This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 7 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 7 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. If $\nabla\phi$ $\nabla^2\phi=$ A) 1. B) -1. C) 2. D) 0. Show Answer Correct Answer: D) 0. 2. If the gradient of a curve is given as $\frac{1}{x^2y}$ $\left(2, 3\right)$ A) $y=\sqrt{\frac{10x-2}{x}}$. B) $y=\sqrt{9-\ln16-\frac{2}{x}}$. C) $y=\sqrt{-\frac{1}{x}+\frac{19}{2}}$. D) None of the above. Show Answer Correct Answer: A) $y=\sqrt{\frac{10x-2}{x}}$. 3. The solution of non-linear partial differential equation $q-p+x-y=0$ A) $z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$. B) $z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$. C) $z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$. D) $z=\left(x+a\right)^2+\left(y+a\right)^2+c$. Show Answer Correct Answer: C) $z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$. 4. In the context of differential equations, what does exponentiating both sides of an equation solve for? A) The variable y. B) The derivative dy/dx. C) The integral of x. D) The constant of integration. Show Answer Correct Answer: A) The variable y. 5. What are the multiples while solving $\left(3z-4y\right)p+\left(4x-2z\right)q=2y-3x$ A) $1, 1, 1$. B) $2, 3, 4$. C) $3, 4, 2$. D) $4, 2, 3$. Show Answer Correct Answer: B) $2, 3, 4$. 6. What is the role of initial conditions in solving partial differential equations? A) Initial conditions are only necessary for linear partial differential equations. B) Initial conditions have no role in solving partial differential equations. C) Initial conditions only apply to ordinary differential equations. D) Initial conditions specify the values of the unknown function and its derivatives at a given point in the domain. Show Answer Correct Answer: D) Initial conditions specify the values of the unknown function and its derivatives at a given point in the domain. 7. Modified Euler's method gives better accuracy than: A) Taylor method. B) Euler method. C) Runge-Kutta method. D) Milne's method. Show Answer Correct Answer: B) Euler method. 8. Lagrange's Auxiliary equation is A) $\frac{\text{d}x}{\text{P}}=\frac{\text{d}z}{\text{Q}}=\frac{\text{d}y}{\text{R}}$. B) $\frac{\text{d}x}{\text{P}}=\frac{\text{d}y}{\text{Q}}$. C) $\frac{\text{d}x}{\text{P}}=\frac{\text{d}y}{\text{Q}}=\frac{\text{d}z}{\text{R}}$. D) None. Show Answer Correct Answer: C) $\frac{\text{d}x}{\text{P}}=\frac{\text{d}y}{\text{Q}}=\frac{\text{d}z}{\text{R}}$. 9. The subsidiary equation for Lagrange's linear equation A) $\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{z}$. B) $\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$ $\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$. C) $\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$. D) None of the above. Show Answer Correct Answer: C) $\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$. 10. Green's theorem connects A) Line integral and double integral. B) Line integral and Surface integral. C) Double integral and Surface integral. D) Surface integral and volume integral. Show Answer Correct Answer: A) Line integral and double integral. 11. Which among the following statements is true? A) A subset of linearly dependent is linear dependent. B) A spanning set should be a basis for a vector space. C) The union of two subspaces is a subspace. D) A subset of a linearly independent is linearly independent. Show Answer Correct Answer: D) A subset of a linearly independent is linearly independent. 12. Which of the following is the solution to the differential equation $\frac{dy}{dx}=3\cos x$ $y\!\left(\frac{\pi}{2}\right)=-1$ A) $y=3\sin x+2$. B) $y=-3\sin x+2$. C) $y=-3\sin x-4$. D) $y=3\sin x-4$. Show Answer Correct Answer: D) $y=3\sin x-4$. 13. If $\frac{dy}{dx}=2y^2$ A) 2/3. B) 1/3. C) -1/3. D) -2/3. Show Answer Correct Answer: C) -1/3. 14. Solve $\left(D^3-DD'^2-D'^3\right)z=0$ A) $Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y-x\right)$. B) $Z=f_1\left(y+x\right)+f_2\left(y-x\right)+xf_3\left(y-x\right)$. C) $Z=f_1\left(y-2x\right)+xf_2\left(y-2x\right)+f_3\left(y+x\right)$. D) $Z=f_1\left(y+x\right)+f_2\left(y-x\right)+f_3\left(y-x\right)$. Show Answer Correct Answer: B) $Z=f_1\left(y+x\right)+f_2\left(y-x\right)+xf_3\left(y-x\right)$. 