This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 19 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 19 (41 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. A population of rabbits grows according to the differential equation dp/dt = 0.08p, where p(0) = 50. What is the population after 5 years? A) 45.12. B) 60.33. C) 74.59. D) 100.25. Show Answer Correct Answer: C) 74.59. 2. Find the order of the differential equation:$\frac{d^2y}{dx^2} + 7\frac{dy}{dx}-2y = 0$ A) 3. B) 0. C) 2. D) 1. Show Answer Correct Answer: C) 2. 3. How can you mitigate Zeno behavior in system design? A) Increase the number of incremental tasks. B) Eliminate deadlines entirely. C) Use synchronous processing exclusively. D) Reduce incremental tasks, prioritize, batch processes, set deadlines, and use asynchronous processing. Show Answer Correct Answer: D) Reduce incremental tasks, prioritize, batch processes, set deadlines, and use asynchronous processing. 4. The differential equation $2ydx-\left(3y-2x\right)dy=0$ A) Exact and homogeneous but not linear. B) Exact, homogeneous and linear. C) Exact and homogeneous but not linear. D) Homogeneous and linear but not excat. Show Answer Correct Answer: B) Exact, homogeneous and linear. 5. What is a finite state machine in CPS? A) A finite state machine in CPS is a graphical representation of system architecture. B) A finite state machine in CPS is a type of algorithm that does not involve states. C) A finite state machine in CPS is a model that represents system behavior through a finite number of states and transitions based on inputs. D) A finite state machine in CPS is a model that uses an infinite number of states. Show Answer Correct Answer: C) A finite state machine in CPS is a model that represents system behavior through a finite number of states and transitions based on inputs. 6. How do the roots of the characteristic equation affect the complementary function? A) They only affect the non-homogeneous part. B) They have no impact on the solution. C) They determine the coefficients of the particular solution. D) They dictate the form of the complementary function. Show Answer Correct Answer: D) They dictate the form of the complementary function. 7. $\frac{dy}{dx}=\frac{x^2}{x^2-y^2}+\frac{y}{x}$ A) Separable. B) Bernoulli. C) Homogeneous. D) Linear. Show Answer Correct Answer: C) Homogeneous. 8. Which of the following differential equations can be classified as separable and exact? A) $\frac{\text{d}y}{\text{d}x}=3\left(y+5\right)x$. B) $\frac{\text{d}y}{\text{d}x}=\frac{x+y}{x}$. C) $\left(x^2+1\right)dy+\left(xy+x\right)dx=0$. D) $y^3dx+3xy^2dy=0$. Show Answer Correct Answer: D) $y^3dx+3xy^2dy=0$. 9. What is the main focus of initial and boundary value problems in differential equations? A) Dealing with non-linear equations. B) Working with partial differential equations. C) Setting conditions for the solution on a given curve or at certain points. D) Finding the solution at a specific point or along a specific curve. Show Answer Correct Answer: C) Setting conditions for the solution on a given curve or at certain points. 10. I.F. of $\frac{\text{d}x}{\text{d}y}+Px=Q$ A) $e^{\int Qdx_{ }}$. B) $e^{\int Pdy_{ }}$. C) $e^{\int Qdy_{ }}$. D) $e^{\int Pdx_{ }}$. Show Answer Correct Answer: B) $e^{\int Pdy_{ }}$. 11. Solve dy/dx = (y+5)(x+2) given the point (0, 1) A) $y=6e^{\frac{1}{2}x^2+2x}-5$. B) $y=4e^{\frac{1}{2}x^2+2x}-5$. C) $y=3e^{\frac{1}{2}x^2+2x}-5$. D) $y=6e^{2x^2+2x}-5$. Show Answer Correct Answer: A) $y=6e^{\frac{1}{2}x^2+2x}-5$. 12. Page 397 #41 CALC A) A. B) B. C) E. D) D. E) C. Show Answer Correct Answer: A) A. 13. What is the solution curve of a differential equation? A) A line that intersects the x-axis. B) A function that satisfies the differential equation. C) A polynomial function. D) A constant function. Show Answer Correct Answer: B) A function that satisfies the differential equation. 