This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 10 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 10 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. The vector $\overrightarrow{f}$ $x^2\overrightarrow{i}+y^2\overrightarrow{j}+z^2\overrightarrow{k}$ A) Harmonic. B) Solenoidal. C) Neither solenoidal or irrotational. D) Irrotational. Show Answer Correct Answer: D) Irrotational. 2. The integrating factor is required when: A) Equation is separable. B) Equation is linear. C) Equation is not exact. D) Equation is already exact. Show Answer Correct Answer: C) Equation is not exact. 3. If $h'\left(x\right)=k\left(x\right)$ $\int_{-1}^1k\left(5x\right)dx=$ A) $\frac{1}{5}h\left(5\right)-\frac{1}{5}h\left(-5\right)$. B) $5h\left(5\right)+5h\left(-5\right)$. C) $\frac{1}{5}h\left(5\right)+\frac{1}{5}h\left(-5\right)$. D) $5h\left(5\right)-5h\left(-5\right)$. Show Answer Correct Answer: A) $\frac{1}{5}h\left(5\right)-\frac{1}{5}h\left(-5\right)$. 4. Find y' for y= sin(2x$^{2}$) A) 2xcos(2x$^{2}$). B) 4xcos(2x$^{2}$). C) Xcos(4x$^{2}$). D) 4sin(2x$^{2}$). Show Answer Correct Answer: B) 4xcos(2x$^{2}$). 5. RK4 method is used to solve: A) Matrices. B) Algebraic equations. C) Ordinary differential equations. D) Integral equations. Show Answer Correct Answer: C) Ordinary differential equations. 6. Determine the general solution of $y"-2y'+5y=0$ A) $y=C_1e^{2x}\cos x+C_2e^{2x}\sin x$. B) $y=C_1e^x+C_2xe^{2x}$. C) $y=C_1e^x\cos2x+C_2e^x\sin2x$. D) $y=C_1e^x+C_2e^{2x}$. Show Answer Correct Answer: C) $y=C_1e^x\cos2x+C_2e^x\sin2x$. 7. $\int_{ }^{ }\sec^2xdx=?$ A) $\sec x+c$. B) $\frac{\sec^3x}{3}+c$. C) $\sec x\tan x+c$. D) $\tan x+c$. Show Answer Correct Answer: D) $\tan x+c$. 8. Pick out one of the conditions of Dirichlet's condition. A) $f\left(x\right)$. B) $f\left(x\right)$. C) $f\left(x\right)$. D) $f\left(x\right)$. Show Answer Correct Answer: A) $f\left(x\right)$. 9. Evaluate the definite integral ( int ..... 1^4 (4x-1) , dx ). A) 20. B) 27. C) 32. D) 15. Show Answer Correct Answer: B) 27. 10. Find the order and degree of the differential equation + A) 4, 2. B) 4, 3. C) 4, 4. D) 4, 1. Show Answer Correct Answer: D) 4, 1. 11. If dy/dt=-2y and if y=1 when t=0, what is the value of t for what y=1/2? A) Ln2. B) Ln(2)/2. C) Sqrt(2)/2. D) -1/4. E) -ln(2)/2. Show Answer Correct Answer: B) Ln(2)/2. 12. L\{(1/48b$^{6}$)(3bt-(bt)$^{3}$(sin bt)-3(bt)$^{2}$cosbt)\} A) S/(s$^{2}$+b$^{2}$)$^{2}$. B) S/(s$^{2}$+b$^{2}$). C) S/(s$^{2}$+b$^{2}$)$^{3}$. D) S/(s$^{2}$+b$^{2}$)$^{4}$. Show Answer Correct Answer: D) S/(s$^{2}$+b$^{2}$)$^{4}$. 13. Classify the PDE:u ..... xx + 2u ..... xy + u ..... yy = 0. A) Elliptic. B) Parabolic. C) Hyperbolic. D) Linear. Show Answer Correct Answer: B) Parabolic. 