This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 16 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 16 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Equations solvable for x are often reduced by: A) Eliminating y. B) Differentiating w.r.t. y. C) Separation of variables. D) None. Show Answer Correct Answer: A) Eliminating y. 2. If $\overrightarrow{f}$ $\overrightarrow{g}$ $\overrightarrow{f}\times\overrightarrow{g}$ A) Irrotational. B) Solenoidal. C) Harmonic. D) Neither irrotational nor solenoidal. Show Answer Correct Answer: B) Solenoidal. 3. Let PDE c$^{2}$(u$_{xx }$+ u$_{yy }$)= u$_{t}$, By is known as A) 2-D heat equation. B) 2-D wave equation. C) Laplace equation. D) None of these. Show Answer Correct Answer: A) 2-D heat equation. 4. Find the linear first order ODE. A) $y' = y^{2} + x$. B) $y' = x y^{2}$. C) $y' + y^{2} = 0$. D) $y' + \sin(x)y =\cos(x)$. Show Answer Correct Answer: D) $y' + \sin(x)y =\cos(x)$. 5. Which mathematician is credited with being the first to solve a differential equation using the method of 'separation of variables' in 1691? A) Isaac Newton. B) Leonhard Euler. C) Gottfried Wilhelm Leibniz. D) Joseph-Louis Lagrange. Show Answer Correct Answer: C) Gottfried Wilhelm Leibniz. 6. What is the integrating factor for the differential equation:$\frac{1}{x}\frac{\text{d}y}{\text{d}x}-\frac{y}{1+x^2}=x^3$ A) $-\frac{1}{\sqrt{x}}$. B) $\ln\left(x^2\right)$. C) $-\frac{1}{x^2}$. D) $\frac{1}{\sqrt{\left(1+x^2\right)}^{ }}$. Show Answer Correct Answer: D) $\frac{1}{\sqrt{\left(1+x^2\right)}^{ }}$. 7. It is number of the highest degree in a differential equation A) Variable. B) Order. C) Derivative. D) Degree. Show Answer Correct Answer: B) Order. 8. Find the particular solution of the differential equation $\frac{dy}{dx}=3e^{\left(x-y\right)}$ $y\left(-1\right)=\ln\left(\frac{\left(e+3\right)}{e}\right)$ A) $y=\ln\left(2e^x+1\right)$. B) $y=\ln\left(2e^x+3\right)$. C) $y=\ln\left(3e^x+1\right)$. D) $y=\ln\left(3e^x+2\right)$. Show Answer Correct Answer: C) $y=\ln\left(3e^x+1\right)$. 9. What is the general solution to the differential equation $\frac{dy}{dx} = 2x$ A) $y = x^2 + C$. B) $y = 2x + C$. C) $y = x^2$. D) None of the above. Show Answer Correct Answer: A) $y = x^2 + C$. 10. What is the role of the particular solution (PS) in a second order ODE? A) It is irrelevant to the overall solution. B) It addresses the non-homogeneous portion of the differential equation. C) It solves the homogeneous equation. D) It determines the roots of the characteristic equation. Show Answer Correct Answer: B) It addresses the non-homogeneous portion of the differential equation. 11. General solution of $\left(D^2+9\right)y=0$ A) Y=x. B) $y=c_1\cos x+c_2\sin x$. C) Y+x. D) $y=c_1\cos x+c_2\sin x+x$. Show Answer Correct Answer: B) $y=c_1\cos x+c_2\sin x$. 12. Find the value of k for which the constant function x(t) = k is a solution of the differential equation $2t^2 \frac{dx}{dt} + 4x + 9 = 0$ A) K = 4. B) K = 0. C) K = 9/4. D) K =-9/4. Show Answer Correct Answer: D) K =-9/4. 13. Singular solution of Partial differential equation can be obtained by A) General solution. B) Complete solution. C) Both a & b. D) None of the above. Show Answer Correct Answer: B) Complete solution. 14. What is the significance of boundary conditions in solving PDEs? A) Boundary conditions are irrelevant to the solution process. B) Boundary conditions determine the uniqueness and behavior of solutions to PDEs. C) Boundary conditions only affect initial values in PDEs. D) Boundary conditions simplify the equations of PDEs. Show Answer Correct Answer: B) Boundary conditions determine the uniqueness and behavior of solutions to PDEs. 15. The solution of Lagrange's partial differential equation xp+yq=z is A) $f\left(\frac{x}{y}, \frac{y}{z}\right)=0$. B) $f\left(\frac{y}{x}, \frac{x}{z}\right)=0$. C) $f\left(\frac{y}{x}, \frac{y}{z}\right)=0$. D) $f\left(\frac{x}{y}, \frac{x}{z}\right)=0$. Show Answer Correct Answer: A) $f\left(\frac{x}{y}, \frac{y}{z}\right)=0$. 