This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 11 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 11 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Cross product is commutative. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 2. If y satisfies $\frac{dy}{dt}=ky$ A) $3e^{kt}$. B) $e^{kt}+5$. C) $kty+7$. D) $3e^{kty}$. Show Answer Correct Answer: A) $3e^{kt}$. 3. How would you CORRECTLY separate the following differential equation? $\frac{dy}{dx}=\frac{2x+1}{3y}$ A) $\frac{dy}{3y}=\frac{dx}{2x+1}$. B) $3y\cdot dy=\left(2x+1\right)dx$. C) $3y^{-1}dy=\left(2x+1\right)dx$. D) $\frac{dy}{3y}=\left(2x+1\right)dx$. Show Answer Correct Answer: B) $3y\cdot dy=\left(2x+1\right)dx$. 4. $L^{-1}\left[\frac{1}{\left(s-a\right)^2}\right]=$ A) $xe^{ax}$. B) $xe^{-ax}$. C) $e^{ax}$. D) $e^x$. Show Answer Correct Answer: A) $xe^{ax}$. 5. Order and degree of the D.E. y$^{" }$+logy$^{" '}$-xy+(y" ')$^{3}$=0 A) 2, 3. B) Order-3, degree not defined. C) 3, 3. D) 3, 1. Show Answer Correct Answer: B) Order-3, degree not defined. 6. Using $v=xy$ $y+x\frac{dy}{dx}=1-xy$ A) $\frac{dv}{dx}=1-v$. B) $\frac{v}{x}+\frac{dv}{dx}=1-v$. C) $\frac{dv}{dx}+x\\frac{dy}{dx}=1-v$. D) $x\frac{dv}{dx}=v$. Show Answer Correct Answer: A) $\frac{dv}{dx}=1-v$. 7. There are two (2) types of conditions in a differential equation:Initial and Boundary. Identify the type of condition for the followings problems.a) $y\left(\pi\right)=-3$ $y'\left(\pi\right)=0$ $y\left(2\right)=1$ $y'\left(1\right)=0$ $y\left(0\right)=0$ $y'\left(0\right)=1.$ A) Initial.b) Boundary.c) Boundary. B) Boundary.b) Initial.c) Boundary. C) Initial.b) Boundary.c) Initial. D) Boundary.b) Initial.c) Initial. Show Answer Correct Answer: C) Initial.b) Boundary.c) Initial. 8. Find the particular solution of the differential equation $\frac{dy}{dx}=2x\sqrt{y}$ $y\left(1\right)=\frac{9}{4}$ A) $y=\tan^{-1}\left(x+3\right)$. B) $y=-\frac{2}{x^2+3}$. C) $y=\tan^{-1}\left(x+2\right)$. D) $y=\left(\frac{x^2}{2}+1\right)^2$. Show Answer Correct Answer: D) $y=\left(\frac{x^2}{2}+1\right)^2$. 9. Which of the following is clairaut's form A) $z=px+qy+\sqrt{pq}$. B) $p+q=1$. C) $p+q=x+y$. D) None of the above. Show Answer Correct Answer: A) $z=px+qy+\sqrt{pq}$. 10. Solve the following differential equations:$\frac{\text{d}y}{\text{d}x}=\frac{x}{y}$ A) $y^2=x^2+2C$. B) $\ln y=\frac{x^2}{2}+C$. C) $y=\frac{x^2}{2}+C$. D) $x^2=y^2+C$. Show Answer Correct Answer: A) $y^2=x^2+2C$. 11. In Clairaut's equation, p denotes: A) Arbitrary constant. B) Derivative dy/dx. C) Particular integral. D) None. Show Answer Correct Answer: B) Derivative dy/dx. 12. The rate of change of M is proportional to M. Which differential equation models this verbal statement? A) $M=Ce^{kt}$. B) $y=Ce^{kt}$. C) $\frac{dM}{dt}=kM$. D) $\frac{dy}{dt}=kt$. Show Answer Correct Answer: C) $\frac{dM}{dt}=kM$. 13. Find the general solution for $\frac{dy}{dx}=\frac{xy}{y+1}$ A) $y+\ln\left|y\right|=-\frac{x^3}{3}+C$. B) $y+\ln\left|y\right|=\frac{x^3}{3}+C$. C) $y+\ln\left|y\right|=\frac{x^2}{2}+C$. D) $y+\ln\left|1\right|=\frac{x^2}{2}+C$. Show Answer Correct Answer: C) $y+\ln\left|y\right|=\frac{x^2}{2}+C$. 