This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 5 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 5 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. A partial differential equation requires A) Exactly one independent variable. B) Two or more independent variables. C) More than one dependent variable. D) Equal number of dependent and independent variables. Show Answer Correct Answer: B) Two or more independent variables. 2. If D2+1y=11+Sinx, then the value of the Wronskin W = A) -1. B) 0. C) 2. D) 1. Show Answer Correct Answer: D) 1. 3. Dy/dx = x-y and y(0) = 1. Use Euler's Method with a step size of 0.5 to approximate y(1.5). A) 0.75. B) 0.8. C) 0.5. D) 1.25. Show Answer Correct Answer: A) 0.75. 4. What is the main method used to solve partial differential equations? A) Boundary element method. B) Finite element method. C) Finite difference method. D) Method of characteristics. Show Answer Correct Answer: C) Finite difference method. 5. Find the Integrating factor of the differential equation $x\frac{\text{d}y}{\text{d}x}-y=\sin x$ A) $\ln\left|x\right|$. B) $\frac{1}{x}$. C) $\frac{1}{x^2}$. D) $-\frac{1}{x}$. Show Answer Correct Answer: B) $\frac{1}{x}$. 6. Part B. Find the particular solution under the given conditions:12. y" +2y'-8y=0, y(0)=1, y'(0)=0 A) $y=e^{2x}+e^{-4x}$. B) $y=2e^{2x}-4e^{-4x}$. C) Y = $\frac{2}{3}e^{2x}+\frac{1}{3}e^{-4x}$. D) $y=e^{2x}-e^{-4x}$. Show Answer Correct Answer: C) Y = $\frac{2}{3}e^{2x}+\frac{1}{3}e^{-4x}$. 7. Find the Particular Integral of (D$^{2}$ +4DD'-4D'$^{2}$ )z = cos(x+y) A) $-\cos\left(x+y\right)$. B) $\cos\left(x-y\right)$. C) $\frac{\cos\left(x+y\right)}{2}$. D) $\frac{\cos\left(x+y\right)}{-2}$. Show Answer Correct Answer: A) $-\cos\left(x+y\right)$. 8. Dy/dx = 2x/e$^{2y }$ find the general solution A) Y = mx+b. B) Y = e$^{2x}$+c. C) $y=\frac{1}{2}\ln\left|2x^2+c\right|$. D) $y=\ln\left|2x^2+c\right|$. Show Answer Correct Answer: C) $y=\frac{1}{2}\ln\left|2x^2+c\right|$. 9. Find the particular integral of (D$^{3}$-3DD'+D'+4)3=e$^{2x+y}$ A) E$^{2x}$. B) F(y+z, x$^{2}$+ +z$^{2}$)=0. C) 1/7. D) 1/7 e$^{2x+y}$. Show Answer Correct Answer: D) 1/7 e$^{2x+y}$. 10. For y" +y=0, y(0)=0, y'(0)=1, the solution is: A) $\sin x$. B) $\cos x$. C) $e^x$. D) None of these. Show Answer Correct Answer: A) $\sin x$. 11. Find the general solution of dy/dx = (x + y)/(x-y). A) X-y = k(x + y). B) X + y = k(x-y). C) Y = kx + c. D) Dy/dx = (x-y)/(x + y). Show Answer Correct Answer: B) X + y = k(x-y). 12. The differential equations are useful to describe anything that changes in time or in space.Is this statement TRUE or FALSE A) TRUE. B) FALSE. C) NOT SURE. D) None of the above. Show Answer Correct Answer: A) TRUE. 13. What is the solution of the differential equation $y'=-3x$ $P\left(3, -4\right)$ A) $y=2x+\frac{12}{5}$. B) $y=-2x+\frac{12}{5}$. C) $y=-\frac{3}{2}x^2-\frac{19}{2}$. D) $y=-\frac{3}{2}x^2+\frac{19}{2}$. Show Answer Correct Answer: D) $y=-\frac{3}{2}x^2+\frac{19}{2}$. 14. Differential equation of family of lines y=mx, where m is arbitrary constant is A) Y=xy'. B) Y"=0. C) Y$^{'}$=0. D) Y=x+y". Show Answer Correct Answer: A) Y=xy'. 15. Let $y=f(x)$ $\frac{dy}{dx}=x-y+1$ $f(0)=3$ $f(2)$ $x=0$ A) $-1$. B) $5$. C) $-2$. D) $2$. Show Answer Correct Answer: D) $2$. 