This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 3 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 3 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Can you explain the concept of boundary conditions in the context of partial differential equations? A) Boundary conditions have no impact on the solution. B) Boundary conditions only apply to ordinary differential equations. C) Boundary conditions define the behavior of the solution at the boundaries of the domain. D) Boundary conditions are only relevant for linear equations. Show Answer Correct Answer: C) Boundary conditions define the behavior of the solution at the boundaries of the domain. 2. What is the general form of a first-order PDE? A) F(x, y, u, p, q) = 0. B) F(x, y, u, q) = 0. C) F(x, y, u, p) = 0. D) F(x, y, u) = 0. Show Answer Correct Answer: A) F(x, y, u, p, q) = 0. 3. State the order and degree of this following Differential Equations $\left(\frac{\text{d}^3x}{\text{d}y^3}\right)^5+xy=1-x$ A) Order=1Degree=0. B) Order=1Degree=1. C) Order=3Degree=5. D) Order=5Degree=3. Show Answer Correct Answer: C) Order=3Degree=5. 4. $\frac{dy}{dx}+x^2y=\left(xy\right)^3$ A) Homogeneous. B) Linear. C) Separable. D) Bernoulli. Show Answer Correct Answer: D) Bernoulli. 5. The solution of simultaneous linear differential equations is of the form ..... A) $\Phi\left(v\right)=0$. B) $\Phi\left(u\right)=0$. C) $\Phi\left(u, v\right)=0$. D) $\Phi\left(x, y\right)=0$. Show Answer Correct Answer: C) $\Phi\left(u, v\right)=0$. 6. Which of the following is a method of solution for separable differential equations? A) Euler's method. B) Method of separation of variables. C) Runge-Kutta method. D) Method of undetermined coefficients. Show Answer Correct Answer: B) Method of separation of variables. 7. Find the integrating factor to solve the equation $\frac{dy}{dx}+6y=x^2$ A) $6x$. B) $e^{7x}$. C) $e^{6x}$. D) $6y$. Show Answer Correct Answer: C) $e^{6x}$. 8. Which method is used to solve first-order linear PDEs? A) Green's function approach. B) Fourier series method. C) Method of characteristics. D) Separation of variables. Show Answer Correct Answer: C) Method of characteristics. 9. Find the general solution of the following DE $\frac{dy}{dx}=\frac{2x}{e^{2y}}$ A) $y=\ln\left(2x^2+C\right)$. B) $y=\frac{1}{2}\ln\left(2x^2+C\right)$. C) $y=e^{2x}+C$. D) $y=\frac{1}{2}\ln\left(x^2+C\right)$. Show Answer Correct Answer: B) $y=\frac{1}{2}\ln\left(2x^2+C\right)$. 10. Wave equation is classified as: A) Elliptic PDE. B) Parabolic PDE. C) Ordinary differential equation. D) Hyperbolic PDE. Show Answer Correct Answer: D) Hyperbolic PDE. 11. What is the significance of the Laplace equation? A) The Laplace equation is significant for modeling steady-state phenomena in physics and engineering. B) The Laplace equation is used to calculate the speed of sound in fluids. C) The Laplace equation describes the motion of celestial bodies in space. D) The Laplace equation is essential for analyzing time-dependent processes in thermodynamics. Show Answer Correct Answer: A) The Laplace equation is significant for modeling steady-state phenomena in physics and engineering. 12. Which method is used to solve homogeneous linear differential equations with constant coefficients? A) Method of Laplace transforms. B) Method of separation of variables. C) Method of undetermined coefficients. D) Method of variation of parameters. Show Answer Correct Answer: B) Method of separation of variables. 13. The highest order derivative present in a differential equation determines its: A) Order. B) Homogeneity. C) Degree. D) Type. Show Answer Correct Answer: A) Order. 14. $L\left(xe^{3x}\right)= ..... $ A) $-\\\frac{1}{\left(s+3\right)^2}$. B) $-\\\\frac{1}{\left(s-3\right)^2}$. C) $\frac{1}{\left(s+3\right)^2}$. D) $\frac{1}{\left(s-3\right)^2}$. Show Answer Correct Answer: D) $\frac{1}{\left(s-3\right)^2}$. 