14. Part B. Find the particular solution under the given conditions:14. y" -5y' + 6y = 0, y(0) = 0, y'(0) = 3 A) Y = $3e^{2x} + e^{3x}$. B) Y = $2e^{2x}-2e^{3x}$. C) $y = e^{2x}-3e^{3x}$. D) $y =-2e^{2x} + 2e^{3x}$. Show Answer Correct Answer: D) $y =-2e^{2x} + 2e^{3x}$. 15. Find the degree for the differential equation. A) 1. B) 3. C) 4. D) 2. Show Answer Correct Answer: A) 1. 16. Find the general solution of the homogeneous equation:y" + 4y = 0. A) Y(t) = C1 cos(t) + C2 sin(t). B) Y(t) = C1 e$^{2t}$ + C2 e$^{-2t}$. C) Y(t) = C1 t + C2 t$^{2}$. D) Y(t) = C1 cos(2t) + C2 sin(2t). Show Answer Correct Answer: D) Y(t) = C1 cos(2t) + C2 sin(2t). 17. The solution of linear differential equation $x\frac{dy}{dx}+2y=x^2$ A) $y=\frac{x^2+c}{4x^2}$. B) $y=\frac{x^4+c}{4x^2}$. C) $y=\frac{x^2}{4}+c$. D) $y=\frac{x^4+c}{x^2}$. Show Answer Correct Answer: B) $y=\frac{x^4+c}{4x^2}$. 18. During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1, 000 people are infected when the epidemic is first discovered, and 1, 200 are infected 7 days later, how many people are infected 12 days after the epidemic is first discovered? A) 2057. B) 343. C) 1343. D) 1367. E) 1400. Show Answer Correct Answer: D) 1367. 19. If you're supposed to find the Average Value of $f\left(x\right)$ A) The same as the unit for $f\left(x\right)$. B) The units of the $\int_{ }f\left(x\right)$. C) The units of the derivative of $f\left(x\right)$. D) None of the above. Show Answer Correct Answer: A) The same as the unit for $f\left(x\right)$. 20. The solution of the system X = AX using the matrix exponential is ..... A) X(t) = [I + At] X(0). B) X(t) = A t X(0). C) $X(t) = A^t X(0)$. D) $e^{At} X(0)$. Show Answer Correct Answer: D) $e^{At} X(0)$. 21. The method of variation of parameters is mainly used to find: A) General solution of homogeneous equations. B) Complementary function only. C) Order of a differential equation. D) Particular solution of a non-homogeneous differential equation. Show Answer Correct Answer: D) Particular solution of a non-homogeneous differential equation. 22. Which of the following is a correct way to verify that $y = e^{2x}$ $\frac{dy}{dx} = 2y$ A) Substitute $y$. B) Integrate both sides of the equation. C) Differentiate both sides of the equation. D) Substitute $x = 0$. Show Answer Correct Answer: A) Substitute $y$. 23. The solution of $\vec{X}' = A \vec{X}$ A) The trace of A only. B) The eigenvalues of A. C) The rank of A. D) The determinant of A only. Show Answer Correct Answer: B) The eigenvalues of A. 24. The degree of a differential equation is define by a A) Positive real number. B) Positive rational number. C) Positive integer. D) All the above. Show Answer Correct Answer: C) Positive integer. 25. If $\frac{\text{d}y}{\text{d}x}=3x^2-2$ $y\left(1\right)=4$ A) $y=x^3-2x+4$. B) $y=x^3-2x+5$. C) $y=x^3-2x$. D) $y=x^3-2x+1$. Show Answer Correct Answer: B) $y=x^3-2x+5$. 26. What is the linear span L(S) of the set $S=\left\{e_1, e_2, \frac{e_1+e_2}{2}\right\}$ $R^3$ A) X-axis. B) Y-axis. C) Yz-plane. D) Xy-plane. Show Answer Correct Answer: D) Xy-plane. 27. If the roots of the auxiliary equation in an ODE are 5 and 6, then complementary function is y= ..... A) $Ae^{5x}+Be^{6x}$. B) $Ae^{-5x}+Be^{-6x}$. C) $Ae^{5x}+xBe^{6x}$. D) None of the above. Show Answer Correct Answer: A) $Ae^{5x}+Be^{6x}$. 28. When sketching a solution curve on a slope field, what does the curve represent? A) The set of all possible slopes at each point. B) A particular solution to the differential equation passing through a given point. C) The general solution to the differential equation. D) The direction of increasing $x$. Show Answer Correct Answer: B) A particular solution to the differential equation passing through a given point. 