This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 12 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 12 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Name the main types of PDEs. A) Discrete, Continuous, Mixed. B) Linear, Quadratic, Cubic. C) Elliptic, Parabolic, Hyperbolic. D) Static, Dynamic, Stochastic. Show Answer Correct Answer: C) Elliptic, Parabolic, Hyperbolic. 2. What type of differential equation is dy/dx = x/y? A) Nonlinear differential equation. B) First-order homogeneous differential equation. C) Exact differential equation. D) Second-order linear differential equation. Show Answer Correct Answer: B) First-order homogeneous differential equation. 3. What is homogeneous A) Same degree. B) Different order. C) Different degree. D) Same order. Show Answer Correct Answer: A) Same degree. 4. If the acceleration function is $a(t) = 6t-4$ $v(t)$ $v(0) = 5$ A) $v(t) = 3t^2 + 4t$. B) $v(t) = 3t^2-4t + 5$. C) $v(t) = 3t^2-4t$. D) $v(t) = 3t^2 + 4t + 5$. Show Answer Correct Answer: B) $v(t) = 3t^2-4t + 5$. 5. Complementary function of $y" +4y'+13y=2e^{-x}$ A) $e^{2x}\left(c_1\cos3x+c_2\sin3x\right)$. B) $e^{3x}\left(c_1\cos2x+c_2\sin2x\right)$. C) $e^{-2x}\left(c_1\cos3x+c_2\sin3x\right)$. D) $\left(c_1\cos3x+c_2\sin3x\right)$. Show Answer Correct Answer: C) $e^{-2x}\left(c_1\cos3x+c_2\sin3x\right)$. 6. For f(x, y) = xy, the total differential df is: A) Y dx + x dy. B) Dx + dy. C) X dx + y dy. D) X dy + y dx. Show Answer Correct Answer: D) X dy + y dx. 7. Determine the general solution of the differential equation $\frac{dy}{dx} = 4y$ A) $y = Ce^{x}$. B) $y = Ce^{3x}$. C) $y = Ce^{2x}$. D) $y = Ce^{4x}$. Show Answer Correct Answer: D) $y = Ce^{4x}$. 8. Wronskian formula for second-order DE $y" +P(x)y'+Q(x)y=0$ A) $W(x)=Ce^{-\int Q(x)dx}$. B) $W(x)=Ce^{\int P(x)dx}$. C) $W(x)=Ce^{-\int P(x)dx}$. D) $W(x)=Ce^{\int Q(x)dx}$. Show Answer Correct Answer: C) $W(x)=Ce^{-\int P(x)dx}$. 9. Dy/dx = 3x-2y and y(0) = k. Starting with x = 0 and a step size of 1, find k if y(2) = 4.5 A) 1. B) 2. C) 2.5. D) 1.5. Show Answer Correct Answer: D) 1.5. 10. The auxiliary equation of the following ODE $D^2y-7Dy+6y=0$ A) $\left(m^2-7m+6\right)y=0$. B) $m^2-3m+5$. C) $m^2+7m-6=0$. D) $m^2-7m+6=0$. Show Answer Correct Answer: D) $m^2-7m+6=0$. 11. In the context of PDEs, what does the term "separation of variables" refer to? A) A way to derive boundary conditions for PDEs. B) A method to convert PDEs into ordinary differential equations. C) A method to solve PDEs by assuming a solution can be written as a product of functions, each depending on a single variable. D) A technique to classify PDEs based on their order. Show Answer Correct Answer: C) A method to solve PDEs by assuming a solution can be written as a product of functions, each depending on a single variable. 12. What is the difference between linear and nonlinear PDEs? A) Linear PDEs involve linear combinations of the unknown function and its derivatives, while nonlinear PDEs include nonlinear terms. B) Linear PDEs do not involve derivatives of the unknown function. C) Nonlinear PDEs cannot be solved using any numerical methods. D) Linear PDEs only have constant coefficients. Show Answer Correct Answer: A) Linear PDEs involve linear combinations of the unknown function and its derivatives, while nonlinear PDEs include nonlinear terms. 13. State the order of given Differential Equations $y^{\left(5\right)}$ A) 5. B) 1. C) 5 and 1. D) Zero. Show Answer Correct Answer: A) 5. 