14. The general solution of (D$^{2}$-5D+6)y=0 is ..... A) C$_{1}$e$^{3x}$+c$_{2}$e$^{-2x}$. B) C$_{1}$e$^{3x}$+c$_{2}$e$^{2x}$. C) C$_{1}$e$^{-3x}$+c$_{2}$e$^{2x}$. D) C$_{1}$e$^{-3x}$+c$_{2}$e$^{-2x}$. Show Answer Correct Answer: B) C$_{1}$e$^{3x}$+c$_{2}$e$^{2x}$. 15. Which of the following initial value problems is solved by $y = 5e^{-2x}$ A) $\frac{dy}{dx} =-2y$ $y(0) =-5$. B) $\frac{dy}{dx} = 2y$ $y(0) = 5$. C) $\frac{dy}{dx} = 2y$ $y(0) =-5$. D) $\frac{dy}{dx} =-2y$ $y(0) = 5$. Show Answer Correct Answer: D) $\frac{dy}{dx} =-2y$ $y(0) = 5$. 16. If A is diagonalizable, then ..... A) $e^{At} = I + A^t$. B) $e^{At} = D^t$. C) $e^{At} = P e^{Dt} P^{-1}$. D) $e^{At} = A^t$. Show Answer Correct Answer: C) $e^{At} = P e^{Dt} P^{-1}$. 17. Determine the order of the differential equation:$\frac{d^3y}{dx^3} + 4\frac{d^2y}{dx^2}-5\frac{dy}{dx} + 6y = 0$ A) 3. B) 2. C) 0. D) 1. Show Answer Correct Answer: A) 3. 18. What are the key components of CPS modeling? A) Key components of CPS modeling include system architecture, data flow, control algorithms, and feedback mechanisms. B) Network protocols. C) User interface design. D) System design principles. Show Answer Correct Answer: A) Key components of CPS modeling include system architecture, data flow, control algorithms, and feedback mechanisms. 19. What is the definition of a homogeneous function? A) A function that is separable. B) A function that is linear in nature. C) A function that contains only one independent variable. D) A function that satisfies the property f(x, y) = f(tx, ty) for all t. Show Answer Correct Answer: D) A function that satisfies the property f(x, y) = f(tx, ty) for all t. 20. $L\left(\sqrt{x}\right)=$ A) $\frac{\sqrt{\pi}}{\left(2s^{\frac{1}{2}}\right)}$. B) $\frac{\sqrt{\pi}}{\left(2s^{\frac{3}{2}}\right)}$. C) $\frac{\pi}{\left(2s^{\frac{3}{2}}\right)}$. D) $\frac{\sqrt{\pi}}{\left(s^{\frac{1}{2}}\right)}$. Show Answer Correct Answer: B) $\frac{\sqrt{\pi}}{\left(2s^{\frac{3}{2}}\right)}$. 21. How do you analyze stability in CPS using differential equations? A) Stability can be determined by examining the system's input-output behavior. B) Differential equations are not applicable for stability analysis in CPS. C) Analyzing stability requires only numerical simulations. D) Stability in CPS can be analyzed by modeling with differential equations, finding equilibrium points, linearizing the system, and examining eigenvalues of the Jacobian. Show Answer Correct Answer: D) Stability in CPS can be analyzed by modeling with differential equations, finding equilibrium points, linearizing the system, and examining eigenvalues of the Jacobian. 22. A company selling Teslas has been using the logistic differential equation to model sales over the last 3 years. That is, $\frac{dN}{dt}=kN\left(2-N\right)$ A) $N=\frac{Ae^{2kt}}{1+Ae^{2kt}}$. B) $N=\frac{2Ae^{2kt}}{1+Ae^{2kt}}$. C) $N=\frac{2Ae^{2kt}}{1-Ae^{2kt}}$. D) $N=\frac{Ae^{2kt}}{1-Ae^{2kt}}$. Show Answer Correct Answer: B) $N=\frac{2Ae^{2kt}}{1+Ae^{2kt}}$. 