16. The derivative of $\sin x$ A) $-\sin x$. B) $\tan x$. C) $-\cos x$. D) $\cos x$. Show Answer Correct Answer: D) $\cos x$. 17. Solve the differential equation(x-y)dy-(x+y)dx=0 A) Tan$^{-1}$(y/x)-(1/2)log|x$^{2}$+y$^{2}$|+c. B) -tan$^{-1}$(y/x)-(1/2)log|x$^{2}$+y$^{2}$|+c. C) Tan$^{-1}$(y/x)+(1/2)log|x$^{2}$+y$^{2}$|+c. D) Tan$^{-1}$(y/x)-(1/3)log|x$^{2}$+y$^{2}$|+c. Show Answer Correct Answer: A) Tan$^{-1}$(y/x)-(1/2)log|x$^{2}$+y$^{2}$|+c. 18. The divergence of a vector valued function is a scalar valued function. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 19. L\{(1/8b$^{4}$)((3-(bt)$^{2}$sin bt-3bt cos bt)\} A) B/(s$^{2}$+b$^{2}$)$^{2}$. B) B/(s$^{2}$+b$^{2}$)$^{3}$. C) B/(s$^{2}$+b$^{2}$). D) B/(s$^{2}$+b$^{2}$)$^{4}$. Show Answer Correct Answer: B) B/(s$^{2}$+b$^{2}$)$^{3}$. 20. A non-exact equation can become exact by: A) Multiplying by integrating factor. B) Adding constants. C) Changing variables arbitrarily. D) Transformation of axes. Show Answer Correct Answer: A) Multiplying by integrating factor. 21. Dy/dx=0.04y(3-y/120). What is the carrying capacity of the population? A) 3. B) 0.04. C) 120. D) 360. Show Answer Correct Answer: D) 360. 22. If the auxiliary equation has complex roots, the complementary function involves: A) Exponential and trigonometric terms together. B) Polynomials only. C) Arbitrary constants. D) None. Show Answer Correct Answer: A) Exponential and trigonometric terms together. 23. Find the particular solution to the equation dy/dx = 4x/y, with the initial condition y(2) =-2? A) $y=-\sqrt[]{4x^2-12}$. B) $y=2x-6$. C) $y=\sqrt[]{4x^2-12}$. D) $y=-\sqrt[]{4x^2-6}$. Show Answer Correct Answer: A) $y=-\sqrt[]{4x^2-12}$. 24. Dy/dy=.04y(1-y/100). At what value of y is population increasing the fastest? A) 50. B) 100. C) .04. D) Cannot be determined. Show Answer Correct Answer: A) 50. 25. The equation ( u ..... {xy} + u ..... {x} = 0 ) is classified as: A) First-order partial differential equation. B) First-order linear equation. C) Second-order ordinary differential equation. D) Second-order partial differential equation. Show Answer Correct Answer: A) First-order partial differential equation. 26. Which of the following are vector spaces under usual addition and scalar multiplication? A) M$_{2}$(R) over R. B) Q over R. C) Z over R. D) R over C. Show Answer Correct Answer: A) M$_{2}$(R) over R. 27. Provide an example of a physical phenomenon modeled by a PDE. A) Heat conduction in a solid, modeled by the heat equation. B) Population dynamics in ecology, modeled by the Laplace equation. C) Electromagnetic waves in a vacuum, modeled by the diffusion equation. D) Fluid flow in a pipe, modeled by the wave equation. Show Answer Correct Answer: A) Heat conduction in a solid, modeled by the heat equation. 28. $\frac{dy}{dx}-\frac{1}{\left(\pi-1\right)x}y=\frac{3}{1-\pi}xy^{\pi}$ A) Separable. B) Linear. C) Bernoulli. D) Homogeneous. Show Answer Correct Answer: C) Bernoulli. 29. Form a PDE by eliminating arbitrary constants from z= a(x+y) + b A) P+q=0. B) Qx = py. C) Px = qy. D) P-q=0. Show Answer Correct Answer: D) P-q=0. 30. A solution containing as many arbitrary constants as there are independent variables is called ..... A) Complete integral. B) General integral. C) Singular integral. D) Particular integral. Show Answer Correct Answer: A) Complete integral. 