29. Explain the concept of boundary conditions in PDEs. A) Boundary conditions are irrelevant in determining the solution's behavior in PDEs. B) Boundary conditions in PDEs define the behavior of solutions at the domain's edges, ensuring uniqueness and stability. C) Boundary conditions in PDEs only apply to linear equations, limiting their use. D) Boundary conditions in PDEs are solely concerned with initial values, not edges. Show Answer Correct Answer: B) Boundary conditions in PDEs define the behavior of solutions at the domain's edges, ensuring uniqueness and stability. 30. What type of differential equation is y" + y = 0? A) Third-order linear non-homogeneous differential equation. B) Second-order non-linear homogeneous differential equation. C) First-order linear homogeneous differential equation with constant coefficients. D) Second-order linear homogeneous differential equation with constant coefficients. Show Answer Correct Answer: D) Second-order linear homogeneous differential equation with constant coefficients. 31. What is a regular singular point? A) A point where the function is not analytic. B) A point where the Wronskian is zero. C) A point where the function has a discontinuity. D) A point where the function is analytic. Show Answer Correct Answer: A) A point where the function is not analytic. 32. CHOOSE THE CORRECT FORM OF PARTICULAR INTEGRAL FOR e$^{ax}$ A) PI=cos$^{ax}$/f(-a$^{2}$, -ab, -b$^{2}$). B) PI=e$^{ax+by}$/f(a, b). C) PI=1f(-a$^{2}$, -ab, -b$^{2}$). D) PI=sinax / f(a, b). Show Answer Correct Answer: B) PI=e$^{ax+by}$/f(a, b). 33. The purpose of state feedback in control is to ..... A) Make the system nonlinear. B) Modify eigenvalues of closed loop system. C) Reduce number of states. D) Increase system order. Show Answer Correct Answer: B) Modify eigenvalues of closed loop system. 34. Solve the Differential equation of ordwer 2y" -4y' + 3y = e$^{2x}$ for y(0) = 2 and y'(0) = 3 and find y when x =1 A) E$^{3x}$+2e$^{x}$-e$^{2x}$. B) 2e$^{3}$. C) E$^{3x}$+2e$^{x}$. D) E$^{3}$+2e-e$^{2}$. Show Answer Correct Answer: D) E$^{3}$+2e-e$^{2}$. 35. A Vector function $\overrightarrow{f}$ A) Div $\overrightarrow{f}\ne0$. B) Div $\overrightarrow{f}$. C) Grad $\overrightarrow{f}$. D) Curl $\overrightarrow{f}$. Show Answer Correct Answer: B) Div $\overrightarrow{f}$. 36. $\frac{dy}{dx}=x^2+3$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 37. The nature of PDE 4u$_{xx }$+3 u$_{xy }$=0 A) Parabolic. B) Elliptic. C) Circular. D) Hyperbolic. Show Answer Correct Answer: D) Hyperbolic. 38. The general form of a first order linear differential equation is: A) $\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} = Q(x)$. B) $dy/dx + P(x)y^2 = Q(x)$. C) Dy/dx + P(x)y = Q(x). D) $\frac{dy}{dx} = P(x)y^2 + Q(x)$. Show Answer Correct Answer: C) Dy/dx + P(x)y = Q(x). 39. A partial differential equation is formed by eliminating: A) Coefficients. B) Arbitrary constants or functions. C) Dependent variables. D) Independent variables. Show Answer Correct Answer: B) Arbitrary constants or functions. 40. The nature of the one-dimensional wave equation is A) Hyperbolic. B) Parabolic. C) Elliptic. D) None of These. Show Answer Correct Answer: A) Hyperbolic. 41. The complementary function of the equation $\frac{d^2y}{dx^2}+2\frac{\text{d}y}{\text{d}x}+y=0$ A) $\left(c_1+c_2x\right)e^{-x}$. B) $c_1e^x+c_2e^x$. C) $c_1e^{-x}+c_2e^{-x}$. D) $c_1e^x+c_2e^{-x}$. Show Answer Correct Answer: A) $\left(c_1+c_2x\right)e^{-x}$. ← PreviousRelated 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