This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 24 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 24 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The rate of change of the volume, V, of oil in a tank with respect to time, t, is directly proportional to the cube root of the volume. Which of the following is a differential equation that describes this relationship? A) $V(t)=k\sqrt[3]{V}$. B) $V(t)=k\sqrt[3]{t}$. C) $\frac{dV}{dt}=\frac{k}{\sqrt[3]{V}}$. D) $\frac{dV}{dt}=k\sqrt[3]{V}$. Show Answer Correct Answer: D) $\frac{dV}{dt}=k\sqrt[3]{V}$. 2. Form a PDE by eliminating arbitrary constants from z= ax + by A) Z = px + qy. B) Z = px-qy. C) Z = qx + py. D) Z = qx-py. Show Answer Correct Answer: A) Z = px + qy. 3. What is the order of the D.E. (d$^{2}$y/dx$^{2}$)$^{2}$+ y = 0 A) 1. B) 2. C) 3. D) 0. Show Answer Correct Answer: B) 2. 4. What is the integrating factor of the differential equation $\left(3x^2y^4+2xy\right)dx+\left(2x^3y^3-x^2\right)dy=0$ A) $\frac{1}{x^2}$. B) $\frac{1}{y^2}$. C) $\frac{1}{xy}$. D) $-\frac{1}{x^2y}$. Show Answer Correct Answer: B) $\frac{1}{y^2}$. 5. Given the differential equation $\frac{dP}{dt}=5P$ $P\left(0\right)=418$ A) $P=e^{5\alpha t}$ $P=e^{418t}$. B) $P=\alpha e^{5t}$ $P=418e^{5t}$. C) $P=\beta e^t$ $P=418e^t$. D) $P=\alpha e^{10t}$ $P=418e^{10t}$. Show Answer Correct Answer: B) $P=\alpha e^{5t}$ $P=418e^{5t}$. 6. If $y" + 6y' + 9y = 0$ A) $y(x) = Ce^{-3x} + Dxe^{-3x}$. B) $y(x) = Ce^{3x} + De^{-3x}$. C) $y(x) = Ce^{-6x} + De^{-9x}$. D) $y(x) = Ce^{-3x} + De^{-6x}$. Show Answer Correct Answer: A) $y(x) = Ce^{-3x} + Dxe^{-3x}$. 7. Find f '(x) if f(x) = x$^{4}$sinx A) 4x$^{3}$cosx. B) X$^{4}$cosx. C) X$^{4}$cosx + 4x$^{3}$sinx. D) -x$^{4}$cosx + 4x$^{3}$sinx. Show Answer Correct Answer: C) X$^{4}$cosx + 4x$^{3}$sinx. 8. Find the particular solution to the differential equation $\frac{dy}{dx}=3xy^2$ $\left(-3, -\frac{1}{14}\right)$ A) $y=\left(3e^x+3\right)^{\frac{1}{3}}$. B) $y=\frac{-1}{x^2+1}$. C) $y=\frac{-2}{3x^2+1}$. D) $y=\frac{-2}{x^2+2}$. Show Answer Correct Answer: C) $y=\frac{-2}{3x^2+1}$. 9. What are the applications of CPS in real-world scenarios? A) CPS applications are limited to mobile gaming. B) CPS is used for traditional farming methods. C) CPS is primarily focused on social media analytics. D) Applications of CPS include smart transportation, healthcare monitoring, industrial automation, and smart grid management. Show Answer Correct Answer: D) Applications of CPS include smart transportation, healthcare monitoring, industrial automation, and smart grid management. 10. Solve the first-order differential equation:$\frac{dy}{dx} = 5x^4$ A) $y = x^5 + C$. B) $y = x^5-C$. C) $y = \frac{5}{4}x^5 + C$. D) $y = x^5 + 5C$. Show Answer Correct Answer: A) $y = x^5 + C$. 11. If the Wronskian is identically zero, then: A) Functions are linearly independent. B) Functions are linearly dependent. C) Equation has no solution. D) None. Show Answer Correct Answer: B) Functions are linearly dependent. 12. An initial value problem requires: A) General solution only. B) Differential equation + conditions on y, y' at some point. C) Only auxiliary equation. D) None of the. Show Answer Correct Answer: B) Differential equation + conditions on y, y' at some point. 13. If the Eigen values of A are 3, 4, 5 then the Eigen values of A$^{2 }$are A) 16, 25, 9. B) 9, 16, 25. C) 1/3, 1/4, 1/5. D) 25, 16, 9. Show Answer Correct Answer: B) 9, 16, 25. 