This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 6 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 6 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Describe the method of characteristics for solving PDEs. A) The method of characteristics is a technique for solving PDEs by reducing them to ODEs along characteristic curves. B) The method of characteristics involves applying Fourier transforms to PDEs directly. C) The method of characteristics is a way to solve PDEs using numerical simulations. D) The method of characteristics simplifies PDEs by integrating them over a fixed domain. Show Answer Correct Answer: A) The method of characteristics is a technique for solving PDEs by reducing them to ODEs along characteristic curves. 2. Which of the following differential equation is a homogeneous equation of degree 2? A) $\frac{\text{d}y}{\text{d}x}=\frac{3xy+y^2}{xy}$. B) $\frac{\text{d}y}{\text{d}x}=2xy$. C) $y\frac{\text{d}y}{\text{d}x}=2x$. D) $xy\frac{\text{d}y}{\text{d}x}=y^2-1$. Show Answer Correct Answer: A) $\frac{\text{d}y}{\text{d}x}=\frac{3xy+y^2}{xy}$. 3. I want to solve the differential equation $\frac{dy}{dx}=6-y$ A) $\frac{dy}{6-y}=dx$. B) $\frac{dy}{dx}+y=6$. C) $dy=\left(6-y\right)dx$. D) $dy+y=6dx$. Show Answer Correct Answer: A) $\frac{dy}{6-y}=dx$. 4. Consider the differential equation dy/dx = e$^{y}$(4x$^{2}$-5x). Let y=f(x) be the particular solution to the differential equation that passes through (1, 0).Write an equation of the line tangent to the graph of f at the point (1, 0). A) Y=x-1. B) Y=2(x-1). C) Y=-(x-1). D) Y=-2(x-1). Show Answer Correct Answer: C) Y=-(x-1). 5. The particular integral of the equation (D$^{2}$+DD$^{'}$-6D$^{'2}$)z = e$^{3x+y}$ A) $\frac{e^{3x}}{6}$. B) $\frac{e^{3x-y}}{6}$. C) $\frac{e^{3x+y}}{5}$. D) $\frac{e^{3x+y}}{6}$. Show Answer Correct Answer: D) $\frac{e^{3x+y}}{6}$. 6. What is the degree of the differential equation:$\left(\frac{d^2y}{dx^2}\right)^2 + 3\left(\frac{dy}{dx}\right)^3-y = 0$ A) 4. B) 3. C) 2. D) 1. Show Answer Correct Answer: C) 2. 7. What is the significance of the Wronskian in differential equations? A) It is used to find particular solutions. B) It provides the roots of the equation. C) It determines the order of the equation. D) It indicates the linear independence of solutions. Show Answer Correct Answer: D) It indicates the linear independence of solutions. 8. Let $y=f\left(x\right)$ $\frac{dy}{dx}=y-10x^2$ $f\left(0\right)=3$ $f\left(0.4\right)$ $x=0$ A) 4.768. B) 4.120. C) 4.240. D) 4.200. Show Answer Correct Answer: C) 4.240. 9. The condition of compatibility of partial differential equations $f\left(x, y, z, p, q\right)=0\, \\g\left(x, y, z, p, q\right)=0$ A) $\frac{\partial\left(f, g\right)}{\partial\left(p, q\right)}\ne0$. B) $\frac{\partial\left(f, p\right)}{\partial\left(g, q\right)}=0$. C) $\frac{\partial\left(f, q\right)}{\partial\left(g, p\right)}\ne0$. D) $\frac{\partial\left(f, g\right)}{\partial\left(p, q\right)}=0$. Show Answer Correct Answer: A) $\frac{\partial\left(f, g\right)}{\partial\left(p, q\right)}\ne0$. 10. What are the orthogonal trajectories of the family of curves $ y = x + c$ ? A) $ y =-x + c*$. B) $ y =-x + c$. C) $ y = x-c$. D) $ y = x + c*$. Show Answer Correct Answer: A) $ y =-x + c*$. 11. What is the full form of C.F. in solution of a linear differential equation? A) Complex Function. B) Continuous Function. C) Complimentary Function. D) None of these. Show Answer Correct Answer: C) Complimentary Function. 12. Solve $m^2-4m+4=0$ A) 0, 2. B) 2, -2. C) 2, 2. D) -2, -2. Show Answer Correct Answer: C) 2, 2. 