This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 13 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 13 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The curl 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: B) False. 2. $\frac{dy}{dx}=\frac{x+1}{y+2}$ A) Homogeneous. B) Nonhomogeneous. C) Not sure. D) None of the above. Show Answer Correct Answer: B) Nonhomogeneous. 3. Which of the following is the formula for growth and decay? A) $y=mx+b$. B) $y=Ce^{kt}$. C) $y=e^{kt}$. D) $y=Ce^t$. Show Answer Correct Answer: B) $y=Ce^{kt}$. 4. If $\frac{\text{d}y}{\text{d}x}=e^y$ A) $y=\ln\left|2-x\right|$. B) $y=-\ln\left|2-x\right|$. C) $y=\ln\left|x\right|$. D) $y=-\ln\left|x\right|$. Show Answer Correct Answer: B) $y=-\ln\left|2-x\right|$. 5. What is the characteristic of first order linear differential equations? A) They involve finding the solution at a specific point or along a specific curve. B) They are non-linear. C) They are partial differential equations. D) They involve initial and boundary value problems. Show Answer Correct Answer: D) They involve initial and boundary value problems. 6. In control theory, the stability of a system modeled by differential equations is determined by ..... A) Rank of A. B) Eigenvalues of the system matrix A. C) Dimension of state vector. D) Determinant of A. Show Answer Correct Answer: B) Eigenvalues of the system matrix A. 7. What is the method of characteristics? A) A process for integrating differential equations directly. B) A technique for solving PDEs by transforming them into ODEs along characteristic curves. C) A technique for graphing functions in three dimensions. D) A method for solving algebraic equations. Show Answer Correct Answer: B) A technique for solving PDEs by transforming them into ODEs along characteristic curves. 8. The expansion for s A) $\frac{\partial^2z}{\partial y^2}$. B) $\frac{\partial^2z}{\partial x^2}$. C) $\frac{\partial^2z}{\partial x\partial y^{ }}$. D) None of the above. Show Answer Correct Answer: C) $\frac{\partial^2z}{\partial x\partial y^{ }}$. 9. A rod of length 10 m has temperature 30$^{0 }$C and 40$^{0 }$C at end points. What is the temperature gradient A) 1$^{0}$C per cm. B) 3$^{0}$C per cm. C) 2$^{0}$C per cm. D) None of these. Show Answer Correct Answer: A) 1$^{0}$C per cm. 10. The PDE u$_{xx }$+ u$_{yy }$= 0, is known as A) 1-D heat equation. B) 1-D wave equation. C) Laplace equation. D) None of these. Show Answer Correct Answer: C) Laplace equation. 11. What is the integral factor, $V\left(x\right)$ $x\frac{dy}{dx}-2y=x^3\ln x$ A) $x^2$. B) $\frac{1}{x^2}$. C) $e^{2x}$. D) $e^{-2x}$. Show Answer Correct Answer: B) $\frac{1}{x^2}$. 12. Find the general solution to $y" -2y' + y = 0$ A) $y(x) = Ce^{x} + De^{-x}$. B) $y(x) = Ce^{x} + Dxe^{x}$. C) $y(x) = C\sin(x) + D\cos(x)$. D) $y(x) = Ce^{2x} + De^{-2x}$. Show Answer Correct Answer: B) $y(x) = Ce^{x} + Dxe^{x}$. 13. The complete solution(or complete integral ) of Linear Differential equations involves A) Complementary function + definite integral. B) Complete function + particular integral. C) Complementary function + particular integral. D) Complete function + indefinite integral. Show Answer Correct Answer: C) Complementary function + particular integral. 14. What are the values of $\alpha$ $\beta$ $dF\left(x, y\right)=\left(\frac{1}{x^2+2}+\frac{\alpha}{y}\right)dx+\left(xy^{\beta}+1\right)dy$ A) $\alpha=1, \beta=-2$. B) $\alpha=-1, \beta=-2$. C) $\alpha=-2, \beta=-1$. D) $\alpha=-1, \beta=2$. Show Answer Correct Answer: B) $\alpha=-1, \beta=-2$. 