This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 9 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 9 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The formula for the Wronskian of two solutions y1 and y2 of y" +Pxy'+Qxy =0 is A) Wx=C e-Qxdx. B) Wx=C eQxdx. C) Wx=C e-Pxdx. D) Wx=C ePxdx. Show Answer Correct Answer: D) Wx=C ePxdx. 2. Which of the following is a logistic differential equation? A) $\frac{dP}{dt}=3P(5-t)$. B) $\frac{dP}{dt}=3P(5-P)$. C) $\frac{dP}{dt}=3t(5-t)$. D) $\frac{dP}{dt}=3P$. Show Answer Correct Answer: B) $\frac{dP}{dt}=3P(5-P)$. 3. Which of the following is a solution to the differential equation $\frac{dy}{dx} = 3y$ A) $y = Ce^{x^3}$. B) $y = 3e^x$. C) $y = 3x$. D) $y = Ce^{3x}$. Show Answer Correct Answer: D) $y = Ce^{3x}$. 4. Differential equation of the circles x$^{2}$+y$^{2 }$=r$^{2 }$, where r >0 is arbitrary constant is A) X-yy"=0. B) X+y.y$^{'}$=0. C) X-y.y$^{'}$=0. D) X$^{2}$+xy'=0. Show Answer Correct Answer: B) X+y.y$^{'}$=0. 5. Let T be a linear transformation from V to W, then T is one to one if and only if A) $N\left(T\right)=\left\{0\right\}$. B) $N\left(T\right)\ne\left\{0\right\}$. C) $R\left(T\right)=\left\{0\right\}$. D) $R\left(T\right)=W$. Show Answer Correct Answer: A) $N\left(T\right)=\left\{0\right\}$. 6. Let V be a vector space, and let S be a subset of V . What does it mean when we say that S spans V? A) The elements of S are all distinct from each other. B) S has at least as many elements as the dimension V. C) S is a basis for V. D) Every vector in V can be expressed as a linear combination of vectors in S. Show Answer Correct Answer: D) Every vector in V can be expressed as a linear combination of vectors in S. 7. Partial differentiation is used when the function has: A) No variables. B) Only dependent variables. C) One variable. D) Two or more independent variables. Show Answer Correct Answer: D) Two or more independent variables. 8. In $\frac{\text{d}x}{\text{d}y}$ A) Y. B) D. C) Secret. D) X. Show Answer Correct Answer: D) X. 9. Which function is the general solution of the differential equation $y" '+y" +y'+y=0$ A) $y=A\sin x+Be^{-x}$. B) $y=A\sin2x-Be^{-2x}$. C) $y=A\sin x+Be^{-2x}$. D) $y=A\sin2x-Be^{-x}$. Show Answer Correct Answer: A) $y=A\sin x+Be^{-x}$. 10. $\int_{ }^{ }\frac{dx}{x}=$ A) $y=\frac{-1}{x^2}+C$. B) $y=\sin x+C$. C) $y=\ln\left|x\right|+C$. D) $y=e^x+C$. Show Answer Correct Answer: C) $y=\ln\left|x\right|+C$. 11. Which statement of differential equation is true? A) An equation that relates one or more functions and their derivatives. B) An equation that has multiple terms made up of numbers and variables. C) All the above. D) None of the above. Show Answer Correct Answer: A) An equation that relates one or more functions and their derivatives. 12. The standard form of a linear differential equation with constant coefficients is: A) $y" +p(x)y'+q(x)y=r(x)$. B) $y" +p(x)y'+qy=r(x)$. C) $y'+p(x)y=q(x)$. D) $y" +py'+qy=r(x)$. Show Answer Correct Answer: D) $y" +py'+qy=r(x)$. 13. An integrating factor of the differential equation $\frac{\text{d}y}{\text{d}x}-\frac{2y}{x}=3x^2$ A) $e^{-2x}$. B) $x^2$. C) $\frac{1}{x^2}$. D) $-\ln\left(x^2\right)$. Show Answer Correct Answer: C) $\frac{1}{x^2}$. 14. What is the general solution of a second order differential equation with independent solutions? A) A linear combination of the solutions. B) A polynomial of degree two. C) An exponential function. D) A constant function. Show Answer Correct Answer: A) A linear combination of the solutions. 