This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 18 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 18 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The integrating factor for the equation y' + $\frac{2}{x}$ A) $x^{2}y$. B) $e^{2x}$. C) $x^{2}$. D) $e^{x}$. Show Answer Correct Answer: C) $x^{2}$. 2. What is the integral factor, $V\left(x\right)$ $\frac{dy}{dx}+y\cot x=\operatorname{cosec}x\tan^3x$ A) $-\operatorname{cosec}^2x$. B) $\sin x$. C) $\tan x$. D) $\frac{1}{\sin x}$. Show Answer Correct Answer: B) $\sin x$. 3. Is the given differential equation linear or nonlinear? $\frac{dy}{dx}=\tan x+15x^2+e^x+\frac{1}{x}$ A) Linear. B) Non linear. C) Not sure. D) None of the above. Show Answer Correct Answer: A) Linear. 4. To solve an equation solvable for dy/dx, we generally treat it as a: A) Linear equation. B) Polynomial in derivative. C) Exact equation. D) None. Show Answer Correct Answer: B) Polynomial in derivative. 5. The complementary function of (D$^{2}$-2D + 2) y = x e$^{x}$ is A) E$^{x}$ (a cos x + b sin x). B) E$^{-x}$ ( a + b x + c x$^{2}$). C) E$^{x}$ (a + b x + c x$^{2}$). D) A e$^{x}$ + b e$^{-x}$ + c e$^{2x}$. Show Answer Correct Answer: A) E$^{x}$ (a cos x + b sin x). 6. $\frac{dy}{dx}+2xy=4x$ A) $y=e^{x^2}$. B) $y=2xe^{x^2}$. C) $y=2+e^{x^2}$. D) $y=2-e^{-x^2}$. Show Answer Correct Answer: D) $y=2-e^{-x^2}$. 7. $\int_{ }^{ }\tan x\cot xdx$ A) $\sec x+c$. B) $\csc x+c$. C) $x+c$. D) $\tan^2x\cot^2x+c$. Show Answer Correct Answer: C) $x+c$. 8. For the homogeneous equation $y" +3y'+2y=0$ A) $m^2-3m+2=0$. B) $m^2+2=0$. C) $m^2+3m+2=0$. D) $m+3=0$. Show Answer Correct Answer: C) $m^2+3m+2=0$. 9. Which method is commonly used to solve Cauchy's and Euler's equations? A) Reduction of order. B) Separation of variables. C) Variation of parameters. D) Method of undetermined coefficients. Show Answer Correct Answer: C) Variation of parameters. 10. Determine the general solution of $25y"-20y'+4y=0$ A) $C_1e^{\frac{2}{5}x}+C_2xe^{\frac{2}{5}x}$. B) $y=C_1e^{\frac{2}{5}x}+C_2xe^{-\frac{2}{5}x}$. C) $y=C_1e^{\frac{2}{5}x}+C_2e^{\frac{2}{5}x}$. D) $y=e^{\frac{2}{5}x}\left(C_1\cos x+C_2\sin x\right)$. Show Answer Correct Answer: A) $C_1e^{\frac{2}{5}x}+C_2xe^{\frac{2}{5}x}$. 11. In Euler's method, what does the step size h represent? A) Value of the derivative. B) Length of the function. C) Number of variables. D) Size of the intervals. Show Answer Correct Answer: D) Size of the intervals. 12. Solve the differential equation. $\frac{dy}{dx}=\frac{2x}{y}$ A) $y^2=x+C$. B) $\frac{y^2}{2}=x^2+C$. C) $y^2=x^2+C$. D) $y=x+C$. Show Answer Correct Answer: B) $\frac{y^2}{2}=x^2+C$. 13. What is the impact of delays in CPS systems? A) Delays in CPS systems enhance performance metrics. B) Delays in CPS systems improve overall efficiency. C) Delays in CPS systems can cause performance degradation, safety risks, and operational inefficiencies. D) Delays in CPS systems have no impact on safety. Show Answer Correct Answer: C) Delays in CPS systems can cause performance degradation, safety risks, and operational inefficiencies. 14. If the PDE is of the type f(z, p, q)=0, then we will take ..... A) P=a. B) F(x, p)=a. C) P=q. D) P=aq. Show Answer Correct Answer: D) P=aq. 15. The initial guess for $f\left(x\right)-4\cos3x+2x^2$ A) $y_p=\left(C\cos3x+Fx^2\right)$. B) $y_p=\left(C\cos3x+D\sin3x+Fx^2\right)$. C) $y_p=\left(C\cos3x-D\sin3x+Fx^2+G\right)$. D) $y_p=\left(C\cos3x+D\sin3x+Fx^2+Gx+H\right)$. Show Answer Correct Answer: D) $y_p=\left(C\cos3x+D\sin3x+Fx^2+Gx+H\right)$. 