14. The equation y" +3y = sinx is: A) Linear and homogeneous. B) Linear and non-homogeneous. C) Non-linear. D) Exact. Show Answer Correct Answer: B) Linear and non-homogeneous. 15. Which of the following differential equation is a separable equation? A) $2ydx=\left(x^2+1\right)dy$. B) $\left(x+y\right)dx-2ydy=0$. C) $y^2dx+\left(2x-3y\right)dy=0$. D) $\left(x+x^2y\right)dy=\left(2x+xy^2\right)dx$. Show Answer Correct Answer: A) $2ydx=\left(x^2+1\right)dy$. 16. $x\left(\frac{\text{d}^2y}{\text{d}x^2}\right)+xy=1$ A) Second order differential equation at first degree. B) First order differential equation at second degree. C) First order differential equation. D) Second order differential equation. Show Answer Correct Answer: A) Second order differential equation at first degree. 17. By usual notations what is the general form of Lagrange's linear PDE. A) Pdx+qdy=0. B) Pp+Qq=R. C) Pdx+Qdy=Rdz. D) Non of the above. Show Answer Correct Answer: B) Pp+Qq=R. 18. $\frac{dy}{dx}+\frac{2x}{1-x^2}y-4x=0$ A) Linear. B) Bernoulli. C) Homogeneous. D) Separable. Show Answer Correct Answer: A) Linear. 19. Find the general solution of the differential equation $e^{-y}\frac{\text{d}y}{\text{d}x}=\left(2-x^3\right)^2$ A) $e^{-y}=\frac{-x^7}{7}+4x^4+4x+c$. B) $-e^{-y}=\frac{x^7}{7}-x^4+4x+c$. C) $-e^{-y}=\frac{-x^7}{7}+x^3-x+c$. D) $e^{-y}=\frac{x^7}{7}-4x^3-x+c$. Show Answer Correct Answer: B) $-e^{-y}=\frac{x^7}{7}-x^4+4x+c$. 20. If S=\{(1, 0, 0), (2, 0, 0), (3, 0, 0)\} is a subset of R$^{3}$, the dim(L(S))= A) 3. B) 4. C) 1. D) 2. Show Answer Correct Answer: C) 1. 21. The complementary function of x$^{2y" -3xy'-5y=sin(logx) is ..... }$ A) C$_{1}$x$^{5}$+c$_{2}$x$^{-1}$. B) C$_{1}$cos5x+c$_{2}$sinx. C) C$_{1e}$$^{5x}$+c$_{2}$x$^{-1}$. D) C$_{1}$x$^{5}$+c$_{2e}$$^{-x}$. Show Answer Correct Answer: A) C$_{1}$x$^{5}$+c$_{2}$x$^{-1}$. 22. Which is Bernoulli's equation? A) Y' + p(x)y = q(x). B) $y' + p(x)y = q(x)y^n$. C) $y = \frac{(x^2 + y^2)}{xy}$. D) None. Show Answer Correct Answer: B) $y' + p(x)y = q(x)y^n$. 23. $L\left(1\right)=$ A) $1$. B) $0$. C) $\frac{1}{s}$. D) $\frac{1}{s^2}$. Show Answer Correct Answer: C) $\frac{1}{s}$. 24. Which of the following is NOT a step in Euler's Method? A) Update $x$ $y$. B) Choose a step size $h$. C) Find the exact analytical solution. D) Use the formula $y_{n+1} = y_n + h f(x_n, y_n)$. Show Answer Correct Answer: C) Find the exact analytical solution. 25. For system $\vec{X}' = A \vec{X}$ A) Always unstable. B) $\vec{X} = 0$. C) Determined by det(A). D) Any constant vector. Show Answer Correct Answer: B) $\vec{X} = 0$. 26. Differential equation of the ellipses having centre at the origin and foci on x axis is A) Xy.y$^{" }$-x (y$^{' }$)$^{2}$=y.y$^{'}$. B) Xy.y$^{"}$+x (y$^{' }$)$^{2}$=y$^{'}$. C) Xy.y$^{"}$+x (y$^{' }$)$^{2}$=y.y$^{'}$. D) Xy.y$^{"}$+x (y$^{' }$)$^{2}$=0. Show Answer Correct Answer: C) Xy.y$^{"}$+x (y$^{' }$)$^{2}$=y.y$^{'}$. 27. Integrating factor of $\frac{d^2y}{dx^2}+\frac{dy}{dx}\tan x=\sec x+\cos x$ A) Sec xTags5. B) Cos x. C) Sin x. D) Tan x. Show Answer Correct Answer: A) Sec xTags5. 28. An integrating factor of the differential equation $\left(e^{2x}+1\right)\frac{\text{d}y}{\text{d}x}+4e^{2x}y=x$ A) $e^{2x}+1$. B) $\left(e^{2x}+1\right)^2$. C) $\left(e^{2x}+1\right)^4$. D) $e^{2e^{2x}}$. Show Answer Correct Answer: B) $\left(e^{2x}+1\right)^2$. 29. Identify the linearly independent set among the following in R$^{3}$. A) {(1, 2, 3), (2, 4, 6), (1, 3, 0)}. B) {(1, -1, 0), (1, 1, 2)}. C) {(1, 1, 2), (1, 0, 3), (0, 0, 0)}. D) {(1, 1, 0), (0, 1, 1), (1, 0, 1), (1, 1, 1)}. Show Answer Correct Answer: B) {(1, -1, 0), (1, 1, 2)}. 