This quiz works best with JavaScript enabled. Home > Class 12 > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 43 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 43 (25 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Let $I$ A) $I$. B) $I$. C) Both $I$. D) Neither $I$. Show Answer Correct Answer: C) Both $I$. 2. If the roots of A.E. are real and distinct then C.F. is $y=c_1e^{m_1x}+c_2e^{m_2x}$ A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: B) True. 3. Determine the general solution of $y"+2y'-8y=0$ A) $y=C_1e^{-4x}+C_2xe^{2x}$. B) $y=C_1e^{-4x}+C_2e^{2x}$. C) $y=e^{4x}\left(C_1\cos2x+C_2\sin2x\right)$. D) $y=e^{2x}\left(C_1\cos4x+C_2\sin4x\right)$. Show Answer Correct Answer: B) $y=C_1e^{-4x}+C_2e^{2x}$. 4. Find the general solution for the differential equation $\frac{dy}{dx}=xy$ A) $y=e^{\frac{x^2}{2}}+c$. B) $y=\alpha e^{\frac{x^2}{2}}$. C) $y=Ae^{x^2}$. D) $y=\sqrt{\frac{1}{x^2+A}}$. Show Answer Correct Answer: B) $y=\alpha e^{\frac{x^2}{2}}$. 5. 2y" +5y'+3y=0 A) UNDERDAMPED. B) OVERDAMPED. C) UNDAMPED. D) CRITICALLY DAMPED. Show Answer Correct Answer: B) OVERDAMPED. 6. Which of the following is not a linear transformation? A) $T\left(x, y\right)=\left(x, -y\right)$. B) $T\left(x, y\right)=\left(-x, y\right)$. C) $T\left(x, y\right)=\left(x, 0\right)$. D) $T\left(x, y\right)=\left(x+2, y+4\right)$. Show Answer Correct Answer: D) $T\left(x, y\right)=\left(x+2, y+4\right)$. 7. Using Euler's method, approximate the value of y at x=2 given the initial condition y(0)=1 and the differential equation dy/dx = x+y A) Approximate value of y at x=0. B) Approximate value of y at x=2. C) Exact value of y at x=2. D) Approximate value of z at x=2. Show Answer Correct Answer: B) Approximate value of y at x=2. 8. What is a complete integral of a PDE? A) A complete integral of a PDE is a numerical approximation of the solution. B) A complete integral of a PDE is a graphical representation of solutions. C) A complete integral of a PDE is a specific solution with fixed parameters. D) A complete integral of a PDE is a solution that encompasses all possible solutions, often expressed with arbitrary functions. Show Answer Correct Answer: D) A complete integral of a PDE is a solution that encompasses all possible solutions, often expressed with arbitrary functions. 9. Let (V, +, .) be a vector space. Then, which one is true? A) There exist an element y inV, Such that x+y=x for all x in V. B) There exist an element y inV, Such that x.y=x for all x in V. C) There exist an element y inV, Such that x+y=y for all x in V. D) There exist an element y inV, Such that x.y=y for all x in V. Show Answer Correct Answer: A) There exist an element y inV, Such that x+y=x for all x in V. 10. Classify the PDE:u ..... xx + u ..... yy + u ..... zz = 0. A) Elliptic. B) Parabolic. C) Hyperbolic. D) Linear. Show Answer Correct Answer: A) Elliptic. 11. At x = 2, which statement is true about the value of differential equation $\frac{\text{d}y}{\text{d}x}=\frac{\left(2-x\right)}{y}$ A) The derivative is positive. B) The derivative is negative. C) The derivative is zero. D) The derivative is undefined. Show Answer Correct Answer: C) The derivative is zero. 12. The integrating factor for the equation y' + $\frac{2}{x}$ A) $e^{2x}$. B) $e^{x}$. C) $x^{2}$. D) $x^{2}y$. Show Answer Correct Answer: C) $x^{2}$. 