This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 4 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 4 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Order of a Differential EquationThe order of a differential equation is determined by: A) The highest power of the derivative in the equation. B) The number of terms in the equation. C) The highest power of the variable in the equation. D) The degree of the function in the equation. Show Answer Correct Answer: A) The highest power of the derivative in the equation. 2. $\frac{\text{d}y}{\text{d}x}=x^2+1$ A) First order differential equation at second degree. B) Second order differential equation at first degree. C) First order differential equation. D) Second order differential equation. Show Answer Correct Answer: C) First order differential equation. 3. $y-x\frac{dy}{dx}=3-2x^2\frac{dy}{dx}$ A) Homogeneous. B) Linear. C) Bernoulli. D) Separable. Show Answer Correct Answer: D) Separable. 4. If $W(y_1, y_2)(x)\ne0$ A) $y_1, y_2$. B) $y_1, y_2$. C) Both are solutions of non-homogeneous DE. D) None. Show Answer Correct Answer: A) $y_1, y_2$. 5. What is the significance of integrating factors in differential equations? A) Integrating factors simplify the solving process of first-order linear differential equations. B) Integrating factors are used only in second-order differential equations. C) Integrating factors are irrelevant to solving linear equations. D) Integrating factors complicate the process of solving equations. Show Answer Correct Answer: A) Integrating factors simplify the solving process of first-order linear differential equations. 6. Which of the following is the solution to the differential equation $\frac{dy}{dt}=4y$ $y(0)=3$ A) $y=2y^2$. B) $y=4ty+3$. C) $y=3+e^{4t}$. D) $y=3e^{4t}$. Show Answer Correct Answer: D) $y=3e^{4t}$. 7. Which of the following statements about second order linear differential equations is true? A) They are only applicable in theoretical mathematics. B) They can only be solved using numerical methods. C) They can be used to model real-world phenomena in various fields. D) They have no solutions if the roots are complex. Show Answer Correct Answer: C) They can be used to model real-world phenomena in various fields. 8. The general solution of the ODE contains ..... A) Complementary function only. B) Particular integral only. C) Both complementary function and particular integral. D) Neither complementary function nor particular integral. Show Answer Correct Answer: C) Both complementary function and particular integral. 9. What is the wave equation and its applications in solving partial differential equations? A) The wave equation is a first-order partial differential equation used to model wave phenomena. B) The wave equation is a third-order partial differential equation used to model wave phenomena. C) The wave equation is a second-order partial differential equation used to model wave phenomena. D) The wave equation is a linear equation used to model wave phenomena. Show Answer Correct Answer: C) The wave equation is a second-order partial differential equation used to model wave phenomena. 10. Find the general solution of the differential equation $y'e^{2x}+2ye^{2x}=4$ A) $y=\frac{2x}{e^{2x}}$. B) $y=\frac{2x+c}{e^{2x}}$. C) $y=\frac{4x+c}{e^{2x}}$. D) $y=\frac{4x}{e^{2x}}$. Show Answer Correct Answer: C) $y=\frac{4x+c}{e^{2x}}$. 11. IVP stand for ..... A) Index Value Problem. B) Index Value Principle. C) Initial Value Problem. D) Initial Value Principle. Show Answer Correct Answer: C) Initial Value Problem. 12. Find the critical points of $f(x, y) = x^2 + y^2-4x-6y + 13$ A) (0, 0). B) (-2, -3). C) (4, 6). D) (2, 3). Show Answer Correct Answer: D) (2, 3). 13. $L\left[f\left(x\right)\right]=\frac{1}{a}F\left(\frac{s}{a}\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 14. Maclaurin series has center at ..... A) X = c. B) X = 1. C) X = e. D) X = 0. Show Answer Correct Answer: D) X = 0. 