15. If a function $f$ $x, $ $a>0, \\b>0, $ $\int_0^af\left(x\right)dx$ $\int_b^{a+b}f\left(x-b\right)dx$ $\int_b^{a+b}f\left(x+b\right)dx$ A) I and II only. B) I and III only. C) II and III only. D) None. Show Answer Correct Answer: A) I and II only. 16. Identify the order and degree of the equation dy/dx + y = sin(x). A) 3, 1. B) 2, 1. C) 0, 1. D) 1, 1. Show Answer Correct Answer: D) 1, 1. 17. If $\frac{dy}{dx}=\frac{x^2}{y}$ $x=0$ $y=4, $ A) $y=\frac{x^3}{3}$. B) $\frac{y^2}{2}=\frac{x^3}{3}$. C) $\frac{y^2}{2}=\frac{x^3}{3}+4$. D) $\frac{y^2}{2}=\frac{x^3}{3}+8$. Show Answer Correct Answer: D) $\frac{y^2}{2}=\frac{x^3}{3}+8$. 18. Clairaut's equation is important because it demonstrates the existence of: A) Singular solutions. B) Particular integrals. C) Complementary functions. D) None. Show Answer Correct Answer: A) Singular solutions. 19. To solve simultaneous equations, we can eliminate variables using: A) Substitution or matrix methods. B) Euler's method. C) Determinants. D) Laplace transforms. Show Answer Correct Answer: A) Substitution or matrix methods. 20. The general solution of a non-homogeneous DE = A) Complementary function. B) Particular solution. C) CF + Particular solution. D) None. Show Answer Correct Answer: C) CF + Particular solution. 21. The following Differential Equation is $x\frac{\text{d}y}{\text{d}x}=yx^2$ A) Separable. B) Non Separable. C) All the above. D) None of the above. Show Answer Correct Answer: A) Separable. 22. Define Zeno behavior in the context of CPS. A) Zeno behavior is the phenomenon of continuous systems achieving instant results. B) Zeno behavior in CPS is the non-terminating or oscillatory behavior caused by discrete events interacting with continuous dynamics. C) Zeno behavior refers to the rapid convergence of discrete events. D) Zeno behavior is a method for optimizing continuous processes. Show Answer Correct Answer: B) Zeno behavior in CPS is the non-terminating or oscillatory behavior caused by discrete events interacting with continuous dynamics. 23. What are the basic types of partial differential equations? A) Elliptic, Parabolic, and Hyperbolic. B) Scalar, Vector, Matrix. C) Linear, Nonlinear, Quadratic. D) Algebraic, Geometric, Trigonometric. Show Answer Correct Answer: A) Elliptic, Parabolic, and Hyperbolic. 24. The velocity of a particle is given by the equation of 8t$^{3}$+9t$^{2}$. If the position at t=0 is 0, find the position of the particle at t=5. Do not use a calculator. A) 1225. B) 750. C) 1625. D) 715. Show Answer Correct Answer: C) 1625. 25. Which of the following is an example for first order linear partial differential equation? A) Lagrange's Partial Differential Equation. B) One-dimensional Wave Equation. C) Clairaut's Partial Differential Equation. D) One-dimensional Heat Equation. Show Answer Correct Answer: A) Lagrange's Partial Differential Equation. 26. Solve the homogeneous equation y" + 2y' + y = 0. A) $y_c=Ae^{-x}+Bxe^x$. B) $y_c=Ae^{-x}+Bxe^{-x}$. C) $y_c=Ae^{-x}+Be^{-x}$. D) $y_c=Ae^x+Bxe^x$. Show Answer Correct Answer: B) $y_c=Ae^{-x}+Bxe^{-x}$. 27. Which of the following function is neither even nor odd? A) $x^2$. B) $x^2+x$. C) $\sin x$. D) $\cos x$. Show Answer Correct Answer: B) $x^2+x$. 28. The sum of a power series is not analytic at the end points inside the interval of convergence. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 29. Solve the initial value problem:$y' = 0.02y + 6$ $y(0) = 100$ $y(t) =$ A) $300e^{0.02t}-200$. B) $200e^{0.02t}-300$. C) $100e^{0.02t}$. D) $200e^{0.02t}$. Show Answer Correct Answer: C) $100e^{0.02t}$. 30. $L\left(e^{ax}\right)= ..... $ A) $\frac{1}{s+a}$. B) $\frac{a}{s+a}$. C) $\frac{1}{s-a}$. D) $\frac{a}{s-a}$. Show Answer Correct Answer: C) $\frac{1}{s-a}$. 