31. Verify that $y(x) = e^{-2x}$ $y" + 4y' + 4y = 0$ A) Only for $x = 1$. B) Yes, it is a solution. C) No, it is not a solution. D) Only for $x = 0$. Show Answer Correct Answer: B) Yes, it is a solution. 32. How is a constant treated when it is raised to the power of e in the context of solving differential equations? A) As a new variable. B) As an unknown variable. C) As an exponent. D) As a coefficient. Show Answer Correct Answer: D) As a coefficient. 33. Solve the partial differential equation $\frac{\partial z}{\partial x}=3.$ A) Z=3x+f(y). B) Z=3y+f(x). C) Z=3+f(x). D) Z=3+f(y). Show Answer Correct Answer: A) Z=3x+f(y). 34. Which method is most commonly used to solve PDEs with boundary conditions? A) Taylor's series. B) Separation of variables. C) Runge-Kutta. D) Newton-Raphson. Show Answer Correct Answer: B) Separation of variables. 35. A curve has a slope of 2x + 3 at each point (x, y) on the curve. Which of the following is an equation for this curve if it passes through the point (1, 2)? A) $y=x^2+3x$. B) $y=x^2+3x-3$. C) $y=x^2+1$. D) $y=x^2+3x-2$. Show Answer Correct Answer: D) $y=x^2+3x-2$. 36. The general solution of the differential equation $\frac{\text{d}y}{\text{d}x}+y=2e^{-x}$ A) $y=e^{-x}\left(A+2x\right)$. B) $y=Ae^{-x}+Bxe^{-x}$. C) $y=Ae^x+2xe^{-x}$. D) $y=Ae^{-x}+2e^{-x}$. Show Answer Correct Answer: A) $y=e^{-x}\left(A+2x\right)$. 37. $x^2\frac{\text{d}^2x}{\text{d}y^2}-5y=x+1$ A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: B) FALSE. 38. Jelaskan signifikansi Wronskian dalam konteks persamaan homogen. A) Wronskian digunakan untuk menyelesaikan persamaan non-homogen. B) Wronskian mengukur konvergensi solusi deret. C) Wronskian membantu menentukan ketergantungan linier solusi untuk persamaan homogen. D) Wronskian menunjukkan jumlah solusi untuk persamaan diferensial. Show Answer Correct Answer: C) Wronskian membantu menentukan ketergantungan linier solusi untuk persamaan homogen. 39. Variation of parameters requires solving for: A) One variable. B) Constant coefficients. C) No additional unknowns. D) Two functions of the independent variable. Show Answer Correct Answer: D) Two functions of the independent variable. 40. Part C. Find the equation (inverse method) from the general solution:19. y = $C_{1}e^{4x} + C_{2}e^{-x}$ A) Y" + 3y' + 4y = 0. B) Y" + y'-4y = 0. C) Y" -4y' + 3y = 0. D) Y" -3y'-4y = 0. Show Answer Correct Answer: D) Y" -3y'-4y = 0. 41. What does the Hessian matrix of a function f(x, y) consist of? A) First-order partial derivatives of f. B) Integral of f. C) Second-order partial derivatives of f. D) Gradient vector of f. Show Answer Correct Answer: C) Second-order partial derivatives of f. 42. The integrating factor of y(1+xy)dx + (2y-x)dy =0 [s A) X$^{2}$y$^{2}$. B) Y$^{2}$. C) X$^{2}$. D) Y$^{-2}$. Show Answer Correct Answer: D) Y$^{-2}$. 43. If the roots of differential equation are real and equal then C. F is C$_{1}$e$^{m1x}$+C$_{2}$e$^{m2x}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 44. Methods to solve DE are ..... A) Integration and Differentiation. B) Integration and Separable Variable Method. C) Integrating Factor and Separable Variable Methods. D) Separable Variable Methods only. Show Answer Correct Answer: C) Integrating Factor and Separable Variable Methods. 45. For what value of k, if any, is $y=e^{2x}+ke^{-3x}$ $4y-y"=10e^{-3x}$ A) 10/3. B) 10. C) There is no such value of k. D) -2. Show Answer Correct Answer: D) -2. 46. Y" = 12x; y'(-1) = 10 and y(2) = 22Find 2y" + 3y'-y A) $-3x^3+18x^2+20x+22$. B) $-3x^3+6x^2+20x-6$. C) $-2x^3+18x^2+28x+10$. D) $-2x^3+18x^2+20x+14$. Show Answer Correct Answer: D) $-2x^3+18x^2+20x+14$. 