This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 20 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 20 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. If $\phi\left(x, y, z\right)=xy^2+yz^3$ $grad\\phi$ $y=0$ A) $z^2\overrightarrow{j}$. B) $z^3\overrightarrow{j}$. C) $z^3\overrightarrow{i}$. D) $z^2\overrightarrow{j}+\overrightarrow{k}$. Show Answer Correct Answer: B) $z^3\overrightarrow{j}$. 2. The complementary function on y ''-4y=0 is A) C1 ex+ C2e-x. B) C1 e2x+ C2e-2x. C) C1 sin2x+ C2cos2x. D) C1 + C2x. Show Answer Correct Answer: B) C1 e2x+ C2e-2x. 3. The highest derivative in the equation. A) Type. B) Degree. C) Order. D) Linearity. Show Answer Correct Answer: C) Order. 4. What is the significance of the indicial roots in the Frobenius method? A) They indicate linear dependence. B) They help in finding the series solution. C) They determine the order of the differential equation. D) They are always zero. Show Answer Correct Answer: B) They help in finding the series solution. 5. A/An ..... to a differential equation is a function $y=f(x)$ $f(x, y)=0$ A) Antiderivative. B) Order. C) Solution. D) Separable. Show Answer Correct Answer: C) Solution. 6. After separating variables and integrating, what is added to both sides of the equation? A) A differential. B) A variable of integration. C) An exponent. D) A constant of integration. Show Answer Correct Answer: D) A constant of integration. 7. Page 393 #22 A) A. B) B. C) C. D) D. E) E. Show Answer Correct Answer: D) D. 8. How do you model a simple CPS using finite state machines? A) Visualize a CPS using FSMs with a flowchart instead of a state transition diagram. B) Model a CPS using FSMs by only defining states without transitions. C) Model a CPS using FSMs by defining states, transitions, and actions, then visualize with a state transition diagram. D) Use FSMs to model a CPS by focusing solely on actions and ignoring states. Show Answer Correct Answer: C) Model a CPS using FSMs by defining states, transitions, and actions, then visualize with a state transition diagram. 9. Form the differential equation of the curve:$y=Ae^x+Be^{-x}$ A) $\frac{\text{d}^2y}{\text{d}x^2}=y$. B) $\frac{\text{d}^2y}{\text{d}x^2}+y=0$. C) $\frac{\text{d}^2y}{\text{d}x^2}=Ae^x-Be^{-x}$. D) $\frac{\text{d}^2y}{\text{d}x^2}=Axe^x-Bxe^{-x}$. Show Answer Correct Answer: A) $\frac{\text{d}^2y}{\text{d}x^2}=y$. 10. What role do differential equations play in system dynamics? A) Differential equations are primarily for solving algebraic equations. B) Differential equations model the behavior and dynamics of systems over time. C) Differential equations are used to calculate static values. D) Differential equations only apply to physical systems without time dependence. Show Answer Correct Answer: B) Differential equations model the behavior and dynamics of systems over time. 11. For each differential equation, indicate the order (as a number) and whether the equation is linear or nonlinear. $\frac{d^4y}{dt^4} = \sin(t+y)$ A) Linear. B) Nonlinear. C) All the above. D) None of the above. Show Answer Correct Answer: B) Nonlinear. 12. Which of the following is an example of a homogeneous differential equation? A) Y' = y-xy. B) Xy' + y = 1. C) Y' = sin(2x). D) Y' = y-x. Show Answer Correct Answer: A) Y' = y-xy. 13. Consider the differential equation dy/dx = x + 2y for which g(x) is the solution. Which of the following statements is true if the particular solution contains (0, -1) A) G(x) is increasing and concave up. B) G(x) is increasing and concave down. C) G(x) is decreasing and concave up. D) G(x) is decreasing and concave down. Show Answer Correct Answer: D) G(x) is decreasing and concave down. 14. $L\left[f\left(x\right)\right]=\int_0^{\infty}e^{-sx}f\left(x\right)dx$ A) $True$. B) $False$. C) All the above. D) None of the above. Show Answer Correct Answer: A) $True$. 15. An equation of first order but higher degree means the equation contains: A) First-order derivative only, but to a higher power. B) Higher-order derivatives. C) Only algebraic terms. D) None. Show Answer Correct Answer: A) First-order derivative only, but to a higher power. 16. What is the heat equation and its applications in solving partial differential equations? A) The heat equation is a differential equation used to model electromagnetic fields. B) The heat equation is a polynomial equation used to model population growth. C) The heat equation is a partial differential equation used to model heat transfer and diffusion processes. D) The heat equation is a linear equation used to model fluid flow. Show Answer Correct Answer: C) The heat equation is a partial differential equation used to model heat transfer and diffusion processes. 17. The formula for the Wronskian of two solutions y1 and y2 of y" +Pxy'+Qxy =0 is A) Wx=C e-Pxdx. B) Wx=C ePxdx. C) Wx=C e-Qxdx. D) Wx=C eQxdx. Show Answer Correct Answer: B) Wx=C ePxdx. 18. Which of the following is a logistic differential equation? A) $\frac{dP}{dt}=3P$. B) $\frac{dP}{dt}=3P(5-P)$. C) $\frac{dP}{dt}=3P(5-t)$. D) $\frac{dP}{dt}=3t(5-t)$. Show Answer Correct Answer: B) $\frac{dP}{dt}=3P(5-P)$. 19. Which of the following is a solution to the differential equation $\frac{dy}{dx} = 3y$ A) $y = 3x$. B) $y = Ce^{3x}$. C) $y = 3e^x$. D) $y = Ce^{x^3}$. Show Answer Correct Answer: B) $y = Ce^{3x}$. 20. Differential equation of the circles x$^{2}$+y$^{2 }$=r$^{2 }$, where r >0 is arbitrary constant is A) X-yy"=0. B) X-y.y$^{'}$=0. C) X+y.y$^{'}$=0. D) X$^{2}$+xy'=0. Show Answer Correct Answer: C) X+y.y$^{'}$=0. 21. Let T be a linear transformation from V to W, then T is one to one if and only if A) $N\left(T\right)\ne\left\{0\right\}$. B) $R\left(T\right)=W$. C) $N\left(T\right)=\left\{0\right\}$. D) $R\left(T\right)=\left\{0\right\}$. Show Answer Correct Answer: C) $N\left(T\right)=\left\{0\right\}$. 22. Let V be a vector space, and let S be a subset of V . What does it mean when we say that S spans V? A) The elements of S are all distinct from each other. B) S has at least as many elements as the dimension V. C) S is a basis for V. D) Every vector in V can be expressed as a linear combination of vectors in S. Show Answer Correct Answer: D) Every vector in V can be expressed as a linear combination of vectors in S. 23. Partial differentiation is used when the function has: A) One variable. B) Two or more independent variables. C) No variables. D) Only dependent variables. Show Answer Correct Answer: B) Two or more independent variables. 24. In $\frac{\text{d}x}{\text{d}y}$ A) X. B) Y. C) D. D) Secret. Show Answer Correct Answer: A) X. 25. Which function is the general solution of the differential equation $y" '+y" +y'+y=0$ A) $y=A\sin x+Be^{-x}$. B) $y=A\sin2x-Be^{-x}$. C) $y=A\sin x+Be^{-2x}$. D) $y=A\sin2x-Be^{-2x}$. Show Answer Correct Answer: A) $y=A\sin x+Be^{-x}$. ← 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 8Class 12 Mathematics Chapter 9 Differential Equations Quiz 9Class 12 Mathematics Chapter 9 Differential Equations Quiz 10 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books