30. What is the order of the equation $y'+y^2=0$ A) 1. B) 3. C) 0. D) 2. Show Answer Correct Answer: A) 1. 31. If dy/dx = 2xy and y = 6 when x = 0, then y= A) $e^{x^2}$. B) $x^2y^2+6$. C) $\sqrt[]{x^2+16}$. D) $6e^{x^2}$. Show Answer Correct Answer: D) $6e^{x^2}$. 32. In differential equations, integration is mainly used to ..... A) Find slopes. B) Recover the function from its rate of change. C) Compute area. D) Simplify algebra. Show Answer Correct Answer: B) Recover the function from its rate of change. 33. Give an example of type III of a PDE A) P$^{2}$-q$^{2}$=1. B) F(z, p, q, a)=0. C) P-x$^{2}$=q+y$^{2}$. D) P(1-q$^{2}$)=q(1-z). Show Answer Correct Answer: D) P(1-q$^{2}$)=q(1-z). 34. The derivative of $\ln x$ A) $\dfrac{1}{x}$. B) $x$. C) $x^{-2}$. D) $e^x$. Show Answer Correct Answer: A) $\dfrac{1}{x}$. 35. Solve the partial differential equation $\frac{\partial z}{\partial y}=0.$ A) Z=xf(y). B) Z=yf(x). C) Z=f(y). D) Z=f(x). Show Answer Correct Answer: D) Z=f(x). 36. The nature of the one-dimensional heat equation is A) Circular. B) Elliptic. C) Parabolic. D) Hyperbolic. Show Answer Correct Answer: C) Parabolic. 37. What are the coefficients in the quadratic equation used to solve homogeneous second-order linear differential equations? A) R, s, t. B) M, n, p. C) A, b, c. D) X, y, z. Show Answer Correct Answer: C) A, b, c. 38. You draw a card at random from a standard deck of 52 cards. Find the probability that the card is a heart given that it is black. A) 0.25. B) 0.333. C) 0 077. D) 0. Show Answer Correct Answer: D) 0. 39. $\frac{dy}{dx}=2y\left(3-x\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 40. The particular solution of the differential equation $\frac{\text{d}y}{\text{d}x}=y^2e^{2x}$ A) $y=\frac{1}{2-e^{2x}}$. B) $y=e^{-2x}$. C) $y=\frac{2}{1+e^{2x}}$. D) $y=\frac{2}{3-e^{2x}}$. Show Answer Correct Answer: D) $y=\frac{2}{3-e^{2x}}$. 41. What do mean by linear A) Degree 2. B) Degree 1. C) Degree 3. D) Degree 4. Show Answer Correct Answer: B) Degree 1. 42. Bismuth-210 has a half life of 5 days. Suppose the sample originally has a mass of 800 mg. Determine how long it will take for the mass to be reduced to 1 mg. A) 36.893 days. B) 48.219 days. C) 53.865 days. D) 79.320 days. Show Answer Correct Answer: B) 48.219 days. 43. The rate at which a chemical element decays is directly proportional to the amount of the element remaining. Suppose 20 grams of the element is present at t = 0. At t = 2, 5 grams of the element is present. Which of the following equations can be used to find the correct constant of proportionality, k? A) $2=20e^{5k}$. B) $20=2e^{5k}$. C) $5=20e^{2k}$. D) $20=5e^{2k}$. Show Answer Correct Answer: C) $5=20e^{2k}$. 44. Which method is used to solve separable differential equations? A) Homogeneous method. B) Substitution method. C) Separation of variables. D) Integration by parts. Show Answer Correct Answer: C) Separation of variables. 45. Solve the homogeneous equation:y" -5y' + 6y = 0. A) Y(t) = C1 e$^{3t}$ + C2 e$^{5t}$. B) Y(t) = C1 e$^{2t}$ + C2 e$^{3t}$. C) Y(t) = C1 e$^{t}$ + C2 e$^{4t}$. D) Y(t) = C1 + C2 e$^{6t}$. Show Answer Correct Answer: B) Y(t) = C1 e$^{2t}$ + C2 e$^{3t}$. 46. Solve $\left(D^3-3D^2D'+2DD^{'2}\right)Z=0$ A) $Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. B) $Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y-2x\right)$. C) $Z=f_1\left(y-x\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. D) $Z=f_1\left(y\right)+xf_2\left(y+x\right)+f_3\left(y+2x\right)$. Show Answer Correct Answer: A) $Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. 