This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 17 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 17 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Find the solution of y' + y = 0. A) $y = e^{x}$. B) $y = Ce^{x}$. C) Y = Cx. D) $y = Ce^{-x}$. Show Answer Correct Answer: D) $y = Ce^{-x}$. 2. What is the first step in forming a partial differential equation? A) Set the equation equal to zero. B) Identify the independent and dependent variables. C) Solve for the constants. D) Differentiate the equation. Show Answer Correct Answer: B) Identify the independent and dependent variables. 3. Find the area under the curve of ( f(x) = x^3 ) from ( x = 1 ) to ( x = 2 ). A) 5.0. B) 2.5. C) 4.0. D) 3.75. Show Answer Correct Answer: D) 3.75. 4. A virus has infected 1.8% of a population. A test detects this virus 95% of the time when it is actually present, but it returns a false positive 3% of the time when the virus is not present.If a person selected at random from this population tests positive for the virus, what is the probability that this person is actually infected? [Round to the nearest percent.] A) 37%. B) 34%. C) 66%. D) 63%. Show Answer Correct Answer: A) 37%. 5. Of the following, which is a solution to the differential equation? Select all that apply:$y" -y=0$ A) $y=\sin\left(x\right)$. B) $y=e^x$. C) All the above. D) None of the above. Show Answer Correct Answer: B) $y=e^x$. 6. A linear differential equation with constant coefficients is one where the coefficients are: A) Constants. B) Variable functions. C) Arbitrary functions. D) None. Show Answer Correct Answer: A) Constants. 7. Which type of differential equations are not suitable for the method of separation of variables? A) Nonlinear. B) First order linear. C) Partial differential equations. D) Second order linear with constant coefficients. Show Answer Correct Answer: A) Nonlinear. 8. Classify the following differential equation:$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=v\frac{\partial^2u}{\partial x^2}$ A) Second order, partial, linear. B) Second order, partial, nonlinear. C) Second order, ordinary, nonlinear. D) Second order, ordinary, linear. Show Answer Correct Answer: B) Second order, partial, nonlinear. 9. Find the complete integral of pq=1 . A) Z=ax+(1/b)y+c. B) Ii) z=ax+(1/a)y+c. C) I) z=z=ax-(1/a)y+c. D) Iii) z=ax+by+c. Show Answer Correct Answer: B) Ii) z=ax+(1/a)y+c. 10. If $\overrightarrow{f}=\left(x+3y\right)i+\left(y-2z\right)j+\left(x+az\right)k$ A) 2. B) -2. C) 1. D) -1. Show Answer Correct Answer: B) -2. 11. If $\frac{dy}{dx} = 4xy$ $y = 1$ $x = 2$ $y =$ A) $e^{2x^2-8}$. B) $4e^{2x^2-8}-3$. C) $2e^{2x^2-8}-1$. D) $8e^{2x^2-8}-7$. Show Answer Correct Answer: A) $e^{2x^2-8}$. 12. What is the result of eliminating arbitrary functions in solving PDEs? A) A solution with a specific boundary condition. B) A general solution with no arbitrary functions. C) A solution with a specific initial condition. D) A particular solution with no arbitrary functions. Show Answer Correct Answer: D) A particular solution with no arbitrary functions. 13. If R(x) is an increasing function, then a Right Hand Riemann sum will be an ..... A) Underapproximation. B) Overapproximation. C) All the above. D) None of the above. Show Answer Correct Answer: B) Overapproximation. 14. Consider the differential equation $\frac{dy}{dx}=y\sec^2x$ $y=f\left(x\right)$ $y\left(0\right)=5$ A) $5e^{\tan x}$. B) $\tan x+5$. C) $e^{\tan x}+4$. D) $\tan x+5e^x$. Show Answer Correct Answer: A) $5e^{\tan x}$. 15. Z-transform is linear A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 16. The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ A) $c_1e^{3x}+c_2e^{-x}$. B) $c_1e^{3x}+c_2e^{-3x}$. C) $c_1e^{-3x}+c_2e^{-x}$. D) $c_1e^{3x}+c_2e^{3x}$. Show Answer Correct Answer: B) $c_1e^{3x}+c_2e^{-3x}$. 