)$. C) $\frac{2\ln\left(3\right)}{\ln\left(2\right)}$. D) $\frac{3\ln\left(3\right)}{\ln\left(2\right)}$. E) $\frac{\ln\left(3\right)}{\ln\left(2\right)}$. Show Answer Correct Answer: D) $\frac{3\ln\left(3\right)}{\ln\left(2\right)}$. 4. L\{(1/2b$^{2}$)(sin bt-bt cos bt)\} A) B/(s$^{2}$+b$^{2}$)$^{4}$. B) B/(s$^{2}$+b$^{2}$). C) B/(s$^{2}$+b$^{2}$)$^{2}$. D) B/(s$^{2}$+b$^{2}$)$^{3}$. Show Answer Correct Answer: C) B/(s$^{2}$+b$^{2}$)$^{2}$. 5. What is the order of the differential equation:$\frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^2 + y = 0$ A) 3. B) 4. C) 1. D) 2. Show Answer Correct Answer: A) 3. 6. Determine the degree of the equation 2y" + 5y' + 3y = 0. A) 2. B) 3. C) 1. D) 4. Show Answer Correct Answer: C) 1. 7. Find the particular integral of (D$^{2}$-2DD$^{' }$+D$^{'2 }$)z = e$^{(x+2y)}$ A) E$^{x-y}$. B) 2e$^{x}$. C) 3e$^{x+2y}$. D) E$^{x+2y}$. Show Answer Correct Answer: D) E$^{x+2y}$. 8. Method of undetermined coefficients is used when: A) DE has polynomial, exponential, sine, cosine RHS. B) DE is nonlinear. C) Initial value problems only. D) None. Show Answer Correct Answer: A) DE has polynomial, exponential, sine, cosine RHS. 9. What are the factors affecting heat conduction according to Fourier's law? A) Shape, texture, and weight. B) Color, density, and volume. C) Time, pressure, and velocity. D) Temperature gradient, cross-sectional area, distance, and material's thermal conductivity. Show Answer Correct Answer: D) Temperature gradient, cross-sectional area, distance, and material's thermal conductivity. 10. If $\phi$ $\nabla\phi$ A) Surface. B) Tangent line. C) Normal. D) Tangent plane. Show Answer Correct Answer: C) Normal. 11. How would you CORRECTLY separate the following differential equation? $\frac{dy}{dx}=2y-xy$ A) $y\cdot dy=\left(2-x\right)dx$. B) $\frac{dy-2y}{y}=xdx$. C) $\frac{1}{y}dy=\left(2-x\right)dx$. D) $\frac{-2y+dy}{y}=x\cdot dx$. Show Answer Correct Answer: C) $\frac{1}{y}dy=\left(2-x\right)dx$. 12. Sinx is analytic at which point? A) 1. B) 2. C) 0. D) -1. Show Answer Correct Answer: C) 0. 13. Why do we add a constant $C$ A) Because derivatives are unique. B) To make answers unique. C) To satisfy boundary conditions only. D) Many functions can have the same derivative. Show Answer Correct Answer: D) Many functions can have the same derivative. 14. What is the method of solution for homogeneous differential equations? A) Method of Solution of Homogeneous Differential Equation. B) Linear Differential Equation. C) Separable differential equation. D) Exact differential equation. Show Answer Correct Answer: A) Method of Solution of Homogeneous Differential Equation. 15. Which method is used to solve nonlinear differential equations? A) Exact equations. B) Separation of variables. C) Integrating factor. D) Substitution. Show Answer Correct Answer: D) Substitution. 16. Solve the following differential equations:$\frac{\text{d}y}{\text{d}x}=e^x$ A) $1=e^x+1$. B) $0=e^x+C$. C) $y=\frac{e^x}{2}+1$. D) $y=e^x+C$. Show Answer Correct Answer: A) $1=e^x+1$. 17. Solve dy/dx = 2x/y A) $y=\pm\sqrt[]{x^2+C}$. B) $y=\pm\sqrt[]{4x^2+C}$. C) $y=\pm\sqrt[]{2x^2+C}$. D) $y=\pm\sqrt[]{4x+C}$. Show Answer Correct Answer: C) $y=\pm\sqrt[]{2x^2+C}$. 18. The annihilator of sin bx is A) D2-b2. B) D2+ b2. C) D-b. D) D2+D. Show Answer Correct Answer: A) D2-b2. 19. The following are the requirements of a DE to be linear except one A) 1st order. B) 1st degree. C) No transcendental functions of the dependent variable and its derivatives. D) No product of the dependent variable and its derivatives. Show Answer Correct Answer: A) 1st order. 