15. For the differential equation (D$^{2}$-4D+4)y=0, the auxiliary equation is: A) $m^2+4m+4=0$. B) $m^2-4m-4=0$. C) $m^2-2m+2=0$. D) $m^2-4m+4=0$. Show Answer Correct Answer: D) $m^2-4m+4=0$. 16. If a slope field for $\frac{dy}{dx} = x-y$ $(x, y)$ A) The value of $y$. B) The slope $x-y$. C) The value of $x$. D) The solution curve passing through that point. Show Answer Correct Answer: B) The slope $x-y$. 17. $L^{-1}\left(\frac{1}{s-a}\right)$ A) $e^{-x}$. B) 1. C) $e^{-ax}$. D) $e^{ax}$. Show Answer Correct Answer: D) $e^{ax}$. 18. A vector $\overrightarrow{f}$ A) $\nabla^2\overrightarrow{f}=\overrightarrow{0}$. B) $\nabla\overrightarrow{f}=0$. C) $\nabla^2\overrightarrow{f}=0$. D) $\nabla\overrightarrow{f}=\overrightarrow{0}$. Show Answer Correct Answer: C) $\nabla^2\overrightarrow{f}=0$. 19. Eigen values of the matrix [ 1 1 1 ; 1 1 1 ; 1 1 1 ] are A) 0, 0, 1. B) 1, 1, 1. C) 0, 0, 0. D) 0, 0, 3. Show Answer Correct Answer: D) 0, 0, 3. 20. The C.F. of the equation $\left(D^2-4\right)y=\sin^2x$ A) $c_1e^{-x}+c_2e^{2x}$. B) $c_1e^x+c_2e^{2x}$. C) $c_1e^x+c_2e^{-2x}$. D) $c_1e^{4x}+c_2e^{2x}$. Show Answer Correct Answer: B) $c_1e^x+c_2e^{2x}$. 21. What is meant by P.I. in solution of a linear differential equation? A) Solution of Auxiliary equation. B) Effect of the terms of equation that are not attached to any dependent variable or its derivatives. C) Proper Involvement. D) Power and Intensity. Show Answer Correct Answer: B) Effect of the terms of equation that are not attached to any dependent variable or its derivatives. 22. In a series LR circuit connected to a DC source (E), the governing differential equation is: A) R (di/dt) + Li = E. B) L (di/dt)-Ri = E. C) L (di/dt) + Ri = E. D) R (di/dt)-Li = E. Show Answer Correct Answer: C) L (di/dt) + Ri = E. 23. The highest derivative in the given pde is called A) Order. B) Degree. C) All the above. D) None of the above. Show Answer Correct Answer: A) Order. 24. What type of differential equations involve finding the solution at a specific point or along a specific curve? A) Initial and boundary value problems. B) Partial differential equations. C) First order linear. D) Nonlinear. Show Answer Correct Answer: A) Initial and boundary value problems. 25. On what interval(s) is the function $f\left(x\right)=x^3+6x^2$ A) $\left(-2, \infty\right)$. B) $\left(-\infty, -2\right)$. C) $\left(0, \infty\right)$. D) $\left(-\infty, -4\right)$. Show Answer Correct Answer: B) $\left(-\infty, -2\right)$. 26. L$^{-1 }$\{1/s\} A) E$^{at}$. B) 1. C) Cos bt. D) T$^{n }$e$^{at}$. E) T$^{n}$. Show Answer Correct Answer: B) 1. 27. The solution of y '" =0 is A) Y=C1 +C2x+ C3x2 ex. B) Y= C1 sinx+C2cosx+ C3. C) Y= C1 ex+C2e-x+ C3. D) Y= C1 +C2x+ C3x2. Show Answer Correct Answer: D) Y= C1 +C2x+ C3x2. 28. Separate the variables for the equation $\frac{dy}{dx}=3y+2$ A) $\frac{dy}{3y}=dx$. B) $\left(3y+2\right)dx=dy$. C) $\frac{dy}{(3y+2)}=dx$. D) $\frac{dy}{y+2}=3dx$. Show Answer Correct Answer: C) $\frac{dy}{(3y+2)}=dx$. 29. $L\left(\sinh ax\right)= ..... $ A) $\frac{s}{s^2+1}$. B) $\frac{a}{s^2-a^2}$. C) $\frac{a}{s^2+1}$. D) $\frac{s}{s^2-1}$. Show Answer Correct Answer: B) $\frac{a}{s^2-a^2}$. 30. The derivative of $e^{3x}$ A) Chain rule. B) Product rule. C) Power rule. D) Quotient rule. Show Answer Correct Answer: A) Chain rule. 31. If P(t) is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population. A) $\frac{dP}{dt}=100P^2$. B) $\frac{dP}{dt}=200P$. C) $\frac{dP}{dt}=100t^2$. D) $\frac{dP}{dt}=200t$. E) $\frac{dP}{dt}=200$. Show Answer Correct Answer: E) $\frac{dP}{dt}=200$. 32. Giving particular values for arbitrary constants in the complete integral is called A) Particular solution. B) General solution. C) Singular solution. D) None of the above. Show Answer Correct Answer: A) Particular solution. 