47. The complete integral of $z=px+qy+f\left(p, q\right)$ A) $z=px+qy$. B) $z=ax+by+f\left(a, b\right)$. C) $z=f\left(a, b\right)$. D) None of the above. Show Answer Correct Answer: B) $z=ax+by+f\left(a, b\right)$. 48. Apply the initial condition $y(1) = 3$ $y = Cx^2$ $\frac{dy}{dx} = \frac{2y}{x}$ $C$ A) $C = 2$. B) $C = 4$. C) $C = 1$. D) $C = 3$. Show Answer Correct Answer: D) $C = 3$. 49. Find the complementary function for the differential equation A) Y=. B) . C) . D) . Show Answer Correct Answer: A) Y=. 50. The roots of the auxiliary equation $\frac{\text{d}x}{\text{d}y}+16y=0$ A) $y=A\cos16x+B\sin16x$. B) $y=A+Be^{-16x}$. C) $y=A\cos4x+B\sin4x$. D) $y=Ae^{4x}+Be^{-4x}$. Show Answer Correct Answer: C) $y=A\cos4x+B\sin4x$. 51. The general solution of $y'(t) + 8y(t) = 0$ A) $y = Ce^{8t}$. B) $y = Ce^{-8t}$. C) $y = e^{8t} + C$. D) None of the above. Show Answer Correct Answer: B) $y = Ce^{-8t}$. 52. What is the order of an equation in differential equations? A) The degree of the unknown function in the equation. B) The number of variables in the equation. C) The sum of the coefficients in the equation. D) The highest power of the unknown function's derivative in the equation. Show Answer Correct Answer: D) The highest power of the unknown function's derivative in the equation. 53. Solve the homogeneous equation 2y" + 3y' + y = 0. A) $y_c=Ae^{0.5x}+Be^x$. B) $y_c=Ae^{2x}+Be^x$. C) $y_c=Ae^{-0.5x}+Be^{-x}$. D) $y_c=Ae^{2x}+Be^{-x}$. Show Answer Correct Answer: C) $y_c=Ae^{-0.5x}+Be^{-x}$. 54. Y is proportional to the product of z and the square root of x A) $y=zkx$. B) $y=kzx^2$. C) $y=kz\sqrt{x}$. D) $y=zx^2$. Show Answer Correct Answer: C) $y=kz\sqrt{x}$. 55. A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy 3weigh when it is 3 months old? A) 4.8 pounds. B) 6.5 pounds. C) 5 pounds. D) 4.6 pounds. Show Answer Correct Answer: D) 4.6 pounds. 56. How can you represent a CPS using state diagrams? A) A CPS can only be represented using flowcharts. B) CPS representation requires only physical components without computational aspects. C) A CPS can be represented using state diagrams by defining states, transitions, and interactions between physical and computational components. D) State diagrams are not applicable to CPS representation. Show Answer Correct Answer: C) A CPS can be represented using state diagrams by defining states, transitions, and interactions between physical and computational components. 57. In the heat equation, the constant c2represents: A) Velocity of wave. B) Frequency. C) Thermal diffusivity. D) Amplitude. Show Answer Correct Answer: C) Thermal diffusivity. 58. $\int_{ }^{ }\frac{\left(\ln x\right)^2}{x}dx=$ A) $\frac{\left(\ln x\right)^3}{3}+C$. B) $2\ln x+C$. C) $\left(\ln x\right)^3+C$. D) $\left(\ln x\right)^2+C$. Show Answer Correct Answer: A) $\frac{\left(\ln x\right)^3}{3}+C$. 59. Which method is commonly used to find the particular solution when f(t) is a polynomial? A) Laplace Transform Method. B) Method of Variation of Parameters. C) Method of Undetermined Coefficients. D) Substitution Method. Show Answer Correct Answer: C) Method of Undetermined Coefficients. 60. Which of the following is a characteristic of a second-order linear PDE? A) It is always non-linear. B) It involves the second derivatives of the unknown function. C) It cannot be solved using separation of variables. D) It can only have one solution. Show Answer Correct Answer: B) It involves the second derivatives of the unknown function. ← 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