17. $.$ $\left(D^3+D^2D'+DD'^2+D'^3\right)Z=0$ A) $1$. B) $0$. C) $e^x$. D) $none$. Show Answer Correct Answer: B) $0$. 18. Find the general solution of the differential equation dy/dx=3y/7x. A) $y=Cx^{\frac{1}{7}}$. B) $y^7=Cx^{\frac{1}{3}}$. C) $y^{\frac{1}{7}}=Cx^3$. D) $y^{\frac{1}{3}}=Cx^{\frac{1}{7}}$. E) $y^3=Cx^7$. Show Answer Correct Answer: D) $y^{\frac{1}{3}}=Cx^{\frac{1}{7}}$. 19. Determine the general solution of $y"-2y'+y=0$ A) $y=C_1e^x+C_2xe^{-x}$. B) $y=C_1e^x+C_2e^x$. C) $y=C_1e^x+C_2xe^x$. D) $y=e^x\left(C_1\cos x+C_2\sin x\right)$. Show Answer Correct Answer: C) $y=C_1e^x+C_2xe^x$. 20. The partial derivative of y(1+xy)dx +(2y-x)dy with respect to y is A) 1. B) -1. C) 2xy + 1. D) 2xy-1. Show Answer Correct Answer: C) 2xy + 1. 21. A vector $\overrightarrow{f}$ $curl\\overrightarrow{f}=0$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 22. Which of the following statements is true regarding Lagrange's equation? A) It can only be applied to linear first-order PDEs. B) It can be applied to any first-order PDE. C) It is only applicable for specific boundary conditions. D) It can only be applied to nonlinear first-order PDEs. Show Answer Correct Answer: A) It can only be applied to linear first-order PDEs. 23. (D$^{2}$+1)y=cosx has ..... A) Particular integral alone as general solution. B) Complementary function alone as general solution. C) Complementary function + particular integral as general solution. D) No solution. Show Answer Correct Answer: C) Complementary function + particular integral as general solution. 24. An equation is non-linear if the coefficients depend at most on the "independent" variables A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 25. Determine whether the functions $y_{1} = e^{4x}$ $y_{2} = e^{3x}$ A) Yes, they are linearly independent. B) Not defined. C) No, they are linearly dependent. D) They are orthogonal. Show Answer Correct Answer: A) Yes, they are linearly independent. 26. A differential equation mainly describes ..... A) A formula for a function. B) A numerical value. C) A relationship between a function and its rate(s) of change. D) A table of data. Show Answer Correct Answer: C) A relationship between a function and its rate(s) of change. 27. In order to solve the partial dierential equation p-x=q-y, the correctoption is ..... A) Dz=(k-x)dx+(k-y)dy. B) Dz=(k-x)dx-(k-y)dy. C) Dz=(k+x)dx-(k+y)dy. D) Dz=(k+x)dx+(k+y)dy. Show Answer Correct Answer: D) Dz=(k+x)dx+(k+y)dy. 28. Separate the variables for the equation $\frac{dy}{dx}=\frac{(x^2+x-1)}{y^2}$ A) $\frac{y^2}{(x^2+x-1)}dy=dx$. B) $y^2dy=(x^2+x-1)dx$. C) $\left(y^2-1\right)dy=(x^2-1)dx$. D) $(x^2+x-1)dx=\frac{1}{y^2}dy$. Show Answer Correct Answer: B) $y^2dy=(x^2+x-1)dx$. 29. Gemma is ready for her driving test. The probability of her passing the first time is 0.65. If she fails her first test, the probability of her passing on the second attempt is 0.85.Calculate the probability Gemma passes in exactly two attempts. A) .0525. B) .2975. C) .9475. D) 0.85. Show Answer Correct Answer: B) .2975. 30. The following Differential Equation is $\frac{\text{d}y}{\text{d}x}=x^2y+x$ A) Separable. B) Non Separable. C) All the above. D) None of the above. Show Answer Correct Answer: B) Non Separable. 31. Which of the following mathematical models CANNOT be solved by using separation of variables A) $L\\frac{dI}{dt}+RI=E\left(t\right)$. B) $\frac{\text{d}P}{\text{dt}}=kP$. C) $\frac{dT}{dt}=k\left(T-Tm\right)$. D) $\frac{dA}{dt}=kA$. Show Answer Correct Answer: A) $L\\frac{dI}{dt}+RI=E\left(t\right)$. 