33. Parametric equation of the line joining (0, 0, 0) and (2, 1, 1) can be taken as ..... A) X=t, y = t, z = t where $0\le t\le1$. B) X = 2, y = 1, z = 1. C) X = 2t, y=t, z = t where $0\le t\le1$. D) X = 2t, y=t, z = t where $0\le t\le2$. Show Answer Correct Answer: C) X = 2t, y=t, z = t where $0\le t\le1$. 34. What is the definition of differential equations? A) Equations with unknown functions and their derivatives. B) Equations with known functions and their integrals. C) Equations with constant values and their derivatives. D) Equations with variables and their limits. Show Answer Correct Answer: A) Equations with unknown functions and their derivatives. 35. Is the differential equation dy/dx = sin(x) linear or nonlinear? A) Cubic. B) Quadratic. C) Nonlinear. D) Linear. Show Answer Correct Answer: C) Nonlinear. 36. Find the complementary function of $\frac{\partial^3z}{\partial x^3}-3\frac{\partial^3z}{\partial x^2\partial y}+4\frac{\partial^3z}{\partial y^3}=0$ A) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y+2x\right)$. B) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. C) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+f_3\left(y-2x\right)$. D) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. Show Answer Correct Answer: B) $Z=f_1\left(y-x\right)+f_2\left(y+2x\right)+xf_3\left(y+2x\right)$. 37. The number of bacteria increases at a rate proportional to the present amount. After 5 hours, there were 80 bacteria and after 8 hours, there were 120 bacteria. How many bacteria were present initially? A) About 53. B) 1. C) About 40. D) About 16. Show Answer Correct Answer: C) About 40. 38. Solve the following differential equations:$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$ A) $\sin y=1+C$. B) $\sin y=x+C$. C) $-\sin y=\frac{x^2}{2}+C$. D) $\sin y=0+C$. Show Answer Correct Answer: B) $\sin y=x+C$. 39. How are partial differential equations different from ordinary differential equations? A) Partial differential equations are the same as ordinary differential equations. B) Partial differential equations involve only one variable. C) Partial differential equations have no variables. D) Partial differential equations involve functions of multiple variables. Show Answer Correct Answer: D) Partial differential equations involve functions of multiple variables. 40. Solve $\left(D^2-5D+6\right)y=0$ A) $y=c_1e^x+c_2e^{2x}$. B) $y=c_1e^{-3x}+c_2e^{-2x}$. C) $y=c_1e^{3x}+c_2e^{2x}$. D) $y=c_1e^{-3x}+c_2e^{2x}$. Show Answer Correct Answer: C) $y=c_1e^{3x}+c_2e^{2x}$. 41. What type of differential equations can be solved using the method of undetermined coefficients? A) Partial differential equations. B) First order linear. C) Nonlinear. D) Higher order linear with constant coefficients. Show Answer Correct Answer: B) First order linear. 42. F(s) = L\{(t$^{n}$e$^{at}$)\} (s) A) 1/(s-a). B) 1/s. C) S/(s$^{2}$+b$^{2}$). D) N!/(s-a)$^{(n+1)}$. E) N!/s$^{(n+1)}$. Show Answer Correct Answer: D) N!/(s-a)$^{(n+1)}$. 43. What do conjugate pairs in complex numbers involve? A) Numbers with the same real part and same imaginary part. B) Numbers with different real parts and different imaginary parts. C) Numbers with the same real part but opposite imaginary parts. D) Numbers with opposite real parts but same imaginary parts. Show Answer Correct Answer: C) Numbers with the same real part but opposite imaginary parts. 44. The population P(t) of a species satisfies the logistic differential equation dP/dt=P(2-P/5000), Where the initial population P(0)=3000 and t is time in years. What is lim(t$\rightarrow$infinity)P(t)? A) 2500. B) 5000. C) 10000. D) 4200. E) 3000. Show Answer Correct Answer: A) 2500. 45. Determine the general solution of $y"+5y'+6y=0$ A) $y=C_1e^{3x}+C_2e^{-2x}$. B) $y=C_1e^{-3x}+C_2xe^{-2x}$. C) $y=e^{-3x}\left(C_1\cos2x+C_2\sin2x\right)$. D) $y=C_1e^{-3x}+C_2e^{-2x}$. Show Answer Correct Answer: D) $y=C_1e^{-3x}+C_2e^{-2x}$. 46. Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by dy/dt = ky. There was initially 10, 000 ft$^{3}$ at t=0. After 4 hours there were 8000 ft$^{3}$ remaining.What is the value of k in the differential equation? A) -.169. B) -0.056. C) -0.050. D) -.200. Show Answer Correct Answer: B) -0.056. 