This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 8 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 8 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The value of $\int_0^a\int_0^axdxdy$ A) $\frac{a^3}{2}$. B) 1/2. C) $\frac{a^3}{3}$. D) $\frac{a^2}{2}$. Show Answer Correct Answer: A) $\frac{a^3}{2}$. 2. For the equation y" +y'+y=0, the roots of the auxiliary equation are: A) Real and distinct. B) Real and equal. C) Complex conjugates. D) All imaginary. Show Answer Correct Answer: C) Complex conjugates. 3. What is the order of the equation y" + 3y'-4y = 0? A) 1. B) 4. C) 3. D) 2. Show Answer Correct Answer: D) 2. 4. What is a second order differential equation? A) Is a differential equation that is equated to two. B) Is any equation that is differentiable. C) Is an equation that is differentiated twice. D) Is a differential equation whose highest derivative is to power two. Show Answer Correct Answer: D) Is a differential equation whose highest derivative is to power two. 5. If the characteristic equation has two distinct real roots $r_1$ $r_2$ $y" + ay' + by = 0$ A) $y(x) = Ce^{r_1x} + De^{r_1x}$. B) $y(x) = Ce^{r_1x} + De^{r_2x}$. C) $y(x) = C\sin(rx) + D\cos(rx)$. D) $y(x) = Ce^{rx}$. Show Answer Correct Answer: B) $y(x) = Ce^{r_1x} + De^{r_2x}$. 6. $\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}=0$ A) $y\left(f, x\right)$. B) $x=f\left(y, t\right)$. C) $f\left(x, y\right)$. D) $f\left(x, t\right)$. Show Answer Correct Answer: C) $f\left(x, y\right)$. 7. Which function grows the fastest as $x$ A) $e^x$. B) $\ln x$. C) $\sin x$. D) $x^2$. Show Answer Correct Answer: A) $e^x$. 8. What are the boundary conditions in the context of solving partial differential equations? A) Conditions specified at the initial time step. B) Conditions specified at random points in the domain. C) Conditions specified on the boundaries of the domain. D) Conditions specified inside the domain. Show Answer Correct Answer: C) Conditions specified on the boundaries of the domain. 9. A set of linear differential equations of first order with constant coefficients, $\frac{d\vec{x}}{dt}=A\\vec{x}$ A) I don't know. B) You mainly have to compute the eigenvalues of the matrix. C) The matrix A must always be invertible. D) It is inhomogeneous. Show Answer Correct Answer: B) You mainly have to compute the eigenvalues of the matrix. 10. $\left(\tan x\right)y'+y=2\sin(x)$ A) $y=\frac{(k-2\cos(2x))}{2\sin(x)}$. B) $y=\frac{(k-2\cos(x))}{\sin(x)}$. C) $y=\frac{(k-2\cos(x))}{(\sin(2x))}$. D) $y=\frac{(k-\cos(x))}{2\sin(2x)}$. Show Answer Correct Answer: A) $y=\frac{(k-2\cos(2x))}{2\sin(x)}$. 11. What is the degree of a differential equation? A) The number of solutions to the equation. B) The number of variables in the equation. C) The highest order derivative present. D) The highest power of the highest order derivative. Show Answer Correct Answer: D) The highest power of the highest order derivative. 12. Consider the differential equation $\frac{dy}{dx}=x+4y+2$ $y=f\left(x\right)$ $f\left(0\right)=k$ $f\left(1\right)\approx\frac{11}{4}$ A) $-\frac{1}{6}$. B) $-\frac{19}{60}$. C) $\frac{3}{20}$. D) $-\frac{5}{36}$. Show Answer Correct Answer: A) $-\frac{1}{6}$. 