14. Write an equation for the description:The rate of change of y with respect to x is proportional to the difference of x and z. A) $\frac{\text{d}y}{\text{d}x}=k\left(\frac{x}{z}\right)$. B) $\frac{\text{d}y}{\text{d}x}=k\left(x-z\right)$. C) $\frac{\text{d}y}{\text{d}x}=\left(x-z\right)$. D) $\frac{\text{d}y}{\text{d}x}=\frac{k}{x-z}$. Show Answer Correct Answer: B) $\frac{\text{d}y}{\text{d}x}=k\left(x-z\right)$. 15. $\int_{ }^{ }f'\left(\frac{1}{2}x\right)dx$ A) $2f\left(\frac{1}{2}x\right)$. B) $\frac{1}{2}f\left(\frac{1}{2}x\right)$. C) $\frac{1}{2}f" \left(\frac{1}{2}x\right)$. D) $2f" \left(\frac{1}{2}x\right)$. Show Answer Correct Answer: A) $2f\left(\frac{1}{2}x\right)$. 16. Which of the following differential equation CAN NOT be separated? A) $\frac{dy}{dx}-e^xy=e^x$. B) $\frac{dy}{dx}+\frac{2}{x}y=\frac{1}{x+1}$. C) $\frac{dy}{dx}+1=4x$. D) $y^2\frac{dy}{dx}=\sqrt{xy}-2x^2\sqrt{y}$. Show Answer Correct Answer: B) $\frac{dy}{dx}+\frac{2}{x}y=\frac{1}{x+1}$. 17. $\int\frac{1}{\csc x}dx$ A) $\cot^2x+c$. B) $\sec x+c$. C) $\sin x+c$. D) $-\cos x+c$. Show Answer Correct Answer: D) $-\cos x+c$. 18. The formation of differential equation can be done by eliminating arbitrary constants in a given expression. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 19. Which of the following differential equations is separable? A) $\frac{dy}{dx} = \frac{y}{x} + x$. B) $\frac{dy}{dx} + y = x$. C) $\frac{dy}{dx} = xy$. D) $\frac{dy}{dx} = y + \sin(x)$. Show Answer Correct Answer: C) $\frac{dy}{dx} = xy$. 20. The rate of change of the volume, V, of oil in a tank with respect to time, t, is directly proportional to the cube root of the volume. Which of the following is a differential equation that describes this relationship? A) $\frac{dV}{dt}=\frac{k}{\sqrt[3]{V}}$. B) $V(t)=k\sqrt[3]{t}$. C) $\frac{dV}{dt}=k\sqrt[3]{V}$. D) $V(t)=k\sqrt[3]{V}$. Show Answer Correct Answer: C) $\frac{dV}{dt}=k\sqrt[3]{V}$. 21. Form a PDE by eliminating arbitrary constants from z= ax + by A) Z = px-qy. B) Z = qx-py. C) Z = qx + py. D) Z = px + qy. Show Answer Correct Answer: D) Z = px + qy. 22. What is the order of the D.E. (d$^{2}$y/dx$^{2}$)$^{2}$+ y = 0 A) 3. B) 1. C) 0. D) 2. Show Answer Correct Answer: D) 2. 23. What is the integrating factor of the differential equation $\left(3x^2y^4+2xy\right)dx+\left(2x^3y^3-x^2\right)dy=0$ A) $\frac{1}{xy}$. B) $\frac{1}{y^2}$. C) $-\frac{1}{x^2y}$. D) $\frac{1}{x^2}$. Show Answer Correct Answer: B) $\frac{1}{y^2}$. 24. If $y" + 6y' + 9y = 0$ A) $y(x) = Ce^{-3x} + De^{-6x}$. B) $y(x) = Ce^{-6x} + De^{-9x}$. C) $y(x) = Ce^{-3x} + Dxe^{-3x}$. D) $y(x) = Ce^{3x} + De^{-3x}$. Show Answer Correct Answer: C) $y(x) = Ce^{-3x} + Dxe^{-3x}$. 25. Find f '(x) if f(x) = x$^{4}$sinx A) 4x$^{3}$cosx. B) X$^{4}$cosx + 4x$^{3}$sinx. C) -x$^{4}$cosx + 4x$^{3}$sinx. D) X$^{4}$cosx. Show Answer Correct Answer: B) X$^{4}$cosx + 4x$^{3}$sinx. 26. Find the particular solution to the differential equation $\frac{dy}{dx}=3xy^2$ $\left(-3, -\frac{1}{14}\right)$ A) $y=\frac{-1}{x^2+1}$. B) $y=\left(3e^x+3\right)^{\frac{1}{3}}$. C) $y=\frac{-2}{x^2+2}$. D) $y=\frac{-2}{3x^2+1}$. Show Answer Correct Answer: D) $y=\frac{-2}{3x^2+1}$. 