47. The value of $\int_{ }\int_{ }dxdy$ A) 2. B) 0. C) 4. D) 3. Show Answer Correct Answer: C) 4. 48. What is the number of arbitrary constant in a particular solution of a differential equation of order 3 and degree2? A) 3$^{2}$. B) 0. C) 2. D) 3. Show Answer Correct Answer: B) 0. 49. Find the complementary function of $\left(D^3-7DD'^2-6D'^3\right)z=0$ A) $Z=f_1\left(y-x\right)+f_2\left(y\right)+f_3\left(y+3x\right)$. B) $Z=f_1\left(y+x\right)+f_2\left(y-2x\right)+xf_3\left(y+3x\right)$. C) $Z=f_1\left(y-x\right)+f_2\left(y-2x\right)+f_3\left(y+3x\right)$. D) $Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+f_3\left(y-3x\right)$. Show Answer Correct Answer: C) $Z=f_1\left(y-x\right)+f_2\left(y-2x\right)+f_3\left(y+3x\right)$. 50. The velocity function is $v(t) = 4t-5$ $s(t)$ $s(0) = 2$ A) $s(t) = 2t^2 + 5t + 2$. B) $s(t) = 2t^2-5t + 2$. C) $s(t) = 2t^2 + 5t$. D) $s(t) = 2t^2-5t$. Show Answer Correct Answer: B) $s(t) = 2t^2-5t + 2$. 51. What are the challenges in modeling Zeno behavior? A) Zeno behavior is easily modeled with simple equations. B) There are no significant challenges in understanding Zeno behavior. C) The challenges in modeling Zeno behavior include dealing with infinite divisibility, reconciling discrete and continuous motion, and addressing philosophical implications of paradoxes. D) Zeno behavior only applies to physical objects in motion. Show Answer Correct Answer: C) The challenges in modeling Zeno behavior include dealing with infinite divisibility, reconciling discrete and continuous motion, and addressing philosophical implications of paradoxes. 52. Given that the acceleration of an object satisfies the differential equation $e^t\frac{dv}{dt}=2\sqrt{v}$ $v=1.$ A) $v=\left(2-e^{-t}\right)^2$. B) $v=\left(3-2e^{-t}\right)^2$. C) $v=\left(2e^{-t}-1\right)^2$. D) None of the above. Show Answer Correct Answer: A) $v=\left(2-e^{-t}\right)^2$. 53. If eigenvalues of system matrix have negative real parts, the system is ..... A) Unstable. B) Saddle. C) Stable. D) Neutral. Show Answer Correct Answer: C) Stable. 54. The nature of PDE 4u$_{xx }$+3 u$_{xy }$+3 u$_{yy}$=0 A) Parabolic. B) Hyperbolic. C) Elliptic. D) Laplace. Show Answer Correct Answer: C) Elliptic. 55. $\frac{dy}{dt}+y=e^t$ A) Linear. B) Nonlinear. C) Not sure. D) None of the above. Show Answer Correct Answer: A) Linear. 56. In the steady state, 2-D heat equation reduces to A) 1-D heat equation. B) 1-D wave equation. C) Laplace equation. D) None of these. Show Answer Correct Answer: C) Laplace equation. 57. Determine the degree of the differential equation:$\left(\frac{d^3y}{dx^3}\right)^4 + \left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0$ A) 1. B) 3. C) 2. D) 4. Show Answer Correct Answer: D) 4. 58. Which of the following is a solution to the differential equation $\frac{dy}{dx} = x^2y$ A) $y = Cx^3$. B) $y = Ce^{x^3/3}$. C) $y = Cx^2$. D) $y = Ce^{x^2}$. Show Answer Correct Answer: B) $y = Ce^{x^3/3}$. 59. What are some common techniques used to solve partial differential equations? A) Euler's method, Newton's method, Runge-Kutta method. B) Separation of variables, method of characteristics, finite difference method, finite element method, and spectral methods. C) Taylor series method, Bisection method, Secant method. D) Gaussian elimination, LU decomposition, Cholesky decomposition. Show Answer Correct Answer: B) Separation of variables, method of characteristics, finite difference method, finite element method, and spectral methods. 60. Findthe general solution to the following DE:$\frac{dy}{dx}-\frac{2y}{x+1}=3x^4$ A) $y=\left(x+1\right)^2\left(\frac{3}{5}x^5\right)+c$. B) $y=\left(x+1\right)^3\left(3x^4+c\right)$. C) $y=\left(x+1\right)^2\left(\frac{3}{5}x^5+c\right)$. D) $y=\left(x+1\right)^2\left(3x^4+c\right)$. Show Answer Correct Answer: C) $y=\left(x+1\right)^2\left(\frac{3}{5}x^5+c\right)$. ← 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 8Class 12 Mathematics Chapter 9 Differential Equations Quiz 9 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books