13. A bacteria culture grows at a constant relative growth rate. The count was 400 after 2 hours and 25, 600 after 6 hours. Find the value of k, the constant relative growth rate. A) $\frac{\ln25, 600}{4}$. B) $\frac{e^{64}}{4}$. C) $\frac{\ln\left(64\right)}{6}$. D) $\frac{\ln\left(64\right)}{4}$. Show Answer Correct Answer: D) $\frac{\ln\left(64\right)}{4}$. 14. Determine whether the functions $y_{1} = e^{2x}$ $y_{2} = e^{-4x}$ A) Not defined. B) Yes, they are linearly independent. C) They are orthogonal. D) No, they are linearly dependent. Show Answer Correct Answer: B) Yes, they are linearly independent. 15. Standard form for a pair of ordinary simultaneous equation of the first order but first degree A) Dx/l+dy/m+dz+n. B) Dx/p=dY/Q=dz/R. C) Dx/p=dy/Q=dz/R. D) Dx/l=dy/m=dz/n. Show Answer Correct Answer: B) Dx/p=dY/Q=dz/R. 16. If dy/dt = ky and k is a nonzero constant, then y could be A) $69e^{kt}$. B) $e^{kt}+3$. C) $kt+5$. D) $2e^{kty}$. Show Answer Correct Answer: A) $69e^{kt}$. 17. Find the complementary function of $\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial x\partial y}-2\frac{\partial^2z}{\partial y^2}=0$ A) $Z=f_1\left(y\right)+f_2\left(y+2x\right)$. B) $Z=f_1\left(y+x\right)+xf_2\left(y-2x\right)$. C) $Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$. D) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)$. Show Answer Correct Answer: C) $Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$. 18. Dy/dx= tanx+15x$^{2}$+e$^{x}$+1/x A) Y= sec$^{2}$(x) + 5x$^{3}$ + e$^{x}$ + ln|x| + c. B) Y=-ln|sinx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. C) Y= secxtanx + 5x$^{3}$ + e$^{x}$ + ln|x| + c. D) Y=-ln|cosx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. Show Answer Correct Answer: D) Y=-ln|cosx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. 19. Number of arbitrary constant in the general solution of a differential equation of degree 3 and order 4 is A) 0. B) 3. C) 4$^{3}$. D) 4. Show Answer Correct Answer: D) 4. 20. Solve the differential equation xdy/dx =y-xtan(y/x) A) Tan(y/x) =c/x. B) Cos(y/x) =c/x. C) Sin(y/x) =c/x. D) -sin(y/x) =c/x. Show Answer Correct Answer: C) Sin(y/x) =c/x. 21. Superposition principle applies to? A) Nonlinear systems. B) Linear homogeneous systems. C) Nonhomogeneous nonlinear systems. D) All differential equations. Show Answer Correct Answer: B) Linear homogeneous systems. 22. What does the natural log function become when exponentiated? A) A new constant. B) The original variable before taking the natural log. C) The constant e. D) The variable y. Show Answer Correct Answer: B) The original variable before taking the natural log. 23. Separate the variables for the equation $(x-1)\frac{dy}{dx}=(y+1)$ A) $\frac{1}{(y+1)}dy=\frac{1}{(x-1)}dx$. B) $\frac{1}{(x-1)}dy=\frac{1}{(y+1)}dx$. C) $\frac{x}{(y+1)}dy=dx$. D) $(y+1)dy=(x-1)dx$. Show Answer Correct Answer: A) $\frac{1}{(y+1)}dy=\frac{1}{(x-1)}dx$. 24. The Wronskian of two functions $y_1, y_2$ A) $y_1y_2$. B) $y_1'y_2-y_1y_2'$. C) $y_1+y_2$. D) None. Show Answer Correct Answer: B) $y_1'y_2-y_1y_2'$. 25. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Find the general solution. A) P=.05e$^{1000t}$. B) P=lne$^{1000t}$. C) P=1000e$^{5t}$. D) P=1000e$^{.05t}$. Show Answer Correct Answer: D) P=1000e$^{.05t}$. 