13. What is the integral factor, $V\left(x\right)$ $\frac{dy}{dx}+y\cot x=\operatorname{cosec}x\tan^3x$ A) $\tan x$. B) $-\operatorname{cosec}^2x$. C) $\frac{1}{\sin x}$. D) $\sin x$. Show Answer Correct Answer: D) $\sin x$. 14. Is the given differential equation linear or nonlinear? $\frac{dy}{dx}=\tan x+15x^2+e^x+\frac{1}{x}$ A) Linear. B) Non linear. C) Not sure. D) None of the above. Show Answer Correct Answer: A) Linear. 15. To solve an equation solvable for dy/dx, we generally treat it as a: A) Linear equation. B) Polynomial in derivative. C) Exact equation. D) None. Show Answer Correct Answer: B) Polynomial in derivative. 16. The complementary function of (D$^{2}$-2D + 2) y = x e$^{x}$ is A) E$^{x}$ (a + b x + c x$^{2}$). B) E$^{x}$ (a cos x + b sin x). C) A e$^{x}$ + b e$^{-x}$ + c e$^{2x}$. D) E$^{-x}$ ( a + b x + c x$^{2}$). Show Answer Correct Answer: B) E$^{x}$ (a cos x + b sin x). 17. $\frac{dy}{dx}+2xy=4x$ A) $y=2-e^{-x^2}$. B) $y=e^{x^2}$. C) $y=2+e^{x^2}$. D) $y=2xe^{x^2}$. Show Answer Correct Answer: A) $y=2-e^{-x^2}$. 18. $\int_{ }^{ }\tan x\cot xdx$ A) $\csc x+c$. B) $\tan^2x\cot^2x+c$. C) $\sec x+c$. D) $x+c$. Show Answer Correct Answer: D) $x+c$. 19. Consider the equation $y' = y(y-1)$ $y(0) = A$ $A$ $t \ge 0$ A) $0 < A < 1$. B) $A > 1$. C) $A < 0$. D) None of the above. Show Answer Correct Answer: A) $0 < A < 1$. 20. For the homogeneous equation $y" +3y'+2y=0$ A) $m^2+3m+2=0$. B) $m^2-3m+2=0$. C) $m+3=0$. D) $m^2+2=0$. Show Answer Correct Answer: A) $m^2+3m+2=0$. 21. Which method is commonly used to solve Cauchy's and Euler's equations? A) Method of undetermined coefficients. B) Variation of parameters. C) Reduction of order. D) Separation of variables. Show Answer Correct Answer: B) Variation of parameters. 22. Determine the general solution of $25y"-20y'+4y=0$ A) $y=C_1e^{\frac{2}{5}x}+C_2e^{\frac{2}{5}x}$. B) $y=C_1e^{\frac{2}{5}x}+C_2xe^{-\frac{2}{5}x}$. C) $y=e^{\frac{2}{5}x}\left(C_1\cos x+C_2\sin x\right)$. D) $C_1e^{\frac{2}{5}x}+C_2xe^{\frac{2}{5}x}$. Show Answer Correct Answer: D) $C_1e^{\frac{2}{5}x}+C_2xe^{\frac{2}{5}x}$. 23. In Euler's method, what does the step size h represent? A) Length of the function. B) Number of variables. C) Size of the intervals. D) Value of the derivative. Show Answer Correct Answer: C) Size of the intervals. 24. Solve the differential equation. $\frac{dy}{dx}=\frac{2x}{y}$ A) $y=x+C$. B) $y^2=x^2+C$. C) $\frac{y^2}{2}=x^2+C$. D) $y^2=x+C$. Show Answer Correct Answer: C) $\frac{y^2}{2}=x^2+C$. 25. What is the impact of delays in CPS systems? A) Delays in CPS systems enhance performance metrics. B) Delays in CPS systems improve overall efficiency. C) Delays in CPS systems can cause performance degradation, safety risks, and operational inefficiencies. D) Delays in CPS systems have no impact on safety. Show Answer Correct Answer: C) Delays in CPS systems can cause performance degradation, safety risks, and operational inefficiencies. ← 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