27. What are the applications of CPS in real-world scenarios? A) CPS applications are limited to mobile gaming. B) CPS is used for traditional farming methods. C) CPS is primarily focused on social media analytics. D) Applications of CPS include smart transportation, healthcare monitoring, industrial automation, and smart grid management. Show Answer Correct Answer: D) Applications of CPS include smart transportation, healthcare monitoring, industrial automation, and smart grid management. 28. Solve the first-order differential equation:$\frac{dy}{dx} = 5x^4$ A) $y = x^5-C$. B) $y = \frac{5}{4}x^5 + C$. C) $y = x^5 + C$. D) $y = x^5 + 5C$. Show Answer Correct Answer: C) $y = x^5 + C$. 29. If the Wronskian is identically zero, then: A) Functions are linearly independent. B) Functions are linearly dependent. C) Equation has no solution. D) None. Show Answer Correct Answer: B) Functions are linearly dependent. 30. An initial value problem requires: A) General solution only. B) Differential equation + conditions on y, y' at some point. C) Only auxiliary equation. D) None of the. Show Answer Correct Answer: B) Differential equation + conditions on y, y' at some point. 31. If the Eigen values of A are 3, 4, 5 then the Eigen values of A$^{2 }$are A) 9, 16, 25. B) 16, 25, 9. C) 1/3, 1/4, 1/5. D) 25, 16, 9. Show Answer Correct Answer: A) 9, 16, 25. 32. Which animal does not hibernate during the winter? A) White-tailed Prarie Dog. B) Polar Bear. C) Eastern Chipmunk. D) Red Squirrel. Show Answer Correct Answer: D) Red Squirrel. 33. Solve:$3\frac{\text{d}y}{\text{d}x}-\\frac{y}{x+2}=x+2$ A) $y=\frac{3}{5}\left(x+2\right)^2+c$. B) $y=\frac{5}{3}\left(x+2\right)^2+c$. C) $y=x(x+2)+c$. D) $y=\frac{2}{3}+c$. Show Answer Correct Answer: A) $y=\frac{3}{5}\left(x+2\right)^2+c$. 34. If the auxiliary equation has repeated roots, the general solution includes terms like: A) Aemx+Be-mx. B) Aemx+Bcos(mx). C) Aemx+Bxemx. D) Acos(mx)+Bsin(mx). Show Answer Correct Answer: C) Aemx+Bxemx. 35. Integrate both sides of the equation $\frac{dy}{dx}=4x^2$ A) $y=\frac{1}{3}x^3+C$. B) $y=4x^3+C$. C) $y=2x^3+C$. D) $y=\frac{4}{3}x^3+C$. Show Answer Correct Answer: D) $y=\frac{4}{3}x^3+C$. 36. Solve the non-homogeneous equation:y" + y = sin(x). A) Y = C1 cos(x) + C2 sin(x) + 1/2. B) Y = C1 cos(x) + C2 sin(x) + x. C) Y = (C1-x/2) cos(x) + C2 sin(x). D) Y = C1 sin(x) + C2 cos(x)-1/2. Show Answer Correct Answer: C) Y = (C1-x/2) cos(x) + C2 sin(x). 37. Solve $\left(D^2+2DD'+D'^2\right)z=0$ A) $Z=f_1\left(y-x\right)+f_2\left(y-x\right)$. B) $Z=f_1\left(y-x\right)+xf_2\left(y-x\right)$. C) $Z=f_1\left(y+x\right)+xf_2\left(y+x\right)$. D) $Z=f_1\left(y+x\right)+xf_2\left(y-x\right)$. Show Answer Correct Answer: B) $Z=f_1\left(y-x\right)+xf_2\left(y-x\right)$. 