26. The Fourier cosine 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) $F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. D) None of the above. Show Answer Correct Answer: C) $F_c\left[e^{-ax}\right]=\sqrt{\frac{2}{\pi}}\\frac{a}{a^2+s^2}$. 27. Which one of the following differential equations is linear? A) $2y\\frac{d^2y}{dx^2}+4\\frac{dy}{dx}+xy=5$. B) $\frac{d^2y}{dx}-\frac{dy}{dx}+\sin y=0$. C) $3\\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2-y=\cos x$. D) $x\\frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=e^{2x}$. Show Answer Correct Answer: D) $x\\frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=e^{2x}$. 28. The matrix exponential method is primarily used to ..... A) Find eigenvalues of matrix. B) Solve nonlinear systems of equations. C) Compute determinants of matrices. D) Solve systems of linear differential equations. Show Answer Correct Answer: D) Solve systems of linear differential equations. 29. The particular integral of the equation $\left(D^3-1\right)y=\left(e^x+1\right)^2$ A) $\frac{e^{-2x}}{7}+\frac{2x}{3}-1$. B) $\frac{e^{2x}}{7}+\frac{2x}{3}-1$. C) $\frac{e^{2x}}{7}+\frac{5x}{3}-1$. D) $\frac{e^{3x}}{7}+\frac{2x}{3}-1$. Show Answer Correct Answer: B) $\frac{e^{2x}}{7}+\frac{2x}{3}-1$. 30. Each ratio in equation dx/P=dy/Q=dz/R is equal to A) Dx/P=dy/Q=dz/R. B) Ldx+mdy+ndz/lP+mQ+nR. C) Pdx+Qdy+Rdz/p$^{2}$+Q$^{2}$+R$^{2}$. D) Dx+dy+dz/P+Q+R. Show Answer Correct Answer: B) Ldx+mdy+ndz/lP+mQ+nR. 31. A population of bacteria grows according to the differential equation dp/dt = 0.05p, where p(0) = 100. What is the population after 10 hours? A) 50.25. B) 200.35. C) 164.87. D) 75.10. Show Answer Correct Answer: C) 164.87. 32. If $\frac{dy}{dx}=4x-1$ $y\left(1\right)=3$ $y$ A) $2x^2+1$. B) $2x^2-x+2$. C) $2x^2-x$. D) $2x^2-x+1$. Show Answer Correct Answer: B) $2x^2-x+2$. 33. What is a homogeneous PDE? A) A homogeneous PDE has no free terms and all terms involve the dependent variable and its derivatives. B) A homogeneous PDE only involves constants and no variables. C) A homogeneous PDE has varying coefficients and free terms. D) A homogeneous PDE is defined by its non-linear terms. Show Answer Correct Answer: A) A homogeneous PDE has no free terms and all terms involve the dependent variable and its derivatives. 34. If $y=\frac{1}{e^{2x}}\left[\frac{x^4}{4}+\ln\left|x\right|+c\right]$ $c$ $x=1, \\y=0$ A) $\ln4$. B) 0. C) Any value in Real Number. D) $-\frac{1}{4}$. Show Answer Correct Answer: D) $-\frac{1}{4}$. 35. Which of the following is not Dirichlet's condition for the Fourier series expansion? A) F(x) is periodic, single valued, finite. B) F(x) has a finite number of discontinuities in only one period. C) F(x) has a finite number of maxima and minima. D) F(x) is a periodic, single valued, infinite. Show Answer Correct Answer: D) F(x) is a periodic, single valued, infinite. 36. When the dependent variable and all of its derivatives are of the first degree, the equation is A) Non linear. B) Linear. C) Both linear and non linear. D) None of the options. Show Answer Correct Answer: B) Linear. 37. An equivalent representation of the definite integral $\int_1^32x\cos\left(x^2\right)dx$ A) $\int_1^92\sqrt[]{u}\cos udu$. B) $\int_1^3\cos udu$. C) $\int_1^9\cos udu$. D) $\int_1^{\sqrt[]{3}}\cos udu$. Show Answer Correct Answer: C) $\int_1^9\cos udu$. 