This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 26 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 26 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. 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) $y=\left(\frac{x^2}{2}+1\right)^2$. C) $y=\tan^{-1}\left(x+2\right)$. D) $y=-\frac{2}{x^2+3}$. Show Answer Correct Answer: B) $y=\left(\frac{x^2}{2}+1\right)^2$. 2. 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) $z=px+qy+\sqrt{pq}$. 3. Solve the following differential equations:$\frac{\text{d}y}{\text{d}x}=\frac{x}{y}$ A) $\ln y=\frac{x^2}{2}+C$. B) $y=\frac{x^2}{2}+C$. C) $x^2=y^2+C$. D) $y^2=x^2+2C$. Show Answer Correct Answer: D) $y^2=x^2+2C$. 4. 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. 5. The rate of change of M is proportional to M. Which differential equation models this verbal statement? A) $\frac{dy}{dt}=kt$. B) $y=Ce^{kt}$. C) $\frac{dM}{dt}=kM$. D) $M=Ce^{kt}$. Show Answer Correct Answer: C) $\frac{dM}{dt}=kM$. 6. Find the general solution for $\frac{dy}{dx}=\frac{xy}{y+1}$ A) $y+\ln\left|1\right|=\frac{x^2}{2}+C$. B) $y+\ln\left|y\right|=\frac{x^3}{3}+C$. C) $y+\ln\left|y\right|=\frac{x^2}{2}+C$. D) $y+\ln\left|y\right|=-\frac{x^3}{3}+C$. Show Answer Correct Answer: C) $y+\ln\left|y\right|=\frac{x^2}{2}+C$. 7. 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: C) C$_{1}$e$^{3x}$+c$_{2}$e$^{2x}$. 8. 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: B) $\frac{dy}{dx} =-2y$ $y(0) = 5$. 9. If A is diagonalizable, then ..... A) $e^{At} = P e^{Dt} P^{-1}$. B) $e^{At} = D^t$. C) $e^{At} = A^t$. D) $e^{At} = I + A^t$. Show Answer Correct Answer: A) $e^{At} = P e^{Dt} P^{-1}$. 10. 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) 0. B) 1. C) 2. D) 3. Show Answer Correct Answer: D) 3. 11. 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. 12. 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. 13. $L\left(\sqrt{x}\right)=$ A) $\frac{\pi}{\left(2s^{\frac{3}{2}}\right)}$. B) $\frac{\sqrt{\pi}}{\left(2s^{\frac{1}{2}}\right)}$. C) $\frac{\sqrt{\pi}}{\left(2s^{\frac{3}{2}}\right)}$. D) $\frac{\sqrt{\pi}}{\left(s^{\frac{1}{2}}\right)}$. Show Answer Correct Answer: C) $\frac{\sqrt{\pi}}{\left(2s^{\frac{3}{2}}\right)}$. 14. 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. 15. 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) $N=\frac{Ae^{2kt}}{1-Ae^{2kt}}$. C) $N=\frac{2Ae^{2kt}}{1+Ae^{2kt}}$. D) $N=\frac{2Ae^{2kt}}{1-Ae^{2kt}}$. Show Answer Correct Answer: C) $N=\frac{2Ae^{2kt}}{1+Ae^{2kt}}$. 16. $2\frac{d^2y}{dt^2}+\frac{dy}{dt}-\sin y=t^2$ A) TRUE. B) FALSE. C) NOT SURE. D) None of the above. Show Answer Correct Answer: A) TRUE. 17. Number of arbitrary constant in the general solution of differential equation (1+x$^{2}$)(y$^{" }$)$^{3}$+y(y$^{'}$)$^{4}$=4x is A) 4. B) 3. C) 2. D) 1. Show Answer Correct Answer: C) 2. 18. F(s) = L\{(sin(bt))\} (s) A) 1/s. B) B/(s-a)$^{2 }$+ b$^{2}$. C) (s-a)/(s-a)$^{2}$+b$^{2}$. D) B/(s$^{2}$ + b$^{2}$). E) S/(s$^{2}$+b$^{2}$). Show Answer Correct Answer: D) B/(s$^{2}$ + b$^{2}$). 19. Population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. If the population doubles every 10 years then the value of k is A) 0.200. B) 0.069. C) 0.301. D) 3.322. Show Answer Correct Answer: B) 0.069. 20. The directional derivative of $\phi$ $3\overrightarrow{i}+4\overrightarrow{j}+5\overrightarrow{k}$ A) $\frac{46}{5}$. B) $\frac{46}{5\sqrt[]{2}}$. C) $\frac{46}{\sqrt[]{2}}$. D) $\frac{23}{5}$. Show Answer Correct Answer: B) $\frac{46}{5\sqrt[]{2}}$. 21. $\frac{\text{d}^2x}{\text{d}y^2}-y=\cot x$ A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: B) FALSE. 22. 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. 23. 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) $c_1y_1+c_2y_2=0$ $c_1, c_2$. 24. 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. 25. $\int_{ }^{ }f'\left(2x\right)dx$ 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: A) $\frac{1}{2}f\left(2x\right)$. ← 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