38. Differential equation of the y=a.e$^{2x}$+b.e$^{-x}$, where a &b are arbitrary constants, is A) Y$^{" }$-y$^{'}$+2y=0. B) Y$^{" }$+y$^{'}$+2y=0. C) Y$^{"}$+y$^{'}$-2y=0. D) Y$^{"}$-y$^{"}$-2y=0. Show Answer Correct Answer: D) Y$^{"}$-y$^{"}$-2y=0. 39. Determine the general solution of $9y"+9y'-4y=0$ A) $y=e^{\frac{1}{3}x}\left(C_1\cos\frac{4}{3}x+C_2\sin\frac{4}{3}x\right)$. B) $y=C_1e^{\frac{1}{3}x}+C_2xe^{-\frac{4}{3}x}$. C) $y=C_1e^{-\frac{1}{3}x}+C_2e^{\frac{4}{3}x}$. D) $y=C_1e^{\frac{1}{3}x}+C_2e^{-\frac{4}{3}x}$. Show Answer Correct Answer: D) $y=C_1e^{\frac{1}{3}x}+C_2e^{-\frac{4}{3}x}$. 40. In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated. A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: A) False. 41. Which of the following represents the solution to the differential equation $\frac{dy}{dx} =-y$ A) A linear function with a negative slope. B) An exponential growth function. C) A linear function with a positive slope. D) An exponential decay function. Show Answer Correct Answer: D) An exponential decay function. 42. How do you classify a second-order PDE? A) A second-order PDE is categorized by its variable coefficients and solution methods. B) A second-order PDE is classified by its boundary conditions and initial values. C) A second-order PDE can be classified as elliptic, parabolic, or hyperbolic based on the discriminant of its coefficients. D) A second-order PDE can only be classified as linear or nonlinear based on its terms. Show Answer Correct Answer: C) A second-order PDE can be classified as elliptic, parabolic, or hyperbolic based on the discriminant of its coefficients. 43. CHOOSE THE CORRECT FORM OF PARTICULAR INTEGRAL FOR sin(ax) A) PI=1/ f(-a$^{2}$, -ab, -b$^{2}$). B) PI=e$^{ax+by }$/f(a, b). C) PI=sin$^{ax}$/ f(-a$^{2}$, -ab, -b$^{2}$). D) PI=sin$^{ax}$ / f(a, b). Show Answer Correct Answer: C) PI=sin$^{ax}$/ f(-a$^{2}$, -ab, -b$^{2}$). 44. What is the order and degree for the differential equation:$\left(\frac{\text{d}y}{\text{d}x}\right)^4-\left(\frac{\text{d}^2y}{\text{d}x^2}\right)=-y$ A) Order:4 Degree:1. B) Order:1 Degree:4. C) Order:2 Degree:4. D) Order:2 Degree:1. Show Answer Correct Answer: D) Order:2 Degree:1. 45. Choose the correct expansion of (1+x)$^{-1}$ A) 1+x+ x$^{2}$-x$^{3}$. B) 1-x+ x$^{2}$-x$^{3}$+x$^{4}$. C) 1-x+ x$^{2}$-x$^{3}$-x$^{4}$. D) 1-x+ x$^{2}$-x$^{3}$+x$^{5}$. Show Answer Correct Answer: B) 1-x+ x$^{2}$-x$^{3}$+x$^{4}$. 46. Any periodic motion can be written as sum of harmonic functions. A) Yes. B) No. C) All the above. D) None of the above. Show Answer Correct Answer: A) Yes. 47. For repeated roots m1=m2=m, the complementary function is: A) $(c_1+c_2e^{mx})$. B) $c_1e^{mx}$. C) $(c_1+c_2)xe^{mx}$. D) $c_1e^{mx}+c_2xe^{mx}$. Show Answer Correct Answer: D) $c_1e^{mx}+c_2xe^{mx}$. 48. The fundamental frequency corresponds to: A) The highest frequency of the signal. B) The first harmonic. C) The average of all harmonics. D) Zero frequency component. Show Answer Correct Answer: B) The first harmonic. 