38. Solve the initial value problem $y'=x^3\left(1-y\right)\\\\, \\\\\\y\left(0\right)=3$ A) $y=e\frac{-x^4}{2}+1$. B) $y=2e^{\frac{-x^4}{4}}+1$. C) $y=\frac{-x^4}{4}+C$. D) $y=e^{-x^4}+8$. Show Answer Correct Answer: B) $y=2e^{\frac{-x^4}{4}}+1$. 39. Apply the initial condition $y(0) = 2$ $y = Ce^{3x}$ $\frac{dy}{dx} = 3y$ $C$ A) $C = 3$. B) $C = 0$. C) $C = 1$. D) $C = 2$. Show Answer Correct Answer: D) $C = 2$. 40. The general solution of the DE $\frac{d^2y}{dx^{2}}+6\frac{dy}{dx}-5y=0$ A) $y=Ae^x+Be^{5x}$. B) $y=A\cos\left(-3+\sqrt[]{14}\right)x+B\sin\left(-3+\sqrt[]{14}\right)x$. C) $y=Ae^{-x}+Be^{-5x}$. D) $y=Ae^{\left(-3+\sqrt[]{14}\right)x}+Be^{\left(-3-\sqrt[]{14}\right)x}$. Show Answer Correct Answer: D) $y=Ae^{\left(-3+\sqrt[]{14}\right)x}+Be^{\left(-3-\sqrt[]{14}\right)x}$. 41. $\frac{\partial^3z}{\partial x^3}-3\frac{\partial^3z}{\partial x^2\partial y}+4\frac{\partial^3z}{\partial y^3}=0$ A) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. B) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y-2x\right)$. C) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. D) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y+2x\right)$. Show Answer Correct Answer: A) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. 42. Find dy/dx if y = sec(8x$^{2}$). A) Sec(16x)tan(16x). B) 16xcos(8x$^{2}$). C) 16xsec(8x$^{2}$)tan(8x$^{2}$). D) 16x$^{}$secxtanx(8x$^{2}$). Show Answer Correct Answer: C) 16xsec(8x$^{2}$)tan(8x$^{2}$). 43. L$^{-1 }$\{s/(s$^{2}$+b$^{2}$)\} A) 1. B) T$^{n}$. C) Cos bt. D) T$^{n }$e$^{at}$. E) E$^{at}$. Show Answer Correct Answer: C) Cos bt. 44. Let PDE u$_{xx }$+ u$_{yy }$= u$_{t}$, By method separation, we consider the solution A) U(x, y)=X(x)Y(y). B) U(x, y, t)=X(x)Y(y)T(t). C) U(x, t)=X(x)T(t). D) None of these. Show Answer Correct Answer: B) U(x, y, t)=X(x)Y(y)T(t). 45. The value of $\nabla\cdot\overrightarrow{r}$ A) 3. B) 2. C) 0. D) 1. Show Answer Correct Answer: A) 3. 46. If $\phi\left(x, y, z\right)=xy^2+yz^3$ $grad\\phi$ $y=0$ A) $z^2\overrightarrow{j}+\overrightarrow{k}$. B) $z^2\overrightarrow{j}$. C) $z^3\overrightarrow{j}$. D) $z^3\overrightarrow{i}$. Show Answer Correct Answer: C) $z^3\overrightarrow{j}$. 47. The complementary function on y ''-4y=0 is A) C1 sin2x+ C2cos2x. B) C1 ex+ C2e-x. C) C1 e2x+ C2e-2x. D) C1 + C2x. Show Answer Correct Answer: C) C1 e2x+ C2e-2x. 48. The highest derivative in the equation. A) Order. B) Type. C) Linearity. D) Degree. Show Answer Correct Answer: A) Order. 49. What is the significance of the indicial roots in the Frobenius method? A) They determine the order of the differential equation. B) They help in finding the series solution. C) They indicate linear dependence. D) They are always zero. Show Answer Correct Answer: B) They help in finding the series solution. 50. A/An ..... to a differential equation is a function $y=f(x)$ $f(x, y)=0$ A) Antiderivative. B) Order. C) Separable. D) Solution. Show Answer Correct Answer: D) Solution. 51. After separating variables and integrating, what is added to both sides of the equation? A) An exponent. B) A variable of integration. C) A constant of integration. D) A differential. Show Answer Correct Answer: C) A constant of integration. 