16. The solution of Clairaut's equation usually consists of: A) A family of straight lines. B) A family of curves. C) Only one curve. D) None. Show Answer Correct Answer: A) A family of straight lines. 17. If $\dfrac{dy}{dx}=2x$ $y$ A) $x^2+C$. B) $\ln x$. C) $2x^2$. D) $x$. Show Answer Correct Answer: A) $x^2+C$. 18. What are the applications of partial differential equations in real life? A) Social media marketing. B) Building construction. C) Modeling physical phenomena in various fields. D) Cooking recipes. Show Answer Correct Answer: C) Modeling physical phenomena in various fields. 19. The solution to the given differential equation (y + xy2)dx + (2y-x)dy = 0 is $\frac{x^2}{2}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 20. $\frac{dy}{dx}+y\cot\left(x\right)=y^3\sin^3\left(x\right)$ A) Linear. B) Separable. C) Bernoulli. D) Homogeneous. Show Answer Correct Answer: C) Bernoulli. 21. $\left(\frac{d^2u}{dt^2}\right)^2+\left(\frac{du}{dt}\right)^3+u=t$ A) First order, second degree. B) Second order, third degree. C) Second order, second degree. D) First order, third degree. Show Answer Correct Answer: C) Second order, second degree. 22. In the differential equation $\frac{dy}{dx}=x^ky^2$ A) $y=\frac{k+1}{k+1-x^{k+1}}$. B) $y=\frac{k+1}{k+1+x^{k+1}}$. C) $y=\sqrt[3]{\frac{k+1-3x^{k+1}}{k+1}}$. D) $y=\sqrt[3]{\frac{k+1+3x^{k+1}}{k+1}}$. Show Answer Correct Answer: A) $y=\frac{k+1}{k+1-x^{k+1}}$. 23. You draw a card at random from a standard deck of 52 cards. Find the probability that the card is a diamond given that it is a queen. A) 0.077. B) 0.25. C) 0.333. D) 0.5. Show Answer Correct Answer: B) 0.25. 24. For $y" + 2y' + y = 0$ A) A repeated real root. B) Two complex conjugate roots. C) No real roots. D) Two distinct real roots. Show Answer Correct Answer: A) A repeated real root. 25. Clairaut's form of a differential equation is written as: A) Y=px+f(p). B) Y=f(x). C) Y=f(p)only. D) None. Show Answer Correct Answer: A) Y=px+f(p). 26. $5\frac{dy}{dx}=3\frac{dy}{dx}+3xy$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 27. $L\left(x^n\right)=\frac{\Gamma\left(n\right)}{s^{\left(n+1\right)}}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 28. Find the general solution of the non-homogeneous equation:y" -y = e$^{-x}$ A) Y = C1 e$^{-x}$ +C2 e$^{2x}$-(1/2) x$^{2}$ e$^{-x}$. B) Y = C1 e$^{x}$-C2 e$^{-x}$-x e$^{-x}$. C) Y = C1 e$^{x}$-C2 e$^{-x}$ +(1/2) e$^{-x}$. D) Y = C1 e$^{x}$ + C2 e$^{-x}$-(1/2)x e$^{-x}$. Show Answer Correct Answer: D) Y = C1 e$^{x}$ + C2 e$^{-x}$-(1/2)x e$^{-x}$. 29. Find the integrating factor to solve the equation $\frac{dy}{dx}-y=x^3.$ A) $-y$. B) $x^3$. C) $e^{-x}$. D) $e^y$. Show Answer Correct Answer: C) $e^{-x}$. 30. Solve dy/dx = (y+5)(x+2) A) $y=Ce^{x^2+2x}$. B) $y=Ce^{x^2+x}+5$. C) $y=Ce^{x+2}-5$. D) $y=Ce^{\frac{1}{2}x^2+2x}-5$. Show Answer Correct Answer: D) $y=Ce^{\frac{1}{2}x^2+2x}-5$. 31. The particular solution of $y" =e^x$ A) $e^{-x}$. B) $xe^x$. C) $x^2$. D) $e^x$. Show Answer Correct Answer: D) $e^x$. 32. The eigenvalues of the coefficient matrix determine: A) The shape of the graph. B) The stability and type of solution. C) The particular integral. D) Nothing. Show Answer Correct Answer: B) The stability and type of solution. 33. At each point on a certain curve, the slope of the curve is $3x^2y$ A) $y=8e^{x^3}$. B) $y=e^{x^3}+7$. C) $y=x^3+8$. D) $y=\ln\left(x+1\right)+8$. Show Answer Correct Answer: A) $y=8e^{x^3}$. 