49. Which of the following is a first-order PDE? A) U x-u y = 0. B) U x + 2u y = 1. C) U x + u y = 0. D) U x u y = 0. Show Answer Correct Answer: C) U x + u y = 0. 50. Which of the following equations represents Clairaut's partial differential equation? A) Z=px+qy+f(p, q). B) Z=f(p, q). C) Z=px+f(p, q). D) Z=p+q+f(p, q). Show Answer Correct Answer: A) Z=px+qy+f(p, q). 51. If a substance decomposes at a rate proportional to the amount of the substance present, and if the amount decreases from 40 g to 5 g in 3 hours, then the constant of proportionality is A) $-\ln\left(\frac{1}{8}\right)$. B) $-\ln\left(\frac{1}{2}\right)$. C) $-\ln2$. D) $-\ln8$. Show Answer Correct Answer: C) $-\ln2$. 52. Find the general solution $\frac{\text{d}y}{\text{d}x}=4x$ A) $y=2x^2+c$. B) $y=4x^2+c$. C) $y=4x+c$. D) $y=2x+c$. Show Answer Correct Answer: A) $y=2x^2+c$. 53. What type of differential equations are Cauchy's and Euler's equations used to solve? A) First order linear. B) Nonlinear. C) Higher order linear with constant coefficients. D) Initial and boundary value problems. Show Answer Correct Answer: C) Higher order linear with constant coefficients. 54. When applying boundary conditions to a PDE, what is the primary goal? A) To determine specific values of the unknown function at the boundaries of the domain. B) To convert the PDE into an ordinary differential equation. C) To classify the PDE as linear or non-linear. D) To find the general solution of the PDE. Show Answer Correct Answer: A) To determine specific values of the unknown function at the boundaries of the domain. 55. The general solution of the DE $\frac{d^2y}{dx^{2}}+4\frac{dy}{dx}-5y=0$ A) $y=Ae^x+Be^{5x}$. B) $y=Ae^x+Be^{-5x}$. C) $y=Ae^{-x}+Be^{5x}$. D) $y=Ae^{-x}+Be^{-5x}$. Show Answer Correct Answer: B) $y=Ae^x+Be^{-5x}$. 56. Which of the following has the trial solution z=ax+by+c A) $z=px+qy+p^2$. B) $p+q=pq$. C) $p\left(1+q\right)=qz$. D) None of the above. Show Answer Correct Answer: B) $p+q=pq$. 57. A differential equation in which the function depends on more than one independent variabls is called a ..... A) Non linear equation. B) Partial differential equation. C) Linear equation. D) Ordinary differential equation. Show Answer Correct Answer: B) Partial differential equation. 58. Determine whether the functions $y_{1} = e^{4x}$ $y_{2} = xe^{4x}$ A) The functions coincide. B) The functions are undefined. C) No, the functions are linearly dependent. D) Yes, the functions are linearly independent. Show Answer Correct Answer: D) Yes, the functions are linearly independent. 59. A particular integral of y ''-4y=ex is A) E-x. B) X2ex. C) Xex. D) 12ex. Show Answer Correct Answer: C) Xex. 60. Integrating factor of $\frac{\text{d}y}{\text{d}x}+\frac{1}{x}=\frac{e^y}{x^2}$ A) $x^2$. B) $\frac{1}{x}$. C) $-\frac{1}{x}$. D) X. Show Answer Correct Answer: B) $\frac{1}{x}$. ← 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