15. In the equation dx/P=dy/Q=dz/R, what does the variable P represent? A) Ratio of changes in x. B) Constant term. C) Sum of all variables. D) Coefficient of dx. Show Answer Correct Answer: D) Coefficient of dx. 16. Part C. Find the equation (inverse method) by the general solution:18. $C_1e^x+C_2e^{-2x}$ A) Y" -2y'-y = 0. B) Y" + 2y' + y = 0. C) Y" + y'-2y = 0. D) Y" -y' + 2y = 0. Show Answer Correct Answer: C) Y" + y'-2y = 0. 17. What are some applications of partial differential equations? A) Modeling population growth, analyzing stock market trends, predicting weather patterns. B) Solving algebraic equations, studying animal behavior, analyzing chemical reactions. C) Designing computer algorithms, studying human anatomy, analyzing geological formations. D) Modeling heat transfer, fluid dynamics, electromagnetic fields, quantum mechanics, and financial mathematics. Show Answer Correct Answer: D) Modeling heat transfer, fluid dynamics, electromagnetic fields, quantum mechanics, and financial mathematics. 18. Solve the first-order differential equation:$\frac{dy}{dx} = 3x^2$ A) $y = x^2 + C$. B) $y = x^3 + C$. C) $y = x^3-C$. D) $y = 3x^3 + C$. Show Answer Correct Answer: B) $y = x^3 + C$. 19. Solve this differential equation:$\frac{dy}{dt} = 0.03y(900-y)$ $y(0) = 2$ $y(t) =$ A) Y(t) = $\frac{900}{1 + 449e^{27t}}$. B) Y(t) = $\frac{900}{1 + 449e^{-27t}}$. C) Y(t) = $\frac{900}{1-449e^{27t}}$. D) Y(t) = $\frac{900}{1-449e^{-27t}}$. Show Answer Correct Answer: B) Y(t) = $\frac{900}{1 + 449e^{-27t}}$. 20. What is the solution of ODE A) Y = Cx. B) Y = $\frac{C}{x}$. C) $y = Cx^{2}$. D) $y = Ce^{x}$. Show Answer Correct Answer: B) Y = $\frac{C}{x}$. 21. $\frac{dx}{dt}+\frac{2}{4-t}x=5$ A) Homogeneous. B) Bernoulli. C) Separable. D) Linear. Show Answer Correct Answer: D) Linear. 22. What does it mean if the Wronskian of a set of functions vanishes identically? A) The functions are not defined. B) The functions are linearly dependent. C) The functions are constant. D) The functions are linearly independent. Show Answer Correct Answer: B) The functions are linearly dependent. 23. What is the general solution to $y" -4y' + 13y = 0$ A) $y(x) = Ce^{2x}\sin(13x) + De^{2x}\cos(13x)$. B) $y(x) = Ce^{2x}\sin(3x) + De^{2x}\cos(3x)$. C) $y(x) = Ce^{2x} + De^{3x}$. D) $y(x) = Ce^{4x}\sin(13x) + De^{4x}\cos(13x)$. Show Answer Correct Answer: B) $y(x) = Ce^{2x}\sin(3x) + De^{2x}\cos(3x)$. 24. Consider the differential equation $\frac{dy}{dx}=\frac{x}{y}$ A) $y=x\ln\left|y\right|+c$. B) $y=\frac{x^2}{2}\ln\left|y\right|+c$. C) $y^2=x^2+c$. D) $y^2=\frac{x^2}{2}+c$. Show Answer Correct Answer: C) $y^2=x^2+c$. 25. In variation of parameters, particular solution is found by: A) Matching coefficients. B) Substituting constants. C) Separation of variables. D) Integrating functions involving Wronskian. Show Answer Correct Answer: D) Integrating functions involving Wronskian. 26. Solve the initial value problem:$y" + 2y' + 5y = 0$ $y(0) = 1$ $y'(0) = 0$ A) $y(x) = e^{-2x}\cos(x)$. B) $y(x) = e^{-x}\sin(2x)$. C) $y(x) = e^{-x}\cos(2x) + e^{-x}\sin(2x)$. D) $y(x) = e^{-x}\cos(2x)$. Show Answer Correct Answer: D) $y(x) = e^{-x}\cos(2x)$. 27. If a function is y = 2x, then its slope field equation would ..... A) Have all slopes of 2. B) Have all slopes of x$^{2}$. C) Have slopes x$^{3}$. D) Have slopes of 0. Show Answer Correct Answer: A) Have all slopes of 2. 28. Consider the general solution y = f(x) to the differential equation $\frac{dy}{dx} = \frac{y}{x}$ A) Quadratic. B) Exponential. C) Logarithmic. D) Linear. Show Answer Correct Answer: D) Linear. 29. Using the separation of variables, what is the solution of the differential equation $\frac{dy}{dx}=\frac{3xy}{1-x^2}$ A) $y=\frac{K}{\left(\frac{3}{2}-x^2\right)^{\frac{5}{2}}}$. B) $y=\frac{K}{\left(1-x^2\right)^{\frac{3}{2}}}$. C) $y=\frac{K}{\left(1-x^2\right)^{\frac{5}{2}}}$. D) $y=\frac{K}{\left(1+x^2\right)^{\frac{3}{2}}}$. Show Answer Correct Answer: B) $y=\frac{K}{\left(1-x^2\right)^{\frac{3}{2}}}$. 