13. The differential equation for a damped harmonic oscillator is given by:$\frac{d^2y}{dt^2}+4\\frac{dy}{dt}+5y=0$ A) Overdamped. B) Underdamped. C) Critically damped. D) Oscillatory without damping. Show Answer Correct Answer: B) Underdamped. 14. If the equation is in the form f(x, p, q)=0, then we assume A) Z=pq. B) Q=a. C) P=aq. D) P=a. Show Answer Correct Answer: B) Q=a. 15. Solve the differential equation. $\frac{dy}{dx}=5y$ A) $y=Ce^{5x}$. B) $y=x^2+C$. C) $\log y=5x+C$. D) $y=5x+C$. Show Answer Correct Answer: A) $y=Ce^{5x}$. 16. Solve the following differential equations:$\frac{\text{d}y}{\text{d}x}=3y$ A) $y=3x+C$. B) $\ln y=x+C$. C) $\frac{y^2}{2}=3x+C$. D) $y=Ce^{3x}$. Show Answer Correct Answer: D) $y=Ce^{3x}$. 17. Solution of a linear differential equation of first order can be obtain by A) Multiplying on both the sides by its integrating factor. B) Adding on both the sides by integrating factor. C) Subtracting integrating factor from both sides. D) None of the above. Show Answer Correct Answer: A) Multiplying on both the sides by its integrating factor. 18. If the velocity function is $v(t) = 2t^3-3t^2 + 4t$ $s(t)$ $s(0) = 7$ A) $s(t) = \frac{1}{2}t^4-t^3 + 2t^2$. B) $s(t) = \frac{1}{2}t^4-t^3 + 4t^2 + 7$. C) $s(t) = \frac{1}{2}t^4-t^3 + 2t^2 + 7$. D) $s(t) = \frac{1}{2}t^4 + t^3 + 2t^2 + 7$. Show Answer Correct Answer: C) $s(t) = \frac{1}{2}t^4-t^3 + 2t^2 + 7$. 19. $\int_{\frac{\pi}{2}}^x\cos\left(t\right)dt=$ A) $\sin x-1$. B) $-\sin x-1$. C) $-\sin x$. D) $\sin x+1$. Show Answer Correct Answer: A) $\sin x-1$. 20. What is the integral of for $f\left(t\right)=t^3$ A) $3$. B) $\frac{6}{ }$. C) $\frac{2}{3}$. D) $\frac{1}{4}$. Show Answer Correct Answer: D) $\frac{1}{4}$. 21. Consider the vector u = (1, 2, 3) and v = (2, 3, 1) in R$^{3}$. If w = (1, k, 4) is a linear combination of u and v, then k= A) 11. B) 2. C) 11/5. D) 3. Show Answer Correct Answer: C) 11/5. 22. Given $y" + 2y' + 5y = 0$ A) $y(x) = Ce^{-x}\sin(2x) + De^{-x}\cos(2x)$. B) $y(x) = Ce^{-x}\sin(2x) + De^{-x}\cos(2x)$. C) $y(x) = Ce^{2x} + De^{-2x}$. D) $y(x) = Ce^{-x} + De^{-5x}$. Show Answer Correct Answer: A) $y(x) = Ce^{-x}\sin(2x) + De^{-x}\cos(2x)$. 23. From the following partial differential equations, the seperableequation is ..... A) $p+q=pq$. B) $p-x=q-y$. C) $p-x=q-y$. D) $p-z=q-y$. Show Answer Correct Answer: B) $p-x=q-y$. 24. $\frac{d^2y}{dx^2}+y=e^xx$ A) $\frac{e^x\left(x+1\right)}{2}$. B) $\frac{e^x\left(x-1\right)}{2}$. C) $\frac{e^xx}{3}$. D) $\frac{e^xx}{2}$. Show Answer Correct Answer: B) $\frac{e^x\left(x-1\right)}{2}$. 25. Suppose that $T:R^2\longrightarrow R^2$ A) (2, 5). B) (5, 11). C) (5, 2). D) (8, 5). Show Answer Correct Answer: B) (5, 11). 26. What is the order of the differential equation $y" +xy'+x^2y=\sin x$ A) 2. B) 4. C) 3. D) 1. Show Answer Correct Answer: A) 2. 27. What is a common error when trying to separate variables in differential equations? A) Forgetting to add the constant of integration. B) Subtracting variables instead of dividing. C) Adding instead of multiplying by dy/dx. D) Incorrectly applying the chain rule. Show Answer Correct Answer: C) Adding instead of multiplying by dy/dx. 28. Dy/dx = x-y-2 and f(-1) = 3. Use Euler's Method with three steps to approximate f(2). A) 0. B) 1. C) -1. D) 2. Show Answer Correct Answer: C) -1. 29. For $y" -y=e^x, $ A) $Ae^{-x}$. B) $12xe^x$. C) $Ae^x$. D) $Axe^x$. Show Answer Correct Answer: D) $Axe^x$. 30. What is the solution method for Bernoulli differential equations? A) Method of Solution of Homogeneous Differential Equation. B) Orthogonal trajectories. C) Separable differential equation. D) Linear Differential Equation. Show Answer Correct Answer: B) Orthogonal trajectories. 