32. Let $y=g\left(x\right)$ $\frac{dy}{dx}=xy-2$ $g\left(2\right)=-1$ A) 3.00. B) 1.50. C) 1.25. D) -6.25. Show Answer Correct Answer: C) 1.25. 33. An equation containing derivatives of one or more dependent variables with one or more independent variables? A) Differential Equation. B) Ordinary Equation. C) Different Equation. D) Partial Equation. Show Answer Correct Answer: A) Differential Equation. 34. The general solution of the equation $y' + 4y = 8$ $y(t) = Ce^{-4t} + 2$ $y(0)=9$ A) $y(t) = 7e^{-4t} + 2$. B) $y(t) = 9e^{-4t} + 2$. C) $y(t) = 11e^{-4t} + 2$. D) None of the above. Show Answer Correct Answer: A) $y(t) = 7e^{-4t} + 2$. 35. Average value is ..... A) $y-y_1=m\left(x-x_1\right)$. B) $\frac{1}{b-a}\int_a^bf\left(x\right)dx$. C) $\int_a^bf\left(x\right)dx$. D) $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. Show Answer Correct Answer: B) $\frac{1}{b-a}\int_a^bf\left(x\right)dx$. 36. The equation $y=e^x-2$ A) Y'-y = 2. B) Y'-y =-2. C) Y' + y =-2. D) Y' + y = 2. Show Answer Correct Answer: A) Y'-y = 2. 37. What does the complementary function (CF) represent in the context of a second order ODE? A) The specific solution to the non-homogeneous equation. B) The general solution to the associated homogeneous equation. C) The derivative of the particular solution. D) The solution to the characteristic equation. Show Answer Correct Answer: B) The general solution to the associated homogeneous equation. 38. $.$ $\left(D^2-4DD'+4D'^2\right)Z=e^{\left(2x+y\right)}$ A) $\frac{x^2}{2}e^{\left(2x+y\right)}$. B) $\\frac{y^2}{2}e^{\left(2x+y\right)}$. C) $\frac{x^2}{2}e^{\left(2y+x\right)}$. D) $\frac{xy^{ }}{2}e^{\left(2x+y\right)}$. Show Answer Correct Answer: A) $\frac{x^2}{2}e^{\left(2x+y\right)}$. 39. If R(x) is an increasing concave up function, then a Trapezoidal Riemann sum will be an ..... A) Underapproximation. B) Overapproximation. C) All the above. D) None of the above. Show Answer Correct Answer: B) Overapproximation. 40. Number of arbitrary constant in particular solution of the differential equation x(y$^{" '}$)$^{4}$-y(y$^{" " }$)$^{3}$=xy is A) 4. B) 1. C) 3. D) 0. Show Answer Correct Answer: D) 0. 41. $.$ $\left(\frac{\partial^2u}{\partial x^2}\right)^3+\left(\frac{\partial u}{\partial y}\right)^4=0$ A) 2. B) 4. C) 3. D) 1. Show Answer Correct Answer: A) 2. 42. Determine the complementary solution for:y" + 2y' + y = 0. A) F= C1 + C2 x. B) F = (C1 + C2 x )e$^{-x}$. C) F= C1 e$^{x}$ + C2 x e$^{x}$. D) F= C1 e$^{-2x}$+ C2 e$^{-x}$. Show Answer Correct Answer: B) F = (C1 + C2 x )e$^{-x}$. 43. What is the degree in the given differential equation:(1-x) y" -4xy' + 5y = cosx A) 2. B) 4. C) 3. D) 1. Show Answer Correct Answer: D) 1. 44. (D$^{2}$+4D+4)y=0 has no complementary function. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 45. The complete solution of partial differential equation 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. 46. Define a variable separable equation. A) A linear equation that relates x and y without derivatives. B) A function that describes the rate of change of y with respect to x. C) An equation that cannot be separated into two distinct variables. D) A differential equation that can be written as f(y) dy = g(x) dx. Show Answer Correct Answer: D) A differential equation that can be written as f(y) dy = g(x) dx. 47. Solve the differential equation. $y'=\frac{-2x}{\sin y}$ A) $\cos y=x^3+C$. B) $\cos y=x+C$. C) $\cos y=-x^2+C$. D) $-\cos y=-x^2+C$. Show Answer Correct Answer: D) $-\cos y=-x^2+C$. 