30. If the exact differential equation:M dx+N dy= 0 is solved using the method of integrating factors, then the integrating factor$\mu$(x, y) is chosen such that: A) $\mu$(x, y) always depends only on x. B) $\mu$(x, y) always depends only on y. C) $\mu$(x, y) must be constant. D) $\mu$(x, y)M dx+$\mu$(x, y)N dy satisfies the exactness condition. Show Answer Correct Answer: D) $\mu$(x, y)M dx+$\mu$(x, y)N dy satisfies the exactness condition. 31. The order of the following pde $\left(z_{xx}\right)^2+\left(z_{xyy}\right)+\left(z_{yy}\right)=\sin z$ A) 1. B) 2. C) 3. D) None of the above. Show Answer Correct Answer: C) 3. 32. What does the term 'particular solution' refer to? A) A solution with arbitrary constants. B) A solution that is always true. C) A solution that cannot be derived. D) A solution that satisfies initial conditions. Show Answer Correct Answer: D) A solution that satisfies initial conditions. 33. A population of flies is modeled by a function $y\left(t\right)$ $\frac{\text{d}y}{\text{d}t}=4y\left(1-\frac{y}{50}\right)$ A) 4. B) 25. C) 2. D) 50. Show Answer Correct Answer: B) 25. 34. Which of the following is a characteristic of non-linear PDEs? A) Only linear terms are present in the equation. B) The solution is always unique and stable. C) The equation can be solved using linear methods. D) Presence of non-linear terms involving the unknown function and its derivatives. Show Answer Correct Answer: D) Presence of non-linear terms involving the unknown function and its derivatives. 35. Let y=f(x) be a particular solution to the differential equation $\frac{\text{d}y}{\text{d}x}=xy^3$ $f\left(1\right)=2$ $x=1$ A) $y-2=8\left(x-1\right)$. B) $y-1=8\left(x-2\right)$. C) $y-2=2\left(x-1\right)$. D) $y-1=2\left(x-2\right)$. Show Answer Correct Answer: A) $y-2=8\left(x-1\right)$. 36. $\frac{d^2y}{dx^2}=2x\frac{dy}{dx}+3$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 37. The C.F. of the equation $\frac{d^2y}{dx^2}+\frac{3dy}{dx}+2y=e^{-x}$ A) $c_1e^{-2x}+c_2e^{-2x}$. B) $c_1e^x+c_2e^{2x}$. C) $c_1e^{-x}+c_2e^{-2x}$. D) $c_1e^{3x}+c_2e^{2x}$. Show Answer Correct Answer: C) $c_1e^{-x}+c_2e^{-2x}$. 38. Differential equation of the curve y=Acosx-Bsinx, where A and B are arbitrary constant is A) Y$^{"}$-y=0. B) Y$^{"}$+y=0. C) Y$^{"}$+x=0. D) Y$^{" }$+xy=0. Show Answer Correct Answer: B) Y$^{"}$+y=0. 39. How many degrees is/are this DE? (y" ')$^{3}$ + 3y" + 6y'-12 = 0 A) 2. B) 3. C) 1. D) 4. Show Answer Correct Answer: B) 3. 40. Explain the concept of time in CPS modeling. A) Time in CPS modeling is irrelevant to system performance. B) Time in CPS modeling is a dimension that influences the behavior and interactions of physical and computational components, represented as discrete or continuous time. C) Time in CPS modeling is always represented as a fixed value. D) Time is only considered in computational components, not physical ones. Show Answer Correct Answer: B) Time in CPS modeling is a dimension that influences the behavior and interactions of physical and computational components, represented as discrete or continuous time. 41. The Fourier Sine Transform of $e^{-ax}$ A) $F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. B) $F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{s}{a^2+s^2}$. C) $F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{s}{a^2+s^2}$. D) None of the above. Show Answer Correct Answer: B) $F_s\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{s}{a^2+s^2}$. 42. Mdx +Ndy = 0 is exact iff A) $\frac{\partial M}{\partial x}=\frac{\partial N}{\partial y}$. B) $\frac{\partial M}{\partial y}=-\frac{\partial N}{\partial x}$. C) $\frac{\partial M}{\partial x}=-\frac{\partial N}{\partial y}$. D) $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. Show Answer Correct Answer: D) $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. 