31. The number of radioactive atoms, N, in a sample is described by the differential equation $\frac{dN}{dt}=-kN$ A) 1.290. B) 1.835. C) 0.545. D) 2.380. Show Answer Correct Answer: B) 1.835. 32. If Y is the complementary function and u is the particular integral then the general solution of (D$^{n}$ + a$_{1}$D$^{n-1}$ + a$_{2}$D$^{n-2}$ + ..... + a$_{n}$) y = X is of the form A) Y = c$_{1}$Y + c$_{2}$u. B) Y = c$_{1}$y+ c$_{2}$u. C) Y = Y + u. D) Y = y + u. Show Answer Correct Answer: C) Y = Y + u. 33. In a garden, the rate flowers bloom F with respect to time t is inversely proportional to the square of the amount of rain R (in inches). Approximately 8 flowers bloom per day with 2.6 inches of rain. A) $\frac{dF}{dt}=\frac{4.96}{R^2}$. B) $\frac{dF}{dt}=1.18\sqrt[]{R}$. C) $\frac{dF}{dt}=\frac{54.08}{R^2}$. D) $\frac{dF}{dt}=\frac{12.9}{\sqrt[]{R}}$. Show Answer Correct Answer: C) $\frac{dF}{dt}=\frac{54.08}{R^2}$. 34. Let $y=f(x)$ $\frac{dy}{dx}=x-2y$ $f(3)=2$ $f(4)$ $x=3$ A) $\tfrac34$. B) $-1$. C) $\tfrac74$. D) $\tfrac14$. Show Answer Correct Answer: C) $\tfrac74$. 35. $\frac{dy}{dx}=6y^2x$ $y\left(1\right)=\frac{1}{25}$ A) $y=28-3x^2$. B) $y=\frac{1}{28-3x^2}$. C) $y=\frac{1}{3x^2-28}$. D) None of the above. Show Answer Correct Answer: B) $y=\frac{1}{28-3x^2}$. 36. Please find $\lim_{x\rightarrow0}\frac{2x+\tan3x}{\sin x}$ A) 5. B) 2. C) $\infty$. D) 0. Show Answer Correct Answer: A) 5. 37. What is the order in the given differential equation:(1-x) y" -4xy' + 5y = cosx A) 2. B) 4. C) 3. D) 1. Show Answer Correct Answer: A) 2. 38. Legendre's type equations are transformed into: A) Algebraic equations. B) Linear equations with constant coefficients. C) First-order equations. D) Separable differential equations. Show Answer Correct Answer: B) Linear equations with constant coefficients. 39. The general form of a linear differential equation with constant coefficients involves: A) Powers of derivatives with constant multipliers. B) Homogeneous functions only. C) Exponential terms. D) None. Show Answer Correct Answer: A) Powers of derivatives with constant multipliers. 40. Solve using Lagrange's multipliers(y-2)p+(z-x)q=(x-y) A) F(y+z, x$^{2}$+ y$^{2}$)=0. B) F(y+z, x$^{2}$+ y$^{2}$+z$^{2}$)=0. C) F(x+y+z, x$^{2}$+ y$^{2}$+z$^{2}$)=0. D) F(x+3y+2z, 6x$^{2}$+ y$^{2}$+z$^{2}$)=0. Show Answer Correct Answer: C) F(x+y+z, x$^{2}$+ y$^{2}$+z$^{2}$)=0. 41. At each point on a certain curve, the slope of the curve is $3x^2=\frac{\text{d}y}{\text{d}x}$ A) $y=x^3+8$. B) $y=8e^{x^3}$. C) $y=\ln\left(x+1\right)+8$. D) $y=e^{x^3}+7$. Show Answer Correct Answer: A) $y=x^3+8$. 42. For the system dx/dt=3x+4y, dy/dt=-4x+3y, the general solution is: A) Real exponentials. B) Oscillatory (sine and cosine terms). C) Polynomials. D) Constant solutions. Show Answer Correct Answer: B) Oscillatory (sine and cosine terms). 43. Which of the following is the graph of the solution to the differential equation $\frac{dy}{dx} =-2y$ A) An exponential decay curve. B) A line with a negative slope. C) A line with a positive slope. D) An exponential growth curve. Show Answer Correct Answer: A) An exponential decay curve. 44. At the point of discontinuity, sum of the series is equal to ..... A) 1/2[f(x+0)+f(x-0)]. B) 1/4[f(x+0)-f(x-0)]. C) 1/4[f(x+0)+f(x-0)]. D) 1/2[f(x+0)-f(x-0)]. Show Answer Correct Answer: A) 1/2[f(x+0)+f(x-0)]. 45. Now that we have separated, what would be the correct integration? $e^{-y}dy=e^xdx$ A) $-e^{-y}=e^x+C$. B) $e^{-y}=e^x+c$. C) $-e^{-y}=-e^x+c$. D) $e^y=e^x+c$. Show Answer Correct Answer: A) $-e^{-y}=e^x+C$. 46. The solution of y" '-y" -y'+y=0 is obtained by: A) Using separation of variables. B) Applying Laplace transforms. C) Direct integration. D) Finding characteristic roots. Show Answer Correct Answer: D) Finding characteristic roots. 