47. A non-empty set W of V(F) is a subspace V if and only if $u, v\\in W$ $\alpha, \\beta\\in F$ A) $\beta u\\in W$. B) $\alpha u\\in W$. C) $u+v\\in W$. D) $\alpha u+\beta v\\in W$. Show Answer Correct Answer: D) $\alpha u+\beta v\\in W$. 48. Find the Partial derivative of f with respect to y for $f\left(x, y\right)=\frac{y^2}{x+y}$ A) $\frac{2x^2+y^2}{\left(x+y\right)^2}$. B) $\frac{2xy+y^2}{\left(x+y\right)^2}$. C) $\frac{2}{\left(x+y\right)^{ }}$. D) $\frac{2x+y}{\left(x+y\right)^2}$. Show Answer Correct Answer: B) $\frac{2xy+y^2}{\left(x+y\right)^2}$. 49. What is a partial differential equation? A) An equation that is always true. B) An equation involving derivatives with respect to one variable. C) An equation that has no solutions. D) An equation involving derivatives with respect to multiple variables. Show Answer Correct Answer: D) An equation involving derivatives with respect to multiple variables. 50. Find N(T) if $T:R^3\longrightarrow R^2$ $T\left(x, y, z\right)=\left(x-y, 2z\right)$ A) $N\left(T\right)=\left\{\left(a, a, 0\right):a\in R\text{ }\right\}$. B) $N\left(T\right)=R^2$. C) $N\left(T\right)=\left\{\left(a, 0, 0\right):a\text{ }\in R\right\}$. D) $N\left(T\right)=\left\{0\text{ }\right\}$. Show Answer Correct Answer: A) $N\left(T\right)=\left\{\left(a, a, 0\right):a\in R\text{ }\right\}$. 51. The derivative of the matrix exponential is ..... A) $d/dt e^{At} = Ae^{At}$. B) $d/dt e^{At} = A$. C) $d/dt e^{At}$. D) $d/dt e^{At} = e^{At} + A$. Show Answer Correct Answer: A) $d/dt e^{At} = Ae^{At}$. 52. Solve the differential equation. $y'=x^2y$ A) $\ln\left|y\right|=\frac{x^3}{3}+C$. B) $y=\frac{x^2}{2}+C$. C) $\left|y\right|=x^2+C$. D) $\ln\left|y\right|=\frac{x^2}{2}+C$. Show Answer Correct Answer: A) $\ln\left|y\right|=\frac{x^3}{3}+C$. 53. Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by $\frac{dy}{dt}=ky$ $ft^3$ $t=0$ $ft^3$ $k$ A) -.200. B) -0.050. C) -.169. D) -0.056. Show Answer Correct Answer: D) -0.056. 54. Using the separation of variables, what is the solution of the differential equation $y'=-\frac{x}{y}$ $y\left(1\right)=4$ A) $y^2=15+x^2$. B) $y^2=17-x^2$. C) $y^2=14+2x^2$. D) $y^2=18-2x^2$. Show Answer Correct Answer: B) $y^2=17-x^2$. 55. Write a differential equation that describes each relationship. If necessary, use k as the constant of proportionality. The number of packets, p, Mr. Sullivan completes for Pre-Calculus is increasing as he nears the end of the school year. The rate of change of p with respect to time t is inversely proportional to the natural log of t. A) Dp/dt = k / ln(t). B) Dp/dt = k * ln(t). C) Dp/dt = k * t. D) Dp/dt = k / t. Show Answer Correct Answer: A) Dp/dt = k / ln(t). 56. The expression 12(1.015)$^{t}$ models the population of elephants in a wildlife refuge after t years since 1975.What does the value 1.015 represent? A) The number of elephants is multiplied by 1.015 each year. B) The number of elephants was 1, 015 in 1975. C) The number of elephants increases by 1.5% each year. D) The number of elephants increases by 101.5% each year. Show Answer Correct Answer: C) The number of elephants increases by 1.5% each year. 57. What is the notation for the first derivative of y with respect to x? A) Y'. B) Dy/dx. C) D/dx(y). D) D(y)/d(x). Show Answer Correct Answer: A) Y'. 58. $L\left[f" \left(x\right)\right]=s^2L\left[f\left(x\right)\right]-f\left(0\right)-f'\left(0\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 59. The value of $j\times k$ A) I. B) -i. C) 1. D) 0. Show Answer Correct Answer: A) I. 60. If the characteristic equation has a repeated root $r$ A) $y(x) = Ce^{rx} + De^{rx}$. B) $y(x) = Ce^{rx}$. C) $y(x) = Ce^{rx} + Dxe^{rx}$. D) $y(x) = Cx^r + Dx^r$. Show Answer Correct Answer: C) $y(x) = Ce^{rx} + Dxe^{rx}$. ← PreviousNext →Related 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