14. Which animal does not hibernate during the winter? A) White-tailed Prarie Dog. B) Eastern Chipmunk. C) Red Squirrel. D) Polar Bear. Show Answer Correct Answer: C) Red Squirrel. 15. Solve:$3\frac{\text{d}y}{\text{d}x}-\\frac{y}{x+2}=x+2$ A) $y=x(x+2)+c$. B) $y=\frac{3}{5}\left(x+2\right)^2+c$. C) $y=\frac{2}{3}+c$. D) $y=\frac{5}{3}\left(x+2\right)^2+c$. Show Answer Correct Answer: B) $y=\frac{3}{5}\left(x+2\right)^2+c$. 16. If the auxiliary equation has repeated roots, the general solution includes terms like: A) Aemx+Be-mx. B) Aemx+Bxemx. C) Aemx+Bcos(mx). D) Acos(mx)+Bsin(mx). Show Answer Correct Answer: B) Aemx+Bxemx. 17. Integrate both sides of the equation $\frac{dy}{dx}=4x^2$ A) $y=\frac{1}{3}x^3+C$. B) $y=4x^3+C$. C) $y=2x^3+C$. D) $y=\frac{4}{3}x^3+C$. Show Answer Correct Answer: D) $y=\frac{4}{3}x^3+C$. 18. Solve the non-homogeneous equation:y" + y = sin(x). A) Y = C1 cos(x) + C2 sin(x) + x. B) Y = C1 sin(x) + C2 cos(x)-1/2. C) Y = (C1-x/2) cos(x) + C2 sin(x). D) Y = C1 cos(x) + C2 sin(x) + 1/2. Show Answer Correct Answer: C) Y = (C1-x/2) cos(x) + C2 sin(x). 19. Solve $\left(D^2+2DD'+D'^2\right)z=0$ A) $Z=f_1\left(y-x\right)+xf_2\left(y-x\right)$. B) $Z=f_1\left(y+x\right)+xf_2\left(y-x\right)$. C) $Z=f_1\left(y+x\right)+xf_2\left(y+x\right)$. D) $Z=f_1\left(y-x\right)+f_2\left(y-x\right)$. Show Answer Correct Answer: A) $Z=f_1\left(y-x\right)+xf_2\left(y-x\right)$. 20. Differential equation of the y=a.e$^{2x}$+b.e$^{-x}$, where a &b are arbitrary constants, is A) Y$^{"}$+y$^{'}$-2y=0. B) Y$^{"}$-y$^{"}$-2y=0. C) Y$^{" }$-y$^{'}$+2y=0. D) Y$^{" }$+y$^{'}$+2y=0. Show Answer Correct Answer: B) Y$^{"}$-y$^{"}$-2y=0. 21. Determine the general solution of $9y"+9y'-4y=0$ A) $y=C_1e^{-\frac{1}{3}x}+C_2e^{\frac{4}{3}x}$. B) $y=C_1e^{\frac{1}{3}x}+C_2e^{-\frac{4}{3}x}$. C) $y=e^{\frac{1}{3}x}\left(C_1\cos\frac{4}{3}x+C_2\sin\frac{4}{3}x\right)$. D) $y=C_1e^{\frac{1}{3}x}+C_2xe^{-\frac{4}{3}x}$. Show Answer Correct Answer: B) $y=C_1e^{\frac{1}{3}x}+C_2e^{-\frac{4}{3}x}$. 22. In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated. A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: A) False. 23. Which of the following represents the solution to the differential equation $\frac{dy}{dx} =-y$ A) A linear function with a positive slope. B) A linear function with a negative slope. C) An exponential decay function. D) An exponential growth function. Show Answer Correct Answer: C) An exponential decay function. 24. How do you classify a second-order PDE? A) A second-order PDE is categorized by its variable coefficients and solution methods. B) A second-order PDE is classified by its boundary conditions and initial values. C) A second-order PDE can be classified as elliptic, parabolic, or hyperbolic based on the discriminant of its coefficients. D) A second-order PDE can only be classified as linear or nonlinear based on its terms. Show Answer Correct Answer: C) A second-order PDE can be classified as elliptic, parabolic, or hyperbolic based on the discriminant of its coefficients. 25. For each differential equation, indicate the order (as a number) and whether the equation is linear or nonlinear. $\frac{d^4 y}{d t^4} = \sin(t + y)$ A) Linear. B) Nonlinear. C) All the above. D) None of the above. Show Answer Correct Answer: B) Nonlinear. ← 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