52. How do you model a simple CPS using finite state machines? A) Visualize a CPS using FSMs with a flowchart instead of a state transition diagram. B) Model a CPS using FSMs by only defining states without transitions. C) Model a CPS using FSMs by defining states, transitions, and actions, then visualize with a state transition diagram. D) Use FSMs to model a CPS by focusing solely on actions and ignoring states. Show Answer Correct Answer: C) Model a CPS using FSMs by defining states, transitions, and actions, then visualize with a state transition diagram. 53. Form the differential equation of the curve:$y=Ae^x+Be^{-x}$ A) $\frac{\text{d}^2y}{\text{d}x^2}=Axe^x-Bxe^{-x}$. B) $\frac{\text{d}^2y}{\text{d}x^2}=Ae^x-Be^{-x}$. C) $\frac{\text{d}^2y}{\text{d}x^2}=y$. D) $\frac{\text{d}^2y}{\text{d}x^2}+y=0$. Show Answer Correct Answer: C) $\frac{\text{d}^2y}{\text{d}x^2}=y$. 54. What role do differential equations play in system dynamics? A) Differential equations are primarily for solving algebraic equations. B) Differential equations model the behavior and dynamics of systems over time. C) Differential equations are used to calculate static values. D) Differential equations only apply to physical systems without time dependence. Show Answer Correct Answer: B) Differential equations model the behavior and dynamics of systems over time. 55. For each differential equation, indicate the order (as a number) and whether the equation is linear or nonlinear. $\frac{d^4y}{dt^4} = \sin(t+y)$ A) Linear. B) Nonlinear. C) All the above. D) None of the above. Show Answer Correct Answer: B) Nonlinear. 56. Which of the following is an example of a homogeneous differential equation? A) Y' = y-x. B) Y' = y-xy. C) Xy' + y = 1. D) Y' = sin(2x). Show Answer Correct Answer: B) Y' = y-xy. 57. Consider the differential equation dy/dx = x + 2y for which g(x) is the solution. Which of the following statements is true if the particular solution contains (0, -1) A) G(x) is increasing and concave up. B) G(x) is increasing and concave down. C) G(x) is decreasing and concave up. D) G(x) is decreasing and concave down. Show Answer Correct Answer: D) G(x) is decreasing and concave down. 58. $L\left[f\left(x\right)\right]=\int_0^{\infty}e^{-sx}f\left(x\right)dx$ A) $True$. B) $False$. C) All the above. D) None of the above. Show Answer Correct Answer: A) $True$. 59. An equation of first order but higher degree means the equation contains: A) First-order derivative only, but to a higher power. B) Higher-order derivatives. C) Only algebraic terms. D) None. Show Answer Correct Answer: A) First-order derivative only, but to a higher power. 60. What is the heat equation and its applications in solving partial differential equations? A) The heat equation is a differential equation used to model electromagnetic fields. B) The heat equation is a polynomial equation used to model population growth. C) The heat equation is a partial differential equation used to model heat transfer and diffusion processes. D) The heat equation is a linear equation used to model fluid flow. Show Answer Correct Answer: C) The heat equation is a partial differential equation used to model heat transfer and diffusion processes. ← 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 9 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books