15. If $f\left(x\right)$ $x^2\sqrt[]{x^3+1}$ $f\left(2\right)=0, $ $f\left(0\right)=$ A) $-\frac{52}{9}$. B) $\frac{56}{9}$. C) $-\frac{56}{9}$. D) $\frac{52}{9}$. Show Answer Correct Answer: A) $-\frac{52}{9}$. 16. The value of f ..... x(1, 2) for f(x, y) = $4x^2y-3y$ A) 13. B) 16. C) 8. D) 4. Show Answer Correct Answer: B) 16. 17. What is the solution to dy/dx = 3? A) Y = x + C. B) Y = 3x + 1. C) Y = 3x + C. D) Y = 3x. Show Answer Correct Answer: C) Y = 3x + C. 18. Dy/dx = x + 3y and y(2) = 3. Use Euler's Method with a step size of 1 to approximate y(5). A) 11. B) 181. C) 240. D) 45. Show Answer Correct Answer: C) 240. 19. For the function f(x, y) = $(x^2 + y^2)(y^2 + 1)$ A) 8. B) 12. C) 16. D) 10. Show Answer Correct Answer: D) 10. 20. Given the differential equation $\frac{dy}{dx} = 3x^2$ A) $y = x^3 + C$. B) $y = x^2 + C$. C) $y = 3x^2 + C$. D) $y = 6x + C$. Show Answer Correct Answer: A) $y = x^3 + C$. 21. The total differential of z = f(x, y) is: A) Dz = f x dx + f y dy. B) Dz = f dx + f dy. C) Dz = dx + dy. D) Dz = dfdy. Show Answer Correct Answer: A) Dz = f x dx + f y dy. 22. Solve $\left(D^2-3DD'+2D'^2\right)z=0$ A) $Z=xf_1\left(y\right)+xf_2\left(y+x\right)$. B) $Z=f_2\left(y-x\right)+f_3\left(y-2x\right)$. C) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)$. D) $Z=f_1\left(y\right)+f_2\left(y+x\right)$. Show Answer Correct Answer: C) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)$. 23. Given $\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}+3y=2e^{-x}.$ $y_p$ A) $Ce^{-x}$. B) $Ce^x$. C) $C\cos x+D\sin x$. D) $Cxe^{-x}$. Show Answer Correct Answer: D) $Cxe^{-x}$. 24. $\int_{ }^{ }3te^{2t}dt$ A) $3e^{2t}-\frac{e^{2t}}{4}+C$. B) $\frac{3te^{2t}}{2}-\frac{3e^{2t}}{4}+C$. C) $\frac{3te^{2t}}{2}-\frac{e^{2t}}{4}+C$. D) $6te^{2t}-12e^{2t}+C$. Show Answer Correct Answer: B) $\frac{3te^{2t}}{2}-\frac{3e^{2t}}{4}+C$. 25. How can Zeno behavior affect system performance? A) Zeno behavior improves system performance by increasing efficiency. B) Zeno behavior can negatively impact system performance by causing delays and inefficiencies. C) Zeno behavior has no effect on system performance whatsoever. D) Zeno behavior only affects hardware, not system performance. Show Answer Correct Answer: B) Zeno behavior can negatively impact system performance by causing delays and inefficiencies. 26. A stone is thrown straight up from the top of a building with initial velocity $40\\frac{ft}{\sec}$ $32\\frac{ft}{\sec^2}$ A) 128. B) 96. C) 112. D) 88. Show Answer Correct Answer: B) 96. 27. Integrate $\int_{ }^{ }\left(\frac{1-\sin x}{\cos^2x}\right)dx$ A) $\tan x+\sec x+c$. B) $\cot x-\operatorname{cosec}x+c$. C) $\tan x-\sec x+c$. D) $\cot x+\operatorname{cosec}x+c$. Show Answer Correct Answer: C) $\tan x-\sec x+c$. 28. Classify the PDE:u ..... xx + u ..... yy = 0. A) Parabolic. B) Hyperbolic. C) Linear. D) Elliptic. Show Answer Correct Answer: D) Elliptic. 29. There were 7 wolves introduced to a reserve in Colorado. The rate of growth of the population of wolves W over time is assumed to be proportional to the number of wolves. A) $W=\sqrt[]{2kt+14}$. B) $W=6e^{kt}$. C) $W=\sqrt[]{2kt+49}$. D) $W=7e^{kt}$. Show Answer Correct Answer: D) $W=7e^{kt}$. 30. $x\frac{dy}{dx}-2y=2x^2\ln\left(x\right)$ A) Bernoulli. B) Linear. C) Homogeneous. D) Separable. Show Answer Correct Answer: B) Linear. 31. What type of differential equations are of the form y' = f(x)g(y)? A) Differential equations with separable variables. B) Differential equations with constant functions. C) Differential equations with polynomial solutions. D) Differential equations with linear coefficients. Show Answer Correct Answer: A) Differential equations with separable variables. 