20. An equation that contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable A) Dependent Differential Equation. B) Ordinary Differential Equation. C) First Order Differential Equation. D) Partial Differential Equation. Show Answer Correct Answer: B) Ordinary Differential Equation. 21. Differential equations in control systems are mainly used to describe ..... A) Static behavior of systems. B) Stability of algebraic equations. C) Geometric modeling only. D) Dynamic behavior of systems. Show Answer Correct Answer: D) Dynamic behavior of systems. 22. What is Euler's method used for? A) Solving algebraic equations. B) Approximating solutions to ordinary differential equations. C) Finding prime numbers. D) Measuring angles in a triangle. Show Answer Correct Answer: B) Approximating solutions to ordinary differential equations. 23. Solutions of differential equations are categorized into general and particular. Identify the type of solution of the equations shown below.a) $\frac{\text{d}y}{\text{d}x}=2e^x+5e^{2x}$ $\frac{\text{d}y}{\text{d}x}=Ae^x+5e^{2x}$ A) Particular.b) Particular. B) General.b) General. C) General.b) Particular. D) Particular.b) General. Show Answer Correct Answer: D) Particular.b) General. 24. Which of the following is a method for solving first-order differential equations? A) Partial fractions. B) Matrix method. C) Integration by substitution. D) Variable separation. Show Answer Correct Answer: D) Variable separation. 25. Solve dy/dx = 5x/y given the point (0, -1) A) $y=-\sqrt[]{5x^2+5}$. B) $y=-\sqrt[]{5x^2+1}$. C) $y=\sqrt[]{5x^2+1}$. D) $y=-\sqrt[]{\frac{5}{2}x^2+1}$. Show Answer Correct Answer: B) $y=-\sqrt[]{5x^2+1}$. 26. F(s) = L\{(e$^{at}$sin(bt))\} (s) A) 1/s. B) B/(s-a)$^{2 }$+ b$^{2}$. C) (s-a)/(s-a)$^{2}$+b$^{2}$. D) S/(s$^{2}$+b$^{2}$). E) B/(s$^{2}$ + b$^{2}$). Show Answer Correct Answer: B) B/(s-a)$^{2 }$+ b$^{2}$. 27. Let PDE c$^{2}$(u$_{xx }$+ u$_{yy }$)= u$_{tt}$, By is known as A) 2-D heat equation. B) 2-D wave equation. C) Laplace equation. D) None of these. Show Answer Correct Answer: B) 2-D wave equation. 28. Part B. Find the particular solution under the given conditions:15. y" + 3y' + 2y = 0, y(0) = 1, y'(0) = 1 A) $y = 2e^{-x}-e^{-2x}$. B) $y = e^{-x}-e^{-2x}$. C) $y = e^{x}-2e^{-2x}$. D) $y = e^{-x} + e^{-2x}$. Show Answer Correct Answer: A) $y = 2e^{-x}-e^{-2x}$. 29. Let A and B be two subspaces of a vector space V. Then A U B is a subspace of V if and only if A) $A\cap B=\left\{0\right\}$. B) $A\cup B=B$. C) $A\cap B=\left\{0, 1\right\}$. D) $A\cap B=\phi$. Show Answer Correct Answer: B) $A\cup B=B$. 30. Which of these is the correct complimentary function for the equation $2\frac{\text{d}^2y}{\text{d}x}+4\frac{\text{d}y}{\text{d}x}-6y=0$ A) $Ae^{3x}+Be^{-x}$. B) $Ae^{-3x}+Be^x$. C) $Ae^{-3x}+Be^{-x}$. D) $Ae^{3x}+Be^x$. Show Answer Correct Answer: B) $Ae^{-3x}+Be^x$. 31. If N(t) is the size of a population at time t, which of the following differential equations describes exponential growth in the size of the population? A) $\frac{dN}{dt}=300$. B) $\frac{dN}{dt}=300N$. C) $\frac{dN}{dt}=300t$. D) $\frac{dN}{dt}=\frac{300}{N}$. Show Answer Correct Answer: B) $\frac{dN}{dt}=300N$. 32. The general solution of Clairaut's equation involves: A) One arbitrary constant. B) Two arbitrary constants. C) No arbitrary constant. D) None. Show Answer Correct Answer: A) One arbitrary constant. 