15. What is the general solution for $\frac{\text{d}^2y}{\text{d}x^2}-2y=0$ A) $y=Ae^{\sqrt{2}x}+Be^{\sqrt{2}x}$. B) $y=\left(Ax+B\right)e^{-\sqrt{2}x}$. C) $y=Ae^{\sqrt{2}x}+Be^{-\sqrt{2}x}$. D) $y=\left(Ax+B\right)e^{\sqrt{2}x}$. Show Answer Correct Answer: C) $y=Ae^{\sqrt{2}x}+Be^{-\sqrt{2}x}$. 16. Eliminate the arbitrary function from z = f(x$^{2}$+y$^{2}$) A) Py+qx=0. B) Py = qx. C) Px = qy. D) Px+qy=0. Show Answer Correct Answer: B) Py = qx. 17. According to Newton's law of cooling, the rate at which an object cools (or warms) is directly proportional to the temperature difference between the environment and the object itself. If a pot of boiling water (100$^\circ$C) is left at room temperature (22$^\circ$C) and after five minutes the water is only 70$^\circ$C, find its temperature after another 5 minutes. A) 63$^\circ$C. B) 62$^\circ$C. C) 51$^\circ$C. D) 61$^\circ$C. E) 52$^\circ$C. Show Answer Correct Answer: E) 52$^\circ$C. 18. If $\frac{dy}{dx} = ky$ A) $300e^{3k}$. B) 100 $e^{k}$. C) $100e^{3k}$. D) $300e^{k}$. Show Answer Correct Answer: C) $100e^{3k}$. 19. Given the differential equation $\frac{\text{d}P}{\text{d}t}=5P$ A) P=Ce$^{5t}$B) P= 5e$^{418t}$. B) P=Ce$^{t}$B) P= 418e$^{t}$. C) P=Ce$^{10t}$B) P= 418e$^{10t}$. D) P=Ce$^{5t}$B) P= 418e$^{5t}$. Show Answer Correct Answer: D) P=Ce$^{5t}$B) P= 418e$^{5t}$. 20. Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^3}{y}$ $y(2)=-3$ A) $y=-3e^{\frac{x^4}{4}-4}$. B) $y=-\sqrt{\frac{x^4}{2}+1}$. C) $y=3e^{\frac{x^4}{4}-4}$. D) $y=\sqrt{\frac{x^4}{2}+1}$. Show Answer Correct Answer: B) $y=-\sqrt{\frac{x^4}{2}+1}$. 21. The general solution of the DE $\frac{d^2y}{dx^{2}}+7\frac{dy}{dx}-8y=0$ A) $y=Ae^{-x}+Be^{8x}$. B) $y=Ae^x+Be^{7x}$. C) $y=Ae^x+Be^{-8x}$. D) $y=Ae^{-x}+Be^{7x}$. Show Answer Correct Answer: C) $y=Ae^x+Be^{-8x}$. 22. $\frac{\text{d}x}{\text{d}y}-4y-5+e^{-2x}=0$ A) $y=-\frac{5}{4}e^{-4x}+\frac{1}{6}e^{-2x}+Ce^{4x}$. B) $y=-20+6e^{-2x}+Ce^{4x}$. C) $y=-\frac{5}{4}+\frac{1}{6}e^{-2x}+Ce^{-4x}$. D) $y=-20e^{-8x}+e^{-10x}+Ce^{-4x}$. Show Answer Correct Answer: A) $y=-\frac{5}{4}e^{-4x}+\frac{1}{6}e^{-2x}+Ce^{4x}$. 23. Hence, roots $r_1$ $r_2$ A) Real and distinct. B) Real and equal. C) Complex numbers. D) None of the above. Show Answer Correct Answer: C) Complex numbers. 24. Which of the following differential equations is linear? (i) $y' = 10y + 1$ $y'-4y = 0$ $y' + \sin y = 1$ A) (i). B) (i) and (iii). C) (i) and (ii). D) None of the above. Show Answer Correct Answer: C) (i) and (ii). 25. What type of differential equation is dy/dx = y + x? A) First-order linear differential equation. B) Second-order linear differential equation. C) Nonlinear differential equation. D) Partial differential equation. Show Answer Correct Answer: A) First-order linear differential equation. 26. Which of the following differential equations is separable? (i) $y' = \dfrac{12t}{y}$ $t^2 y'-y^2 \sin t = 0$ $y' + ty = 9$ A) (i). B) (i) and (iii). C) (i) and (ii). D) None of the above. Show Answer Correct Answer: C) (i) and (ii). 27. Given the differential equation $\frac{dy}{dx} = 2x + 1$ $y(0) = 3$ A) $y = x^2 + x + 3$. B) $y = x^2 + 3$. C) $y = x^2 + x$. D) $y = 2x + 3$. Show Answer Correct Answer: A) $y = x^2 + x + 3$. 28. What is the characteristic equation for $y" -5y' + 6y = 0$ A) $r^2 + 5r + 6 = 0$. B) $r^2 + 6r + 5 = 0$. C) $r^2-6r + 5 = 0$. D) $r^2-5r + 6 = 0$. Show Answer Correct Answer: D) $r^2-5r + 6 = 0$. 