34. $\frac{du}{dx}=cux$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 35. General solution of the equation $\left(x^3+3xy^2\right)dx+\left(3x^2y+y^3\right)dy=0$ A) $\frac{1}{12}\left(x^3+6xy^2+y^4\right)=c$. B) $x^2+y^2=c$. C) $\frac{1}{5}\left(8x^4+6x^3y+y^3\right)=c$. D) $\frac{1}{4}\left(x^4+6x^2y^2+y^4\right)=c$. Show Answer Correct Answer: D) $\frac{1}{4}\left(x^4+6x^2y^2+y^4\right)=c$. 36. Suppose you deposit $ 1000 in an account that pays 1.5% annual interest and it increases at a rate proportional to the present amount. How much money will you have after 7 years if the interest is compounded continuously? A) $ 1110.71. B) $ 7, 000. C) $ 36, 315, 502.67. D) $ 10, 500. Show Answer Correct Answer: A) $ 1110.71. 37. If R(x) is a decreasing function, then a Left 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. 38. The solution of linear differential equation $\frac{\text{d}y}{\text{d}x}+y=e^x$ A) $y=\frac{e^2+c}{4x^2}$. B) $y=\frac{ex^2}{4}+c$. C) $y=\frac{e^x}{2}+Ce^{-x}$. D) $y=\frac{e^{4x}+c}{x^2}$. Show Answer Correct Answer: C) $y=\frac{e^x}{2}+Ce^{-x}$. 39. What is the criterion for linear dependence in a set of functions? A) All functions must be zero. B) At least one function can be expressed as a linear combination of others. C) All functions must be independent. D) The Wronskian must be non-zero. Show Answer Correct Answer: B) At least one function can be expressed as a linear combination of others. 40. How would you CORRECTLY separate the following differential equation? $\frac{dy}{dx}=e^{\left(x+y\right)}$ A) $\frac{dy}{e^{\left(x+y\right)}}=1dx$. B) $-\frac{dy}{e^y}=e^xdx$. C) $1dy=e^{\left(x+y\right)}dx$. D) $e^{-y}dy=e^xdx$. Show Answer Correct Answer: D) $e^{-y}dy=e^xdx$. 41. The complementary function represents the solution of the: A) Homogeneous part. B) Non-homogeneous part. C) Singular equation. D) None. Show Answer Correct Answer: A) Homogeneous part. 42. If the roots of the auxiliary equation in an ODE are 1, -1, i, -i, then complementary function is y= ..... A) $Ae^{ix}+Be^{-ix}+C\cos x+D\sin x$. B) $Ae^x+Be^{-x}+C\cos x+D\sin x$. C) $Ae^x+xBe^{-x}+C\cos x+D\sin x$. D) $Ae^x+Be^{-x}+C\cos x-D\sin x$. Show Answer Correct Answer: B) $Ae^x+Be^{-x}+C\cos x+D\sin x$. 43. When an equation is solvable for y, it often gives: A) Explicit solution for y. B) A relation involving x. C) Multiple roots. D) None. Show Answer Correct Answer: A) Explicit solution for y. 44. Solve the non-homogeneous equation:y" + 9y = cos(3x). A) Y = C1 cos(3x) + C2 sin(3x) +(1/6) xsin(3x). B) Y = C1 sin(3x) + C2 cos(3x)-(1/9) xsin(3x). C) Y = C1 cos(3x) + C2 sin(3x)-(1/6) xcos(3x). D) Y = C1 cos(3x) + C2 sin(3x) + (1/9) x$^{2}$sin(3x). Show Answer Correct Answer: A) Y = C1 cos(3x) + C2 sin(3x) +(1/6) xsin(3x). 45. Which of the following represents a particular integral? A) The general solution to a differential equation. B) The specific solution to the non-homogeneous part of a differential equation. C) The derivative of the non-homogeneous part of a differential equation. D) The solution to the homogeneous part of a differential equation. Show Answer Correct Answer: B) The specific solution to the non-homogeneous part of a differential equation. 46. The singular solution is obtained by eliminating: A) P from equations. B) Y from equations. C) Xxx only. D) None. Show Answer Correct Answer: A) P from equations. 