32. What is the integral for $e^{-2t}$ A) $\frac{1}{2}$. B) $\frac{1}{2}-\frac{\frac{ }{1}}{2e^2}$. C) $\frac{1}{2-e}$. D) $\frac{2}{e^2}$. Show Answer Correct Answer: B) $\frac{1}{2}-\frac{\frac{ }{1}}{2e^2}$. 33. By separation of variables, solve the resulting equations $\int\\frac{v}{v+1}dv=\int_{ }^{ }\\frac{1}{y}dy\text{}$ A) $\frac{x}{y}+\frac{x^2}{2y^2}=\ln y+C$. B) $\frac{x}{y}-\ln\left|\frac{x}{y}+1\right|=\ln y+C$. C) $\frac{x}{y}-\frac{x^2}{2y^2}=\ln y+C$. D) $\frac{x}{y}+\ln\left|\frac{x}{y}+1\right|=\ln y+C$. Show Answer Correct Answer: B) $\frac{x}{y}-\ln\left|\frac{x}{y}+1\right|=\ln y+C$. 34. Why are initial or boundary conditions important in solving second order ODEs? A) They only apply to first order ODEs. B) They help find the constants of the complementary function. C) They are not relevant to the solution. D) They determine the form of the non-homogeneous term. Show Answer Correct Answer: B) They help find the constants of the complementary function. 35. The complete solution of Linear Differential equations involves A) Complete function + particular integral. B) Complete function + indefinite integral. C) Complementary function + particular integral. D) Complementary function + definite integral. Show Answer Correct Answer: C) Complementary function + particular integral. 36. Unit normal to the surface $\phi\left(x, y, z\right)=c$ A) $\left|\nabla\phi\right|$. B) $\nabla\phi$. C) $\nabla$. D) $\frac{\nabla\phi}{\left|\nabla\phi\right|}$. Show Answer Correct Answer: D) $\frac{\nabla\phi}{\left|\nabla\phi\right|}$. 37. Y is proportional to the difference of x and z A) Y= k(z-x). B) Y= z-x. C) Y= k(x-z). D) Y= x-z. Show Answer Correct Answer: C) Y= k(x-z). 38. Determine the degree of the differential equation:$\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} + y = 0$ A) 1. B) 2. C) 3. D) 4. Show Answer Correct Answer: C) 3. 39. What is the general solution to Cauchy's and Euler's equations? A) Exponential functions. B) Polynomial functions. C) Power series. D) Trigonometric functions. Show Answer Correct Answer: C) Power series. 40. State the order and degree for differential equation below:$\left(\frac{\text{d}^2y}{\text{d}x^2}\right)^3+5\left(\frac{\text{d}y}{\text{d}x}\right)^4=\cos3x$ A) Order:2Degree:4. B) Order:3Degree:2. C) Order:2Degree:3. D) Order:4Degree:2. Show Answer Correct Answer: C) Order:2Degree:3. 41. If $f\left(x\right)=\int_1^xt\left(t^3+1\right)^{\frac{3}{2}}dt, $ $f'\left(2\right)$ A) 54. B) $2^{\frac{3}{2}}$. C) 135. D) 18. Show Answer Correct Answer: A) 54. 42. Given a differential equation as $y\frac{\text{d}y}{\text{d}x}=3$ A) $y^2=3+\frac{1}{2}$. B) $\frac{y^2}{2}=3x+\frac{1}{2}$. C) $\frac{y^2}{2}=3x+c$. D) $\frac{y^2}{2}=\frac{3x^2}{3}+\frac{1}{2}$. Show Answer Correct Answer: B) $\frac{y^2}{2}=3x+\frac{1}{2}$. 43. Find the particular solution to the differential equation $\frac{dy}{dx}=2xy+x^2y$ $\left(-3, -1\right)$ A) $y=e^{\frac{x^2}{2}+\frac{x^3}{3}}$. B) $y=-2e^{x^2+\frac{x^3}{3}}$. C) $y=-e^{\frac{x^2}{2}+\frac{x^3}{3}}$. D) $y=-e^{x^2+\frac{x^3}{3}}$. Show Answer Correct Answer: D) $y=-e^{x^2+\frac{x^3}{3}}$. 44. If we eliminate a and b from z = (x+a)(y+b) A) Z = pq. B) Z = p/q. C) Z = p-q. D) Z = p + q. Show Answer Correct Answer: A) Z = pq. 45. What is the condition for a system of equations to have infinitely many solutions? A) All equations are independent. B) Determinant of coefficients is zero. C) At least one equation is redundant. D) Determinant of coefficients is non-zero. Show Answer Correct Answer: B) Determinant of coefficients is zero. 46. The roots of m$^{2}$+1=0 are real and equal. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 47. What does it mean if a function is analytic at a point? A) It has a discontinuity. B) It is not defined at that point. C) It is a constant function. D) It has a Taylor series representation. Show Answer Correct Answer: D) It has a Taylor series representation. 48. What type of PDE is the heat equation? A) Hyperbolic partial differential equation. B) Parabolic partial differential equation. C) Elliptic partial differential equation. D) Linear ordinary differential equation. Show Answer Correct Answer: B) Parabolic partial differential equation. 49. $L\left(\cosh ax\right)= ..... $ A) $\frac{s}{s^2+1}$. B) $\frac{a}{s^2+1}$. C) $\frac{s}{s^2-a^2}$. D) $\frac{a}{s^2-1}$. Show Answer Correct Answer: C) $\frac{s}{s^2-a^2}$. 50. What is the degree of the equation $(y" ')^2+2y'+3=0$ A) 4. B) 1. C) 2. D) 3. Show Answer Correct Answer: C) 2. 51. How is a PDE formed by eliminating arbitrary constants? A) A PDE is formed by differentiating a family of functions and eliminating arbitrary constants. B) A PDE is formed by substituting values into a constant equation. C) A PDE is formed by integrating a single function and adding constants. D) A PDE is formed by multiplying two functions and introducing variables. Show Answer Correct Answer: A) A PDE is formed by differentiating a family of functions and eliminating arbitrary constants. 52. If $3x^2 + 2y$ $2x + 4y$ A) Exact. B) Not exact. C) Linear. D) Homogeneous. Show Answer Correct Answer: A) Exact. 53. Find the solution of the second order differential equation y ''+y=0 is A) Y= C1 ex+ C2e-2x. B) Y= C1sinx+ C2cosx. C) Y= C1 + C2x. D) Y= C1 e2x+ C2e-2x. Show Answer Correct Answer: A) Y= C1 ex+ C2e-2x. 54. The solution of the differential equation x$^{2}$y" +xy'-y=0 is y=c$_{1}$x+c$_{2}$x$^{-1}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 55. Which of the following functions is a solution of the equation $y" -y = 0$ $y(t) = Ce^{t}$ $C$ $y(t) = Ce^{-t} + 2$ $C$ $y(t) =\cos t$ A) (i) and (ii). B) (i) and (iii). C) (i). D) None of the above. Show Answer Correct Answer: C) (i). 56. The value of $\nabla\left(\overrightarrow{a}\cdot\overrightarrow{r}\right)$ A) $\left|\overrightarrow{a}\right|$. B) $\overrightarrow{a}$. C) 0. D) A. Show Answer Correct Answer: B) $\overrightarrow{a}$. 57. Given the logistic differential equation $\frac{\text{d}P}{\text{d}t}=P\left(2-\frac{P}{900}\right)$ $\lim_{t\rightarrow\infty}P\left(t\right)$ A) 1800. B) 900. C) 2. D) 450. Show Answer Correct Answer: A) 1800. 58. How do you solve a separable differential equation? A) Multiply the variables and then integrate each side separately. B) Take the derivative of each side separately. C) Separate the variables and then integrate each side separately. D) Combine the variables and then integrate each side separately. Show Answer Correct Answer: C) Separate the variables and then integrate each side separately. 59. Solvability for y or x is usually done by differentiating with respect to: A) Independent variable only. B) The variable being eliminated. C) Constant. D) None. Show Answer Correct Answer: A) Independent variable only. 60. Which of the following differential equations is not linear? A) $\frac{1}{x}\frac{\text{d}y}{\text{d}x}-y=x+1$. B) $x\frac{\text{d}y}{\text{d}x}+y=x-x^2y$. C) $\frac{\text{d}y}{\text{d}x}=1-xy$. D) $\frac{\text{d}y}{\text{d}x}+xy^2=e^x$. Show Answer Correct Answer: D) $\frac{\text{d}y}{\text{d}x}+xy^2=e^x$. ← 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