29. What is the method to solve a linear differential equation? A) Employ numerical methods and approximations. B) Use integrating factors and integration. C) Apply substitution and differentiation. D) Use Laplace transforms and series. Show Answer Correct Answer: B) Use integrating factors and integration. 30. If the auxiliary equation has equal roots, the complementary function involves: A) Polynomial factor multiplied by exponential. B) Trigonometric terms. C) Only constants. D) None. Show Answer Correct Answer: A) Polynomial factor multiplied by exponential. 31. CHOOSE THE CORRECT FORM OF PARTIAL DIFFERENTIAL EQUATION BY ELIMINATING ARBITARY FUNCTIONz=f(sinx+cosy) A) -Sinyp+cosxq=0. B) -cosx+siny=0. C) Cosxp-sinyq=0. D) Tanxpq=0. Show Answer Correct Answer: A) -Sinyp+cosxq=0. 32. The complementary function of (D$^{2}$-6D+9)y=0 is ..... A) E$^{3x}$(ax-b). B) E$^{3x}$(ax+b). C) Ax+b. D) E$^{-3x}$(ax+b). Show Answer Correct Answer: B) E$^{3x}$(ax+b). 33. What is the degree of the differential equation:$\left(\frac{d^3y}{dx^3}\right)^2 + 5y = 0$ A) 4. B) 2. C) 3. D) 1. Show Answer Correct Answer: B) 2. 34. The general solution of $y'(t)-6y(t) = 12$ A) $y = Ce^{-6t} + 12$. B) $y = e^{6t} + C$. C) $y = Ce^{6t}-2$. D) None of the above. Show Answer Correct Answer: C) $y = Ce^{6t}-2$. 35. In IVPs, uniqueness of solutions is guaranteed if: A) Equation is linear. B) Coefficients are continuous. C) Both A & B. D) None. Show Answer Correct Answer: C) Both A & B. 36. Analyze the stability of the equilibrium solution $y = 0$ $\frac{dy}{dt} = y^2-4$ A) Unstable. B) Semi-stable. C) Stable. D) Cannot be determined. Show Answer Correct Answer: A) Unstable. 37. Which subject is the best? A) Math. B) Math. C) C'mon, u don't like it. D) Math. Show Answer Correct Answer: C) C'mon, u don't like it. 38. A population is modeled by a function P that satisfies the logistic differential equation $\frac{dp}{dt}=2P\left(1-\frac{P}{90}\right), $ A) 45. B) 55. C) 30. D) 90. Show Answer Correct Answer: A) 45. 39. The set $S=\left\{1, x+x^2, 4x^2\right\}$ A) Linearly independent. B) Linearly dependent. C) All the above. D) None of the above. Show Answer Correct Answer: A) Linearly independent. 40. Part B. Find the particular solution under the given conditions:11. y" -3y' + 2y = 0, y(0) = 4, y'(0) = 2 A) Y = $2e^{x} + 4e^{2x}$. B) $y = 4e^{x} + 4e^{2x}$. C) $y = 2e^{x} + 2e^{2x}$. D) $y = 4e^{x} + 2e^{2x}$. Show Answer Correct Answer: C) $y = 2e^{x} + 2e^{2x}$. 41. The governing differential equation for charge qin a series LCR circuit is: A) Ldqdt+Rq+1Cq=E(t). B) D2qdt2+Rq=E(t). C) Ld2qdt2+Rq+1Cq=E(t). D) Ld2qdt2+Rdqdt+1Cq=E(t). Show Answer Correct Answer: C) Ld2qdt2+Rq+1Cq=E(t). 42. What is the order and degree of the equation $(\frac{\text{d}y}{\text{d}x})^2+y=0$ A) 1, 2. B) 3, 2. C) 4, 1. D) 2, 1. Show Answer Correct Answer: A) 1, 2. 43. Dim M$_{4x3}$(R) is A) 64. B) 81. C) 12. D) 43. Show Answer Correct Answer: C) 12. 44. What is the primary use of differential equations? A) To solve algebraic equations. B) To find roots of polynomials. C) To calculate limits. D) To model real-world phenomena. Show Answer Correct Answer: D) To model real-world phenomena. 45. Classify the following differential equation:$\frac{\text{d}^3y}{\text{d}x^3}+y\frac{\text{d}^2y}{\text{d}x^2}=0$ A) Third-order, nonlinear, ordinar. B) Second-order, ordinary, nonlinear. C) Third order, ordinary, linear. D) Third-order, partial, linear. Show Answer Correct Answer: A) Third-order, nonlinear, ordinar. 