23. $2\frac{d^2y}{dt^2}+\frac{dy}{dt}-\sin y=t^2$ A) TRUE. B) FALSE. C) NOT SURE. D) None of the above. Show Answer Correct Answer: A) TRUE. 24. Number of arbitrary constant in the general solution of differential equation (1+x$^{2}$)(y$^{" }$)$^{3}$+y(y$^{'}$)$^{4}$=4x is A) 1. B) 2. C) 3. D) 4. Show Answer Correct Answer: B) 2. 25. F(s) = L{(sin(bt))} (s) A) S/(s$^{2}$+b$^{2}$). B) B/(s$^{2}$ + b$^{2}$). C) (s-a)/(s-a)$^{2}$+b$^{2}$. D) B/(s-a)$^{2 }$+ b$^{2}$. E) 1/s. Show Answer Correct Answer: B) B/(s$^{2}$ + b$^{2}$). 26. The directional derivative of $\phi$ $3\overrightarrow{i}+4\overrightarrow{j}+5\overrightarrow{k}$ A) $\frac{46}{\sqrt[]{2}}$. B) $\frac{46}{5\sqrt[]{2}}$. C) $\frac{23}{5}$. D) $\frac{46}{5}$. Show Answer Correct Answer: B) $\frac{46}{5\sqrt[]{2}}$. 27. $\frac{\text{d}^2x}{\text{d}y^2}-y=\cot x$ A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: B) FALSE. 28. Which of the following is not a differential equation? A) $\frac{\text{d}y}{\text{d}x}=2x+5$. B) $\frac{\text{d}y}{\text{d}x}=-\frac{\text{}x}{\text{}y}$. C) $y" +y=0$. D) None of these. Show Answer Correct Answer: D) None of these. 29. Two functions y1, y2 are linearly dependent if: A) $c_1y_1+c_2y_2=0$ $c_1, c_2$. B) Their Wronskian is nonzero. C) Auxiliary equation has real roots. D) None. Show Answer Correct Answer: A) $c_1y_1+c_2y_2=0$ $c_1, c_2$. 30. In the context of simultaneous equations, what does the term 'homogeneous' refer to? A) Equations have no solutions. B) All terms are equal. C) All variables are positive. D) All constant terms are zero. Show Answer Correct Answer: D) All constant terms are zero. 31. $\int_{ }^{ }f'\left(2x\right)dx$ A) $\frac{1}{2}f" \left(2x\right)$. B) $2f\left(2x\right)$. C) $2f" \left(2x\right)$. D) $\frac{1}{2}f\left(2x\right)$. Show Answer Correct Answer: D) $\frac{1}{2}f\left(2x\right)$. 32. The annihilator of eax is A) D + a. B) D-a. C) D2+a. D) D2-a. Show Answer Correct Answer: B) D-a. 33. The particular solution of the differential equation $\left(x^2+1\right)\frac{\text{d}y}{\text{d}x}+xy=x\left(x^2+1\right)$ A) $y=\frac{1}{3}\left(x^2+1\right)^{\frac{3}{2}}-\frac{1}{3}$. B) $y=\frac{1}{3}\left(x^2+1\right)-\frac{1}{3}\left(x^2+1\right)^{-\frac{1}{2}}$. C) $y=\frac{x^2\left(x^2+2\right)}{4\left(x^2+1\right)}$. D) $y=\frac{1}{3}\left(x^2+1\right)+\frac{1}{3}\left(x^2+1\right)^{\frac{1}{2}}$. Show Answer Correct Answer: B) $y=\frac{1}{3}\left(x^2+1\right)-\frac{1}{3}\left(x^2+1\right)^{-\frac{1}{2}}$. 34. What is the Frobenius method used for? A) To solve linear equations. B) To find power series solutions. C) To determine the Wronskian. D) To solve non-homogeneous equations. Show Answer Correct Answer: B) To find power series solutions. 35. Apa bentuk umum dari persamaan diferensial linear homogen orde kedua? A) Y" + b(x)y' + c(x)y" = 0. B) A(x)y" + b(x)y' + y = 0. C) Y" + y' + c = 0. D) A(x)y" + b(x)y' + c(x)y = 0. Show Answer Correct Answer: D) A(x)y" + b(x)y' + c(x)y = 0. 