47. Find the general solution of the differential equation y'-ysin(x)=0. A) $|y|=Ce^{-\sin\left(x\right)}$. B) $|y|=-Ce^{\cos\left(x\right)}$. C) $|y|=Ce^{\cos\left(x\right)}$. D) $|y|=Ce^{\sin\left(x\right)}$. E) $|y|=Ce^{-\cos\left(x\right)}$. Show Answer Correct Answer: E) $|y|=Ce^{-\cos\left(x\right)}$. 48. The singular solution of Clairaut's equation is always: A) An envelope curve. B) Parallel lines. C) Polynomial. D) None. Show Answer Correct Answer: A) An envelope curve. 49. Integrating factor of $\cos^2x\\frac{dy}{dx}+y=\tan x$ A) $e^{-\tan x}$. B) $e^{\tan x}$. C) $e^{\sec^2x}$. D) $e^{\tan^2x}$. Show Answer Correct Answer: B) $e^{\tan x}$. 50. Given the de y'=2x+1 find y in terms of x for y(0)=1 A) Y=x$^{2}$+x+1. B) X$^{2}$+x+c=y. C) Y=x$^{3}$+x+1. D) Y=2x+1. Show Answer Correct Answer: A) Y=x$^{2}$+x+1. 51. For repeated roots, applying initial conditions helps to determine: A) Auxiliary roots. B) Arbitrary constants. C) Order of DE. D) Wronskian. Show Answer Correct Answer: B) Arbitrary constants. 52. $\frac{dt}{dx}=2x$ A) YES. B) NO. C) All the above. D) None of the above. Show Answer Correct Answer: A) YES. 53. Find the exact solution to the following equation given that y = 6, dy/dx =-5 when x = 0. $\frac{\text{d}^2y}{\text{d}x^2}+4\frac{\text{d}x}{\text{d}y}+5y=x^2+2x+4$ A) $y=e^x(4.6-2.4x)+0.5x^2+3.14$. B) $y=e^x(4.6\cos\left(-2x\right)+2.22\sin\left(-2x\right))+0.34x^2+0.8x+3.5$. C) $y=e^{-2x}(2.72\cos x+0.36\sin x)+0.2x^2+0.08x+3.28$. D) $y=e^{-2x}(2.14+0.86x)+1.2x^2+0.56x$. Show Answer Correct Answer: C) $y=e^{-2x}(2.72\cos x+0.36\sin x)+0.2x^2+0.08x+3.28$. 54. Bagaimana solusi berubah jika akar dari persamaan karakteristik adalah kompleks? A) Solusi mencakup istilah osilasi yang melibatkan fungsi sinus dan kosinus. B) Solusi disederhanakan menjadi persamaan linier. C) Solusi menjadi nilai konstan. D) Solusi tidak memiliki akar real. Show Answer Correct Answer: A) Solusi mencakup istilah osilasi yang melibatkan fungsi sinus dan kosinus. 55. The method of variation of parameters is preferred when: A) The equation is of first order. B) The non-homogeneous term is complex. C) Initial conditions are not given. D) The complementary function is unknown. Show Answer Correct Answer: B) The non-homogeneous term is complex. 56. What type of differential equations can be solved using the method of separation of variables? A) First order linear with constant coefficients. B) Second order linear. C) Nonlinear. D) Partial differential equations. Show Answer Correct Answer: A) First order linear with constant coefficients. 57. Solve the differential equation. $\frac{dy}{dx}=y^2$ A) $y^3=x^2+C$. B) $y=x+C$. C) $-\frac{1}{y}=x+C$. D) $\frac{1}{y^2}=x+C$. Show Answer Correct Answer: C) $-\frac{1}{y}=x+C$. 58. The functions that are analytic at all points ..... A) $e^x$. B) $\sin x$. C) $\cos x$. D) All the above. Show Answer Correct Answer: D) All the above. 59. What is the solution to the PDE ( u ..... {t} = k u ..... {xx} ) called? A) Laplace equation. B) Wave equation. C) Poisson equation. D) Heat equation. Show Answer Correct Answer: D) Heat equation. 60. Determine the value of "c" that satisfies the differential equation $\frac{dy}{dx}=\frac{x+1}{y+2}$ A) 1. B) 5/2. C) -3/2. D) -1/2. Show Answer Correct Answer: C) -3/2. ← 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 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