16. State the order and degree of the differential function $\frac{\text{d}y}{\text{d}x}+\frac{\text{d}^2y}{\text{d}x^2}=10$ A) 2nd order and degree 1. B) 1st order and degree 2. C) 2nd order and degree 2. D) 1st order and degree 1. Show Answer Correct Answer: A) 2nd order and degree 1. 17. The particular integral represents the solution of the: A) Non-homogeneous part. B) Homogeneous part. C) Exact part. D) None. Show Answer Correct Answer: A) Non-homogeneous part. 18. If the roots of A.E are real and equal then C.F is $y=e^{m_1x}\left(c_1+c_2\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 19. The particular integral of (D$^{3}$+3D$^{2}$+3D+1)y=e$^{-x}$ is ..... A) X$^{3}$e$^{-x}$/3. B) X$^{3}$e$^{-2x}$/6. C) X$^{3}$e$^{-x}$/6. D) X$^{3}$e$^{-3x}$/6. Show Answer Correct Answer: C) X$^{3}$e$^{-x}$/6. 20. Let the function $y=f\left(x\right)$ $\frac{dy}{dx}=4-x$ $y\left(-1\right)=2.$ A) $y=2x+\frac{x^2}{2}-\frac{11}{2}$. B) $y=4x-\frac{x^2}{2}+\frac{13}{2}$. C) $y=-4x+\frac{x^2}{2}-\frac{13}{2}$. D) $y=2x-\frac{x^2}{2}-\frac{11}{2}$. Show Answer Correct Answer: B) $y=4x-\frac{x^2}{2}+\frac{13}{2}$. 21. What is the form of a first-order linear differential equation? A) Dy/dx = P(x)y + Q(x). B) Dy/dx + P(x)y = Q(x). C) Dy/dx = Q(x) + P(y). D) Dy/dx = P(y)x + Q(y). Show Answer Correct Answer: B) Dy/dx + P(x)y = Q(x). 22. Consider the differential equation $\frac{dy}{dx}=e^y\left(4x^2-5x\right)$ A) Y=-2(x-1). B) Y=2(x-1). C) Y=x-1. D) Y=-(x-1). Show Answer Correct Answer: D) Y=-(x-1). 23. If $\frac{dy}{dx}=2y^2$ $y=-1$ $x=1, $ $x=2$ $y=?$ A) $-\frac{1}{3}$. B) $\frac{2}{3}$. C) $\frac{1}{3}$. D) $0$. E) $-\frac{2}{3}$. Show Answer Correct Answer: A) $-\frac{1}{3}$. 24. Sum of the Eigen values of a matrix is equal to ..... of the matrix A) Determinant. B) Sum. C) Trace. D) Product. Show Answer Correct Answer: C) Trace. 25. Which of the following statements is true about slope fields? A) Slope fields provide a graphical representation of the slopes of solutions at given points. B) Slope fields are only used for linear differential equations. C) Slope fields show the exact solution to a differential equation. D) Slope fields are used to solve algebraic equations. Show Answer Correct Answer: A) Slope fields provide a graphical representation of the slopes of solutions at given points. 26. The complementary function of the equation $\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}=0$ A) $\left(c_1+c_2x\right)+c_3e^{-2x}$. B) $\left(c_1+c_2x\right)+c_3e^{-x}$. C) $\left(c_1+c_2\right)+c_3e^x$. D) $\left(c_1+c_2x\right)+c_3e^x$. Show Answer Correct Answer: B) $\left(c_1+c_2x\right)+c_3e^{-x}$. 27. Evaluate the integral ( int ..... 0^{pi} sin(x) , dx ). A) 1. B) 2. C) (frac{pi}{2}). D) 3. Show Answer Correct Answer: B) 2. 28. $\frac{dy}{dx}=x+y$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 29. If y satisfies $\frac{dy}{dx}=\frac{1}{2y}$ A) $\tfrac12\ln|y|+3$. B) $\tfrac12\ln|x|+3$. C) $\sqrt{x}+3$. D) $\sqrt{x+3}$. Show Answer Correct Answer: D) $\sqrt{x+3}$. 30. If $f'\left(a\right)=0$ $f'\left(x\right)$ $x=a$ $f\left(x\right)$ A) A relative minimum at x=a. B) No relative extrema at x=a. C) A relative maximum at x=a. D) A vertical tangent line at x=a. Show Answer Correct Answer: C) A relative maximum at x=a. 31. If u is homogeneous function of order n then $\frac{\partial u}{\partial x}\, \\frac{\partial u}{\partial y}$ A) N+1. B) N-1. C) N-2. D) N. Show Answer Correct Answer: B) N-1. 