48. $L\left[f'\left(x\right)\right]=$ A) $L\left[f\left(x\right)\right]-sf\left(0\right)$. B) $sL\left[f\left(x\right)\right]-sf\left(0\right)-f'\left(0\right)$. C) $sL\left[f\left(x\right)\right]-f\left(0\right)$. D) $sL\left[f\left(x\right)\right]-f'\left(0\right)$. Show Answer Correct Answer: C) $sL\left[f\left(x\right)\right]-f\left(0\right)$. 49. Find the complementary function of $\left(D^3-3D^2+3D-1\right)y=x^2e^x$ A) $C.F=e^x\left(c_1x+c_2x^2+c_3x^3\right)$. B) $C.F=e^{-x}\left(c_1+c_2x+c_3x^2\right)$. C) $C.F=e^{-x}\left(c_1x+c_2x^2+c_3x^3\right)$. D) $C.F=e^x\left(c_1+c_2x+c_3x^2\right)$. Show Answer Correct Answer: D) $C.F=e^x\left(c_1+c_2x+c_3x^2\right)$. 50. Determine the order of the differential equation:$\frac{d^6y}{dx^6} + \left(\frac{dy}{dx}\right)^2 + y = 0$ A) 5. B) 2. C) 6. D) 4. Show Answer Correct Answer: C) 6. 51. What is the order of the differential equation:$\frac{d^4y}{dx^4} + \left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{dy}{dx}\right)^5 = 0$ A) 1. B) 2. C) 4. D) 3. Show Answer Correct Answer: C) 4. 52. If the roots of A.E. are real and distinct then C.F. is $y=c_1e^{m_1x}+c_2e^{m_2x}$ A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: B) True. 53. Determine the general solution of $y"+2y'-8y=0$ A) $y=C_1e^{-4x}+C_2e^{2x}$. B) $y=e^{2x}\left(C_1\cos4x+C_2\sin4x\right)$. C) $y=C_1e^{-4x}+C_2xe^{2x}$. D) $y=e^{4x}\left(C_1\cos2x+C_2\sin2x\right)$. Show Answer Correct Answer: A) $y=C_1e^{-4x}+C_2e^{2x}$. 54. Find the general solution for the differential equation $\frac{dy}{dx}=xy$ A) $y=\sqrt{\frac{1}{x^2+A}}$. B) $y=Ae^{x^2}$. C) $y=\alpha e^{\frac{x^2}{2}}$. D) $y=e^{\frac{x^2}{2}}+c$. Show Answer Correct Answer: C) $y=\alpha e^{\frac{x^2}{2}}$. 55. Which of the following is not a linear transformation? A) $T\left(x, y\right)=\left(x, -y\right)$. B) $T\left(x, y\right)=\left(x+2, y+4\right)$. C) $T\left(x, y\right)=\left(-x, y\right)$. D) $T\left(x, y\right)=\left(x, 0\right)$. Show Answer Correct Answer: B) $T\left(x, y\right)=\left(x+2, y+4\right)$. 56. Using Euler's method, approximate the value of y at x=2 given the initial condition y(0)=1 and the differential equation dy/dx = x+y A) Approximate value of z at x=2. B) Exact value of y at x=2. C) Approximate value of y at x=2. D) Approximate value of y at x=0. Show Answer Correct Answer: C) Approximate value of y at x=2. 57. What is a complete integral of a PDE? A) A complete integral of a PDE is a numerical approximation of the solution. B) A complete integral of a PDE is a graphical representation of solutions. C) A complete integral of a PDE is a specific solution with fixed parameters. D) A complete integral of a PDE is a solution that encompasses all possible solutions, often expressed with arbitrary functions. Show Answer Correct Answer: D) A complete integral of a PDE is a solution that encompasses all possible solutions, often expressed with arbitrary functions. 58. Let (V, +, .) be a vector space. Then, which one is true? A) There exist an element y inV, Such that x+y=x for all x in V. B) There exist an element y inV, Such that x.y=x for all x in V. C) There exist an element y inV, Such that x+y=y for all x in V. D) There exist an element y inV, Such that x.y=y for all x in V. Show Answer Correct Answer: A) There exist an element y inV, Such that x+y=x for all x in V. 59. Classify the PDE:u ..... xx + u ..... yy + u ..... zz = 0. A) Hyperbolic. B) Elliptic. C) Parabolic. D) Linear. Show Answer Correct Answer: B) Elliptic. 60. At x = 2, which statement is true about the value of differential equation $\frac{\text{d}y}{\text{d}x}=\frac{\left(2-x\right)}{y}$ A) The derivative is negative. B) The derivative is zero. C) The derivative is positive. D) The derivative is undefined. Show Answer Correct Answer: B) The derivative is zero. ← 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