46. Find the solution for $\cos x\\frac{\text{d}y}{\text{d}x}+y\sin x=e^x\cos^2x$ $x=0$ $y=2$ A) $y=\left(e^x+A\right)\cos x$. B) $y=\cos x\left(1+e^x\right)$. C) $y=-e^x\left(2\tan x+1\right)$. D) None of the above. Show Answer Correct Answer: B) $y=\cos x\left(1+e^x\right)$. 47. What type of differential equations can be solved using the method of variation of parameters? A) Nonlinear. B) First order linear. C) Higher order linear with constant coefficients. D) Partial differential equations. Show Answer Correct Answer: A) Nonlinear. 48. Identify the type of differential equation:y' + 2xy = 0 A) Quadratic. B) Linear. C) Trigonometric. D) Exponential. Show Answer Correct Answer: B) Linear. 49. If Dim V=4, and if $S=\left\{v_1, v_2, v_3\right\}\subseteq V$ A) $L\left(S\right)=V$. B) $L\left(S\right)\subseteq V$. C) $\dim(L(S))=3$. D) None of the others. Show Answer Correct Answer: B) $L\left(S\right)\subseteq V$. 50. If the initial conditions are y(0) = 1, the solution of y ''+y=0 is A) Y = 1 + x. B) Y = cos x. C) Y = ex. D) Y = sin x. Show Answer Correct Answer: B) Y = cos x. 51. Which of the following equations is partial differential equation? A) $\frac{dy}{dt}+y=e^t$. B) $u_{xx}+u_t=0$. C) $\frac{d^2y}{ds^2}+\frac{dy}{ds}+y=2s$. D) $P_{tt}+P_t=t^2$. Show Answer Correct Answer: B) $u_{xx}+u_t=0$. 52. The value of a car is $ 15, 000 and depreciates at a rate of 8% per year. What is the decay factor? A) $^{.08}$. B) .92. C) 8. D) $^{1.08}$. Show Answer Correct Answer: B) .92. 53. What are the characteristic roots of Cauchy's and Euler's equations? A) Real and distinct. B) Complex conjugate. C) Repeated real. D) Imaginary. Show Answer Correct Answer: C) Repeated real. 54. What notation is used to represent the derivative in the separation of variables? A) Euler's notation. B) Newton's notation. C) Leibniz notation. D) Lagrange's notation. Show Answer Correct Answer: C) Leibniz notation. 55. Why is the formation of partial differential equations important in the field of mathematics? A) To make simple systems more complicated. B) To confuse students studying mathematics. C) To have a better understanding of basic arithmetic. D) To describe the behavior of complex systems in physics, engineering, and other scientific fields. Show Answer Correct Answer: D) To describe the behavior of complex systems in physics, engineering, and other scientific fields. 56. Determine the general solution of $4y"+12y'+9y=0$ A) $y=e^{-\frac{3}{2}x}\left(C_1\cos\frac{3}{2}x+C_2\sin\frac{3}{2}x\right)$. B) $y=C_1e^{-\frac{3}{2}x}+C_2xe^{-\frac{3}{2}x}$. C) $y=C_1e^{-\frac{3}{2}}+C_2xe^{-\frac{3}{2}}$. D) $C_1e^{-\frac{3}{2}x}+C_2e^{-\frac{3}{2}x}$. Show Answer Correct Answer: B) $y=C_1e^{-\frac{3}{2}x}+C_2xe^{-\frac{3}{2}x}$. 57. A slope field is a pictorial representation of all of the possible solutions to a given differential equation. A) True. B) False. C) I don't know. D) None of the above. Show Answer Correct Answer: A) True. 58. Pick out the even function. A) $\sin x$. B) $x^3$. C) $\tan x$. D) $x^2$. Show Answer Correct Answer: D) $x^2$. 59. It is a type of differential equation wherein every term has the same degree. A) Non-exact D.E. B) Variable Separable. C) Homogenous D.E. D) Exact D.E. Show Answer Correct Answer: C) Homogenous D.E. 60. The main idea behind equations of first order but higher degree is to reduce them to: A) Linear equations. B) Standard solvable forms. C) Exact equations. D) None. Show Answer Correct Answer: B) Standard solvable forms. ← 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