43. Which of the following is a general form of a second order homogeneous differential equation with constant coefficients? A) $y" + ay' + by = f(x)$. B) $y" + y' + y = f(x)$. C) $y" + ay' + by = 0$. D) $y" + p(x)y' + q(x)y = 0$. Show Answer Correct Answer: C) $y" + ay' + by = 0$. 44. The ends A and B of a rod of length10cm are at 30$^{0}$C and 80$^{0}$C at end points until steady state prevails. Then Initial temperature distribution in the rod A) 10+(5x)/2. B) 30+(5x)/2. C) 30+5x. D) None of these. Show Answer Correct Answer: C) 30+5x. 45. An antiderivative of $\dfrac{1}{x}$ A) $x^{-1}$. B) $x$. C) $\ln|x|$. D) $e^x$. Show Answer Correct Answer: C) $\ln|x|$. 46. What is the characteristic equation / auxiliary equation for $4\frac{\text{d}^2y}{\text{d}x^2}+3\frac{dy}{dx}=0$ A) $4\lambda^2+3\lambda=0$. B) $4\lambda^2+3=0$. C) $\lambda^2+3\lambda=0$. D) None of the above. Show Answer Correct Answer: A) $4\lambda^2+3\lambda=0$. 47. In growth/decay, the sign of k determines: A) The initial value. B) Solution type only. C) Whether solution increases or decreases. D) The domain of solution. Show Answer Correct Answer: C) Whether solution increases or decreases. 48. How many independent variables of an ordinary differential equation? A) One variable. B) Many variable. C) More than one variable. D) Two variables. Show Answer Correct Answer: A) One variable. 49. Determine the order of the differential equation:$\frac{d^5y}{dx^5} + \left(\frac{d^2y}{dx^2}\right)^3 + y = 0$ A) 5. B) 2. C) 4. D) 3. Show Answer Correct Answer: A) 5. 50. $L\left(x^n\right)$ A) $\frac{n}{s^n+1}$. B) $\frac{n!}{s^n-1}$. C) $\frac{n!}{s^n+1}$. D) $\frac{n!}{s+1}$. Show Answer Correct Answer: C) $\frac{n!}{s^n+1}$. 51. Which of the following is an equilibrium solution for the differential equation $\frac{dy}{dt} = y^3-3y$ A) $y = 1$. B) $y = 0$. C) $y =-1$. D) $y = \sqrt{3}$. Show Answer Correct Answer: B) $y = 0$. 52. Find the complementary function of $\left(D^3-3D^2D'+2DD^{'2}\right)Z=0$ A) $Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. B) $Z=f_1\left(y-x\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. C) $Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y-2x\right)$. D) $Z=f_1\left(y\right)+xf_2\left(y+x\right)+f_3\left(y+2x\right)$. Show Answer Correct Answer: A) $Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. 53. Can you give an example of a partial differential equation? A) The quadratic equation. B) The law of gravity. C) The heat equation. D) The Pythagorean theorem. Show Answer Correct Answer: C) The heat equation. 54. A PDE consists of A) Only one independent variable. B) More than one independent variable. C) More than one dependent varible. D) None of the above. Show Answer Correct Answer: B) More than one independent variable. 55. For $y" +y=\sin x$ A) $12x\sin x$. B) $12x\cos x$. C) $A\sin x$. D) $Ax\cos x$. Show Answer Correct Answer: B) $12x\cos x$. 56. Evaluate the definite integral ( int ..... 0^1 (3x^2 + 2) , dx ). A) 2. B) 4. C) 3. D) 5. Show Answer Correct Answer: C) 3. 57. Which differential equation has the solution $y=\sin x$ A) $y-y" =0$. B) $y+y'-y" =0$. C) $y+y" =0$. D) $y-y'+y" =0$. Show Answer Correct Answer: C) $y+y" =0$. 58. A particular solution for $y" +y=1$ A) 1. B) -1. C) Constant = 0. D) None. Show Answer Correct Answer: A) 1. 59. If $\overrightarrow{f}=xyi+yzj+zxk$ $curl\\overrightarrow{f}$ A) -yi-zj-xk. B) Xi+yj+zk. C) Yi+zj-xk. D) Yi+zj+xk. Show Answer Correct Answer: A) -yi-zj-xk. 60. What is the Eigen value of Upper triangular matrix of A= [ 2 3 6 0 1 8 0 0 3] A) 1, 2, 3. B) 1, 2, 1. C) 1, 1, 1. D) None of the above. Show Answer Correct Answer: A) 1, 2, 3. ← 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 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 9 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books