36. The difference between a particular solution to a separable differential equation and a general solution is that in a particular solution you must ..... A) Use $\ln$. B) Solve for C. C) Separate the variables. D) Integrate. Show Answer Correct Answer: B) Solve for C. 37. Find the general solution. $\frac{dy}{dx}=6xy$ A) $y=\sqrt[]{6x^2+C}$. B) $y=Ce^{6x}$. C) $y=Ce^{3x^2}$. D) $y=\ln\left(3x^2+C\right)$. Show Answer Correct Answer: C) $y=Ce^{3x^2}$. 38. The general solution of the DE $\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}-5y=0$ A) $y=Ae^x+Be^{-5x}$. B) $y=Ae^{-x}+Be^{-5x}$. C) $y=Ae^x+Be^{5x}$. D) $y=Ae^{-x}+Be^{5x}$. Show Answer Correct Answer: A) $y=Ae^x+Be^{-5x}$. 39. $L^{-1}\left[F'\left(s\right)\right]=$ A) $-\frac{d}{ds}\left[xF\left(s\right)\right]$. B) $xL^{-1}\left[F\left(s\right)\right]$. C) $-xL^{-1}\left[F\left(s\right)\right]$. D) $-\frac{d}{ds}\left[F\left(s\right)\right]$. Show Answer Correct Answer: C) $-xL^{-1}\left[F\left(s\right)\right]$. 40. The general solution of the differential equation $\frac{\text{d}y}{\text{d}x}=2xy$ A) $y=Ae^{x^2}$. B) $y=e^{x^2}+A$. C) $y=\ln\left(x^2+A\right)$. D) $y=e^{x^2}$. Show Answer Correct Answer: A) $y=Ae^{x^2}$. 41. What is the first step in solving differential equations using separation of variables? A) Separate variables to different sides. B) Solve for the constant of integration. C) Apply implicit differentiation. D) Integrate both sides immediately. Show Answer Correct Answer: A) Separate variables to different sides. 42. What are slope fields used for in differential equations? A) Visualizing the behavior of solutions to differential equations. B) Solving algebraic equations. C) Studying chemical reactions. D) Graphing linear equations. Show Answer Correct Answer: A) Visualizing the behavior of solutions to differential equations. 43. The general solution to $e^y\\frac{dy}{dx}=3x^2$ A) $y=e^y+6x+c$. B) $e^y=x^3+c$. C) $Ae^y=x^3+c$. D) $e^y=6x+c$. Show Answer Correct Answer: B) $e^y=x^3+c$. 44. Suppose that a cup of coffee begins at 183 degrees and, after sitting in room temperature of 69 degrees for 19 minutes, the coffee reaches 173 degrees. How long will it take before the coffee reaches 163 degrees? A) 38 minutes. B) 48 minutes. C) 58 minutes. D) 28 minutes. Show Answer Correct Answer: D) 28 minutes. 45. The solution of the equation z = px + qy + pq is ..... A) Z = ax + by + ab. B) Z = ax + by + pq. C) Z = px + qy + ab. D) Z = bx + ay + ab. Show Answer Correct Answer: A) Z = ax + by + ab. 46. Identify the type of PDE:u ..... xx-u ..... yy = 0. A) Elliptic. B) Parabolic. C) Linear. D) Hyperbolic. Show Answer Correct Answer: D) Hyperbolic. 47. Find the solution to dy/dx = sin(x) with y(0) = 1. A) Y =-cos(x) + 2. B) Y = cos(x) + 1. C) Y = sin(x) + 1. D) Y =-sin(x) + 1. Show Answer Correct Answer: A) Y =-cos(x) + 2. 