33. Find the general solution to the following non-homogeneous DE:$\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=\cos2x$ A) $y=(A+Bx)e^x+(3/31)\cos2x+(4/31)\sin2x$. B) $y=(A+Bx)e^{-x}+(3/31)\cos2x+(4/31)\sin2x$. C) $y=(A+Bx)e^{-x}+P\cos2x+Q\sin2x$. D) $y=Ae^{-x}+Be^x+(3/31)\cos2x+(4/31)\sin2x$. Show Answer Correct Answer: B) $y=(A+Bx)e^{-x}+(3/31)\cos2x+(4/31)\sin2x$. 34. Find the order for the differential equation. A) 2. B) 6. C) 3. D) 4. Show Answer Correct Answer: C) 3. 35. What do you understand by the term 'partial derivative'? A) A derivative of a function with respect to a constant. B) A derivative of a function with respect to one of its variables, with all other variables held constant. C) A derivative of a function with respect to a new variable. D) A derivative of a function with respect to all of its variables. Show Answer Correct Answer: B) A derivative of a function with respect to one of its variables, with all other variables held constant. 36. Solve the partial differential equation $\frac{\partial^2z}{\partial x\partial y}=0$ A) $z=f\left(y\right)g\left(x\right)$. B) $z=xf\left(x\right)+g\left(y\right)$. C) $z=f\left(y\right)+g\left(x\right)$. D) $z=xf\left(y\right)+g\left(x\right)$. Show Answer Correct Answer: C) $z=f\left(y\right)+g\left(x\right)$. 37. Given the differential equation $\frac{dP}{dt}=5P$ A) P=Ce$^{10t}$B) P= 418e$^{10t}$. B) P=Ce$^{5t}$B) P= 5e$^{418t}$. C) P=Ce$^{5t}$B) P= 418e$^{5t}$. D) P=Ce$^{t}$B) P= 418e$^{t}$. Show Answer Correct Answer: C) P=Ce$^{5t}$B) P= 418e$^{5t}$. 38. Roots of the auxiliary equation determine: A) The form of the complementary solution. B) The method of solving algebraic equations. C) The Wronskian of the functions. D) The non-homogeneous term. Show Answer Correct Answer: A) The form of the complementary solution. 39. A solution obtained by giving particular values to the arbitrary constants in a complete integral is called ..... A) Singular integral. B) Particular integral. C) Complete integral. D) General integral. Show Answer Correct Answer: B) Particular integral. 40. Solve the differential equation. $\frac{dy}{dt}=\frac{\sqrt{t}}{y^3}$ A) $y=t^2+C$. B) $\frac{y^4}{4}=t^{\frac{3}{2}}+C$. C) $\frac{y^4}{4}=\frac{2}{3}t^{\frac{3}{2}}+C$. D) $y^3=t^2+C$. Show Answer Correct Answer: C) $\frac{y^4}{4}=\frac{2}{3}t^{\frac{3}{2}}+C$. 41. Find the order of the differential equation:$\frac{d^4y}{dx^4} + 2\frac{d^2y}{dx^2}-y = 0$ A) 1. B) 2. C) 4. D) 3. Show Answer Correct Answer: C) 4. 42. The power of the differential equation's highest derivative, after the equation has been made rational and integral in all of its derivatives A) Order. B) Variable. C) Degree. D) Exponent. Show Answer Correct Answer: C) Degree. 43. General solution of (D$^{2}$+9) y= 0 is A) Y =c$_{1}$ cos x + c$_{2}$ sin x +x. B) Y = x. C) Y = c$_{1}$ cos 3x + c$_{2}$ sin 3x. D) None. Show Answer Correct Answer: C) Y = c$_{1}$ cos 3x + c$_{2}$ sin 3x. 44. When can the method of separation of variables be used to solve differential equations? A) When the equation is non-homogeneous. B) When the equation is homogeneous. C) When the equation is nonlinear. D) When the equation is linear. Show Answer Correct Answer: D) When the equation is linear. 45. Which of the following is the solution to the differential equation $\frac{dy}{dx}=3y^2$ $y\left(0\right)=3$ A) $y=\frac{3}{1-9x}$. B) $y=\sqrt{9e^{3x}}$. C) $y=\frac{3}{9x-1}$. D) $y=\sqrt{\frac{1}{9}e^{3x}}$. Show Answer Correct Answer: A) $y=\frac{3}{1-9x}$. 46. An equation containing partial derivatives of one or more dependent variables of two or more independent variables A) Ordinary Partial Differential Equation. B) Partial Differential Equation. C) Ordinary Differential Equation. D) Partially Ordinary Differential Equation. Show Answer Correct Answer: B) Partial Differential Equation. 