47. What is the order & degree of the p.d.e $x^2\left(\frac{\partial z}{\partial x}\right)^2=z\left(x-y\frac{\partial z}{\partial y}\right)$ A) 1 & 2. B) 2 & 1. C) 2 & 2. D) None of the above. Show Answer Correct Answer: A) 1 & 2. 48. $\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial x\partial y}-2\frac{\partial^2z}{\partial y^2}=0$ A) $Z=f_1\left(y+x\right)+xf_2\left(y-2x\right)$. B) $Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$. C) $Z=f_1\left(y\right)+f_2\left(y+2x\right)$. D) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)$. Show Answer Correct Answer: B) $Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$. 49. (D$^{2}$-4)y=0 has particular integral alone as general solution. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 50. $\overrightarrow{a}\times\overrightarrow{b}=\overrightarrow{0}$ $\overrightarrow{a}$ $\overrightarrow{b}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 51. If A has distinct real eigenvalues, then solution trajectories are A) Spiral. B) Always unstable. C) Straight line combination of eigenvalues. D) Circles. Show Answer Correct Answer: C) Straight line combination of eigenvalues. 52. The solution of y" -4y'+4y=0 is: A) $c_1e^{2x}+c_2xe^{2x}$. B) $c_1e^{4x}+c_2e^{-4x}$. C) $c_1e^{-2x}+c_2e^{2x}$. D) $c_1e^{2x}+c_2e^{-2x}$. Show Answer Correct Answer: A) $c_1e^{2x}+c_2xe^{2x}$. 53. Which of the following differential equations best models exponential growth? A) $\frac{dy}{dt} =-ky$. B) $\frac{dy}{dt} = k$. C) $\frac{dy}{dt} = ky$. D) $\frac{dy}{dt} = y^2$. Show Answer Correct Answer: C) $\frac{dy}{dt} = ky$. 54. What would the sign of the square root after finding c be given the initial condition g(1)=-5 A) Positive. B) Negative. C) All the above. D) None of the above. Show Answer Correct Answer: B) Negative. 55. Describe a method for verifying CPS properties. A) Use simulation to verify CPS properties. B) Apply static analysis for CPS verification. C) Conduct manual code reviews for CPS properties. D) Use model checking to verify CPS properties. Show Answer Correct Answer: D) Use model checking to verify CPS properties. 56. SOLVE USING LAGRANGE'S LINEAR EQUATON2p+3q=1 A) F(3x-2y, y-3x). B) F(3x-2y, y-3z). C) F(3x-2y, x-3z). D) F(3x-2y, y+3z). Show Answer Correct Answer: B) F(3x-2y, y-3z). 57. Kinetic energy, E, changes with respect to t at a rate that is proportional to the square of the velocity, v, and inversely proportional to the mass, m. A) $\frac{dE}{dt}=\frac{k\sqrt[]{v}}{m}$. B) $\frac{dE}{dt}=\frac{km}{\sqrt[]{v}}$. C) $\frac{dE}{dt}=\frac{kv^2}{m}$. D) $\frac{dE}{dt}=\frac{km}{v^2}$. Show Answer Correct Answer: C) $\frac{dE}{dt}=\frac{kv^2}{m}$. 58. A radioactive substance decays according to the differential equation dN/dt =-0.03N, where N(0) = 500. What is the amount of substance left after 20 days? A) 200.50. B) 550.75. C) 301.194. D) 450.25. Show Answer Correct Answer: C) 301.194. 59. The Cayley Hamilton theorem is useful for computing $e^{At}$ A) It allows expressing $e^{At}$. B) It provides the eigenvalues of A directly. C) It simplifies matrix multiplication. D) It determines the rank of A. Show Answer Correct Answer: A) It allows expressing $e^{At}$. 60. The Legendre differential equation is a special case of: A) Exact differential equations. B) Linear differential equations with variable coefficients. C) Nonlinear differential equations. D) Partial differential equations. Show Answer Correct Answer: B) Linear differential equations with variable coefficients. ← 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 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 8Class 12 Mathematics Chapter 9 Differential Equations Quiz 9 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books