32. If $\frac{dy}{dx}=\sin x\\cos^2x$ $\frac{\pi}{2}$ A) -1/3. B) 1. C) 1/3. D) -1. Show Answer Correct Answer: A) -1/3. 33. Use separation of variables to solve the differential equation $\frac{dy}{dx} = \frac{y}{x}$ A) $y = Cx^2$. B) $y = Cx^3$. C) $y = Cx$. D) $y = C\ln(x)$. Show Answer Correct Answer: C) $y = Cx$. 34. Clairaut's equation geometrically represents: A) A set of tangents. B) A set of circles. C) Straight lines with slope p. D) None. Show Answer Correct Answer: A) A set of tangents. 35. $L^{-1}\left[\frac{1}{\left(s+3\right)^2+25}\right]=$ A) $e^{-x}\left(1-x\right)$. B) $\frac{1}{5}e^{-3x}\sin5x$. C) $\frac{1}{3}e^{-3x}\sin5x$. D) $e^{-2x}\left(1-2x\right)$. Show Answer Correct Answer: B) $\frac{1}{5}e^{-3x}\sin5x$. 36. Degree of Differential equation is A) Power of highest order derivative. B) Highest power. C) Highest order derivative in DE. D) None of above. Show Answer Correct Answer: A) Power of highest order derivative. 37. Suppose $y(x)$ $\frac{dy}{dx} = 2y$ $y(0) = 3$ A) $y = 2e^{3x}$. B) $y = 2x + 3$. C) $y = 3x^2$. D) $y = 3e^{2x}$. Show Answer Correct Answer: D) $y = 3e^{2x}$. 38. Classify the PDE:u ..... t + a u ..... x = 0. A) Linear. B) Hyperbolic. C) Elliptic. D) Parabolic. Show Answer Correct Answer: B) Hyperbolic. 39. It pertains to the highest order of the derivative that appears in the differential equation. A) Solution. B) Differential Equation. C) Separable. D) Order. Show Answer Correct Answer: D) Order. 40. What is the general solution to $y" + 9y = 0$ A) $y(x) = C\sin(3x) + D\cos(3x)$. B) $y(x) = Cx^3 + Dx^9$. C) $y(x) = Ce^{3ix} + De^{-3ix}$. D) $y(x) = Ce^{3x} + De^{-3x}$. Show Answer Correct Answer: A) $y(x) = C\sin(3x) + D\cos(3x)$. 41. Which of the following is NOT a solution to $y" + y = 0$ A) $y(x) = e^{x}$. B) $y(x) = e^{ix}$. C) $y(x) =\cos(x)$. D) $y(x) = \sin(x)$. Show Answer Correct Answer: A) $y(x) = e^{x}$. 42. What type of differential equations are discussed in the text? A) Second-order linear differential equations. B) First-order linear differential equations. C) Partial differential equations. D) Non-linear differential equations. Show Answer Correct Answer: A) Second-order linear differential equations. 43. Which of the following initial value problems can be solved using Euler's Method? A) $y = 2x + 3$. B) $y = x^2 + 1$. C) $\frac{dy}{dx} = x^2$ $y(0) = 1$. D) $x^2 + y^2 = 1$. Show Answer Correct Answer: C) $\frac{dy}{dx} = x^2$ $y(0) = 1$. 44. What is the role of simulation in CPS modeling? A) Simulation is only used for visual representation of CPS models. B) Simulation has no impact on the performance of CPS models. C) Simulation is primarily for aesthetic purposes in CPS modeling. D) Simulation enables testing, validation, and optimization of CPS models. Show Answer Correct Answer: D) Simulation enables testing, validation, and optimization of CPS models. 45. What are the Lagrange's multipliers while solving the PDE $p\left(y-z\right)-q\left(2x+y\right)=2x+z$ A) $x, -2y, z$. B) $2x, y, z$. C) $2x, -z, -y$. D) $x, y, z$. Show Answer Correct Answer: C) $2x, -z, -y$. 46. $\frac{dy}{dt}=\frac{4t}{y}$ $y\left(0\right)=1$ A) An initial value problem. B) A boundary value problem. C) All the above. D) None of the above. Show Answer Correct Answer: A) An initial value problem. 