47. The value of $\int_0^1\int_0^1\left(x^2+y^2\right)dxdy$ A) 2/3. B) 2. C) 1/2. D) 2. Show Answer Correct Answer: A) 2/3. 48. Find the roots of $m^3-m^2+m-1=0$ A) 1, -1, i. B) 1, i, -i. C) 1, -1, 0. D) I, -i, 0. Show Answer Correct Answer: B) 1, i, -i. 49. If $\frac{dy}{dx}=y\\sec^2x$ A) $e^{\tan x}+4$. B) $e^{\tan x}+5$. C) $\tan x+5$. D) $5e^{\tan x}$. Show Answer Correct Answer: D) $5e^{\tan x}$. 50. Choose the Differential Equations A) $5y+3y" -x^2=1$. B) $\frac{x}{y}+1=xy$. C) $6x^3+2y^4=xy-4$. D) $3x-4y+xy=1$. Show Answer Correct Answer: A) $5y+3y" -x^2=1$. 51. For y" -y=0, y(0)=2, y'(0)=0, the solution is: A) $2\cosh x$. B) $2e^{-x}$. C) $2\sinh x$. D) $2e^x$. Show Answer Correct Answer: A) $2\cosh x$. 52. The general solution of the differential equation $\frac{\text{d}y}{\text{d}x}-\left(\cot x\right)y=\sin\left(2x\right)$ A) $y=\frac{2}{3}\sin^2\left(x\right)+A\operatorname{cosec}\left(x\right)$. B) $y=\frac{1}{2}\sin\left(2x\right)+A\sin x$. C) $y=2\sin^2\left(x\right)+\sin x+A$. D) $y=2\sin^2\left(x\right)+A\sin x$. Show Answer Correct Answer: D) $y=2\sin^2\left(x\right)+A\sin x$. 53. Solve the differential equation sec$^{2}$xtanydx+sec$^{2}$ytanxdy=0 A) -tanx tany=c. B) -sinx tany=c. C) Tanx tany=c. D) Sinx tany=c. Show Answer Correct Answer: C) Tanx tany=c. 54. $\left(\frac{\text{d}y}{\text{d}x}\right)^2-\tan x=0$ A) First order differential equation at second degree. B) Second order differential equation. C) First order differential equation. D) Second order differential equation at first degree. Show Answer Correct Answer: A) First order differential equation at second degree. 55. What is the degree of the differential equation:$\left(\frac{d^2y}{dx^2}\right)^5 + 4\frac{dy}{dx}-3y = 0$ A) 2. B) 4. C) 5. D) 3. Show Answer Correct Answer: C) 5. 56. Analyze the stability of the equilibrium solution $y = 2$ $\frac{dy}{dt} = (y-2)(y+1)$ A) Stable. B) Semi-stable. C) Cannot be determined. D) Unstable. Show Answer Correct Answer: A) Stable. 57. Based on the slope field, what can you infer about the behavior of the solution curves for the differential equation dy/dx = x-y? A) The behavior of the solution curves is oscillating between two values. B) The behavior of the solution curves is diverging to infinity. C) The behavior of the solution curves is random and unpredictable. D) The behavior of the solution curves is approaching a stable equilibrium. Show Answer Correct Answer: D) The behavior of the solution curves is approaching a stable equilibrium. 58. The other name of characteristic roots A) Root. B) Eigen vector. C) Eigen value. D) Zero. Show Answer Correct Answer: C) Eigen value. 59. The value of $\int_0^1\int_x^1\left(x^2+y^2\right)dydx$ A) 1/2. B) 1/3. C) 1. D) 2/3. Show Answer Correct Answer: B) 1/3. 60. Dp/dy = (4y)$^{-1}$find the general solution A) P= 1 + c 4y. B) P= 1/5y$^{5/4}$ + c. C) P= 1/8y$^{2}$ + c. D) P= 1/4ln|4y| + c. Show Answer Correct Answer: D) P= 1/4ln|4y| + c. ← PreviousNext →Related QuizzesScience QuizzesClass 12 QuizzesClass 12 Mathematics Chapter 9 Differential Equations Quiz 1Class 12 Mathematics Chapter 9 Differential Equations Quiz 2Class 12 Mathematics Chapter 9 Differential Equations Quiz 3Class 12 Mathematics Chapter 9 Differential Equations Quiz 4Class 12 Mathematics Chapter 9 Differential Equations Quiz 5Class 12 Mathematics Chapter 9 Differential Equations Quiz 6Class 12 Mathematics Chapter 9 Differential Equations Quiz 7Class 12 Mathematics Chapter 9 Differential Equations Quiz 8 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books