This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 8 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 8 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Describe a method for verifying CPS properties. A) Use model checking to verify CPS properties. B) Apply static analysis for CPS verification. C) Use simulation to verify CPS properties. D) Conduct manual code reviews for CPS properties. Show Answer Correct Answer: A) Use model checking to verify CPS properties. 2. SOLVE USING LAGRANGE'S LINEAR EQUATON2p+3q=1 A) F(3x-2y, y+3z). B) F(3x-2y, y-3x). C) F(3x-2y, x-3z). D) F(3x-2y, y-3z). Show Answer Correct Answer: D) F(3x-2y, y-3z). 3. 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) $\frac{dE}{dt}=\frac{km}{v^2}$. D) $\frac{dE}{dt}=\frac{kv^2}{m}$. Show Answer Correct Answer: D) $\frac{dE}{dt}=\frac{kv^2}{m}$. 4. 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) 450.25. B) 200.50. C) 301.194. D) 550.75. Show Answer Correct Answer: C) 301.194. 5. 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 $e^{At}$. 6. 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. 7. Order of a Differential EquationThe order of a differential equation is determined by: A) The highest power of the variable in the equation. B) The number of terms in the equation. C) The highest power of the derivative in the equation. D) The degree of the function in the equation. Show Answer Correct Answer: C) The highest power of the derivative in the equation. 8. $\frac{\text{d}y}{\text{d}x}=x^2+1$ A) Second order differential equation. B) First order differential equation. C) First order differential equation at second degree. D) Second order differential equation at first degree. Show Answer Correct Answer: B) First order differential equation. 9. $y-x\frac{dy}{dx}=3-2x^2\frac{dy}{dx}$ A) Linear. B) Separable. C) Bernoulli. D) Homogeneous. Show Answer Correct Answer: B) Separable. 10. If $W(y_1, y_2)(x)\ne0$ A) $y_1, y_2$. B) $y_1, y_2$. C) Both are solutions of non-homogeneous DE. D) None. Show Answer Correct Answer: A) $y_1, y_2$. 11. What is the significance of integrating factors in differential equations? A) Integrating factors simplify the solving process of first-order linear differential equations. B) Integrating factors complicate the process of solving equations. C) Integrating factors are irrelevant to solving linear equations. D) Integrating factors are used only in second-order differential equations. Show Answer Correct Answer: A) Integrating factors simplify the solving process of first-order linear differential equations. 12. Which of the following is the solution to the differential equation $\frac{dy}{dt}=4y$ $y(0)=3$ A) $y=2y^2$. B) $y=4ty+3$. C) $y=3+e^{4t}$. D) $y=3e^{4t}$. Show Answer Correct Answer: D) $y=3e^{4t}$. 13. Which of the following statements about second order linear differential equations is true? A) They can only be solved using numerical methods. B) They are only applicable in theoretical mathematics. C) They can be used to model real-world phenomena in various fields. D) They have no solutions if the roots are complex. Show Answer Correct Answer: C) They can be used to model real-world phenomena in various fields. 14. The general solution of the ODE contains ..... A) Complementary function only. B) Particular integral only. C) Both complementary function and particular integral. D) Neither complementary function nor particular integral. Show Answer Correct Answer: C) Both complementary function and particular integral. 15. What is the wave equation and its applications in solving partial differential equations? A) The wave equation is a first-order partial differential equation used to model wave phenomena. B) The wave equation is a linear equation used to model wave phenomena. C) The wave equation is a third-order partial differential equation used to model wave phenomena. D) The wave equation is a second-order partial differential equation used to model wave phenomena. Show Answer Correct Answer: D) The wave equation is a second-order partial differential equation used to model wave phenomena. 16. Find the general solution of the differential equation $y'e^{2x}+2ye^{2x}=4$ A) $y=\frac{4x+c}{e^{2x}}$. B) $y=\frac{4x}{e^{2x}}$. C) $y=\frac{2x+c}{e^{2x}}$. D) $y=\frac{2x}{e^{2x}}$. Show Answer Correct Answer: A) $y=\frac{4x+c}{e^{2x}}$. 17. IVP stand for ..... A) Index Value Principle. B) Initial Value Principle. C) Initial Value Problem. D) Index Value Problem. Show Answer Correct Answer: C) Initial Value Problem. 18. Find the critical points of $f(x, y) = x^2 + y^2-4x-6y + 13$ A) (2, 3). B) (-2, -3). C) (4, 6). D) (0, 0). Show Answer Correct Answer: A) (2, 3). 19. $L\left[f\left(x\right)\right]=\frac{1}{a}F\left(\frac{s}{a}\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 20. Maclaurin series has center at ..... A) X = 0. B) X = 1. C) X = e. D) X = c. Show Answer Correct Answer: A) X = 0. 21. In the equation dx/P=dy/Q=dz/R, what does the variable P represent? A) Ratio of changes in x. B) Constant term. C) Sum of all variables. D) Coefficient of dx. Show Answer Correct Answer: D) Coefficient of dx. 22. Part C. Find the equation (inverse method) by the general solution:18. $C_1e^x+C_2e^{-2x}$ A) Y" + y'-2y = 0. B) Y" -y' + 2y = 0. C) Y" + 2y' + y = 0. D) Y" -2y'-y = 0. Show Answer Correct Answer: A) Y" + y'-2y = 0. 23. What are some applications of partial differential equations? A) Modeling population growth, analyzing stock market trends, predicting weather patterns. B) Solving algebraic equations, studying animal behavior, analyzing chemical reactions. C) Designing computer algorithms, studying human anatomy, analyzing geological formations. D) Modeling heat transfer, fluid dynamics, electromagnetic fields, quantum mechanics, and financial mathematics. Show Answer Correct Answer: D) Modeling heat transfer, fluid dynamics, electromagnetic fields, quantum mechanics, and financial mathematics. 24. Solve the first-order differential equation:$\frac{dy}{dx} = 3x^2$ A) $y = x^3 + C$. B) $y = x^3-C$. C) $y = 3x^3 + C$. D) $y = x^2 + C$. Show Answer Correct Answer: A) $y = x^3 + C$. 25. Solve this differential equation:$\frac{dy}{dt} = 0.03y(900-y)$ $y(0) = 2$ $y(t) =$ A) Y(t) = $\frac{900}{1 + 449e^{-27t}}$. B) Y(t) = $\frac{900}{1 + 449e^{27t}}$. C) Y(t) = $\frac{900}{1-449e^{-27t}}$. D) Y(t) = $\frac{900}{1-449e^{27t}}$. Show Answer Correct Answer: A) Y(t) = $\frac{900}{1 + 449e^{-27t}}$. ← PreviousNext →Related 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 9Class 12 Mathematics Chapter 9 Differential Equations Quiz 10Class 12 Mathematics Chapter 9 Differential Equations Quiz 11 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books