48. What is the main purpose of the method of variation of parameters? A) To find the general solution of a homogeneous equation. B) To solve linear equations. C) To determine the Wronskian. D) To find a particular solution of a non-homogeneous equation. Show Answer Correct Answer: D) To find a particular solution of a non-homogeneous equation. 49. L$^{-1 }$\{n!/(s-a)$^{n+1}$\} A) T$^{n }$e$^{at}$. B) Cos bt. C) E$^{at}$. D) 1. E) T$^{n}$. Show Answer Correct Answer: A) T$^{n }$e$^{at}$. 50. $7x=\int3x++2dx_{ }^{ }$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: B) NO. 51. Which of the following differential equations can NOT be solved by separation of the variables? A) $\frac{dy}{dx}=x+y$. B) $\frac{dy}{dx}=xy$. C) $\frac{dy}{dx}=\frac{x}{y}$. D) $\frac{dy}{dx}=x$. Show Answer Correct Answer: A) $\frac{dy}{dx}=x+y$. 52. Find a particular solution for:y" -3y' + 2y = e$^{x}$ A) Xe$^{2x}$. B) 2 xe$^{x}$. C) -xe$^{x}$. D) E$^{x}$. Show Answer Correct Answer: C) -xe$^{x}$. 53. What is the order of a differential equation that involves finding the solution at a specific point? A) Higher order. B) First order. C) Third order. D) Second order. Show Answer Correct Answer: B) First order. 54. Sketch the slope field for the differential equation dy/dx = x-y A) Slope field sketching is a visual process and cannot be represented as a single correct answer. B) The slope field is a parabola. C) The slope field is a straight line. D) The slope field cannot be sketched for this differential equation. Show Answer Correct Answer: A) Slope field sketching is a visual process and cannot be represented as a single correct answer. 55. Part C. Find the equation (inverse method) from the general solution:20. y = $C_{1}e^{7x} + C_{2}e^{-3x}$ A) Y" + 10y' + 21y = 0. B) Y" -10y' + 21y = 0. C) Y" + 4y'-21y = 0. D) Y" -4y'-21y = 0. Show Answer Correct Answer: D) Y" -4y'-21y = 0. 56. Find the area between the curves ( y = x^2 ) and ( y = x ) from ( x = 0 ) to ( x = 1 ). A) 0.25. B) 0.1. C) 0.1667. D) 0.5. Show Answer Correct Answer: C) 0.1667. 57. $ydx-\left(x-2\right)dy=0$ A) Linear. B) Separable. C) Homogeneous. D) Bernoulli. Show Answer Correct Answer: B) Separable. 58. If $\frac{dy}{dx}=3e^{2x}, $ $x=0, \\y=\frac{5}{2}, $ A) $\frac{3}{2}e^{2x}+1$. B) $3e^{2x}+2$. C) $\frac{3}{2}e^{2x}+5$. D) $\frac{3}{2}e^{2x}+\frac{1}{2}$. Show Answer Correct Answer: A) $\frac{3}{2}e^{2x}+1$. 59. Find the specific solution to the following homogeneous DE, given that x = 0 when y = 1:$3\frac{d^2y}{dx^2}+4y=0$ A) $y=Ae^{-\frac{4x}{3}}$. B) $y=e^{-\frac{4x}{3}}$. C) $y=\cos((\frac{2\sqrt{3}}{3})x)$. D) $y=\cos((\frac{2\sqrt{3}}{3})x)+\sin((\frac{2\sqrt{3}}{3})x)$. Show Answer Correct Answer: C) $y=\cos((\frac{2\sqrt{3}}{3})x)$. 60. Find the particular solution to the differential equation $\frac{dy}{dx}=\frac{2x^3}{y^2}$ $\left(2, 3\right)$ A) $y=\left(\frac{3x^4}{2}+1\right)^{\frac{1}{3}}$. B) $y=\left(\frac{3x^4}{4}+3\right)^{\frac{1}{3}}$. C) $y=\left(\frac{3x^4}{2}+3\right)^{\frac{1}{3}}$. D) $y=\ln\left(e^x+3\right)$. Show Answer Correct Answer: C) $y=\left(\frac{3x^4}{2}+3\right)^{\frac{1}{3}}$. ← 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