47. The general solution of a nonhomogeneous differential equation is A) CF-PI. B) CF + PI. C) PI alone. D) CF alone. Show Answer Correct Answer: B) CF + PI. 48. If R(x) is an increasing concave down function, then a Trapezoidal Riemann sum will be an ..... A) Underapproximation. B) Overapproximation. C) All the above. D) None of the above. Show Answer Correct Answer: A) Underapproximation. 49. Which method is commonly used to solve second order homogeneous differential equations with constant coefficients? A) Integrating factor. B) Characteristic equation. C) Separation of variables. D) Variation of parameters. Show Answer Correct Answer: B) Characteristic equation. 50. Degree of $\left[1+\left(\frac{\text{d}y}{\text{d}x}\right)^2\right]^3=\left(\frac{\text{d}^2y}{\text{d}x^2}\right)^2$ A) 4. B) 3. C) 2. D) 6. Show Answer Correct Answer: C) 2. 51. Which of the following is a limitation of the Variation of Parameters method? A) It cannot be applied when the Wronskian is zero. B) It cannot be applied to second-order differential equations. C) It cannot be applied to equations with constant coefficients. D) It requires the forcing function f(x) to be a polynomial. Show Answer Correct Answer: A) It cannot be applied when the Wronskian is zero. 52. The roots of the auxiliary equation help in determining the: A) Complementary function. B) Particular integral. C) Singular solution. D) None. Show Answer Correct Answer: A) Complementary function. 53. Determine the order of the following equation $5\left(\frac{\text{d}^4x}{\text{d}y^4}\right)^3+7\left(\frac{\text{d}x}{\text{d}y}\right)^{10}+y=x$ A) 7. B) 12. C) 10. D) 4. Show Answer Correct Answer: D) 4. 54. Solves this following DE by using Separable Variable Method $y^5\frac{\text{d}y}{\text{d}x}=e^x$ A) $\frac{y^6}{6}=e^x+c$. B) $\frac{y^6}{5}=xe^x+c$. C) $y^5=\frac{e^x}{x}+c$. D) $y^6=e^x+c$. Show Answer Correct Answer: A) $\frac{y^6}{6}=e^x+c$. 55. $\frac{dy}{dx}=\frac{x^3}{y}$ A) $y=\sqrt{x^4+50}$. B) $y=\frac{x^4+50}{2}$. C) $y=\sqrt{\frac{x^4+50}{2}}$. D) None of the above. Show Answer Correct Answer: C) $y=\sqrt{\frac{x^4+50}{2}}$. 56. Clairaut's form is a special case of equations solvable for: A) Y. B) X. C) Dy/dx. D) None. Show Answer Correct Answer: C) Dy/dx. 57. Which of the following situations do not require differential equations? A) Multi-storey building subjected to earthquake. B) Velocity of a vehicle. C) Concentration of pollutant. D) Amount of cement in concrete. Show Answer Correct Answer: D) Amount of cement in concrete. 58. $\frac{1}{y}dy=\left(2-x\right)dx$ A) $\ln\left|y\right|=2x-\frac{1}{2}x^2+C$. B) $\ln\left(y\right)=x-x^2+C$. C) $-\frac{1}{2}y^{-2}=x-\frac{1}{2}x^2+C$. D) $\ln\left|y\right|=\frac{1}{2}x^2-2x+C$. Show Answer Correct Answer: A) $\ln\left|y\right|=2x-\frac{1}{2}x^2+C$. 59. Find the general solution. $\frac{dy}{dx}=\frac{x+3}{y}$ A) $y=Ce^{\left(x^2+3x\right)}$. B) $y=\sqrt[]{x^2+6x+C}$. C) $y=\ln\left(x^2+3x+C\right)$. D) $y=\sqrt[]{2x^2+6x+C}$. Show Answer Correct Answer: B) $y=\sqrt[]{x^2+6x+C}$. 60. Average rate of change is ..... A) $y-y_1=m\left(x-x_1\right)$. B) $\int_a^bf\left(x\right)dx$. C) $\frac{1}{b-a}\int_a^bf\left(x\right)dx$. D) $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. Show Answer Correct Answer: D) $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. ← 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