31. Solve the homogeneous equation y" -6y' + 9y = 0. A) $y_c=A+Bxe^{-3x}$. B) $y_c=Ae^{3x}+Bxe^{3x}$. C) $y_c=Ae^x+Be^{3x}$. D) $y_c=Ae^{3x}+Be^{3x}$. Show Answer Correct Answer: B) $y_c=Ae^{3x}+Bxe^{3x}$. 32. What is the first step in the method of variation of parameters? A) Finding the Wronskian. B) Solving the homogeneous equation. C) Finding the general solution. D) Determining the roots of the equation. Show Answer Correct Answer: B) Solving the homogeneous equation. 33. What is the general solution for the system of equations represented by dx/P=dy/Q=dz/R? A) Dx=Qdy+Rdz. B) Dx=ldx+mdy+ndz. C) Dx=ky+lz. D) Dx/P=dy/Q=dz/R. Show Answer Correct Answer: D) Dx/P=dy/Q=dz/R. 34. The unit normal to the surface $x^3-xyz+z^3=1$ A) $\frac{1}{3}\left(2i-j+k\right)$. B) $\frac{1}{3}\left(2i-j+2k\right)$. C) $\frac{1}{3}\left(2i-2j+2k\right)$. D) $\frac{2}{3}\left(2i-j+2k\right)$. Show Answer Correct Answer: B) $\frac{1}{3}\left(2i-j+2k\right)$. 35. Which of the following is true about the Cayley-Hamilton theorem? A) It diagonalizes any matrix. B) It reduces higher powers of A to linear combinations of lower powers. C) It Avoids the need for eigenvalues. D) It makes the determinant zero. Show Answer Correct Answer: B) It reduces higher powers of A to linear combinations of lower powers. 36. $L^{-1}\left(\frac{a}{s^2+a^2}\right)$ A) Cos ax. B) $e^{ax}$. C) $e^{-ax}$. D) Sin ax. Show Answer Correct Answer: D) Sin ax. 37. The particular integral of the equation $\left(D-1\right)y=e^{3x}$ A) $\frac{e^{-3x}}{2}$. B) $\frac{e^{3x}}{4}$. C) $\frac{e^x}{2}$. D) $\frac{e^{3x}}{2}$. Show Answer Correct Answer: D) $\frac{e^{3x}}{2}$. 38. What is a first order differential equation? A) A first order differential equation is an equation involving the first derivative of a function. B) A first order differential equation is an equation with no derivatives. C) A first order differential equation is a polynomial equation without variables. D) A first order differential equation involves second derivatives of a function. Show Answer Correct Answer: A) A first order differential equation is an equation involving the first derivative of a function. 39. The value of $curl\\overrightarrow{r}$ A) $\overrightarrow{0}$. B) 0. C) 1. D) $\overrightarrow{r}$. Show Answer Correct Answer: A) $\overrightarrow{0}$. 40. A differential equation in which the function depends on only one independent variable is called a ..... A) Partial differential equation. B) Ordinary differential equation. C) Non linear equation. D) Linear equation. Show Answer Correct Answer: B) Ordinary differential equation. 41. The auxiliary equation of the following ODE $\left(D^2-3D+5\right)y=0$ A) $m^2-3m+5$. B) $\left(m^2-3m+5\right)y=0$. C) $m^2+3m-5=0$. D) $m^2-3m+5=0$. Show Answer Correct Answer: D) $m^2-3m+5=0$. 42. In the method of multipliers, what is the primary goal? A) To solve constrained optimization problems by transforming them into unconstrained problems. B) To analyze the behavior of functions in different dimensions. C) To simplify linear equations for easier calculations. D) To identify the maximum value of a function without constraints. Show Answer Correct Answer: A) To solve constrained optimization problems by transforming them into unconstrained problems. 43. F(s) = L\{(e$^{at}$)\} (s) A) 1/(s-a). B) 1/s. C) S/(s$^{2}$+b$^{2}$). D) N!/(s-a)$^{(n+1)}$. E) N!/s$^{(n+1)}$. Show Answer Correct Answer: A) 1/(s-a). 44. Which of the following equations solves dy/dt = ky? A) $y=a\left(x-h\right)^2+k$. B) $y=A\sin\left[B\left(x-C\right)\right]+D$. C) $y=y_0e^{kt}$. D) $y=mx+b$. Show Answer Correct Answer: C) $y=y_0e^{kt}$. 45. Eliminate the arbitrary constants c and r from x$^{2}$+y$^{2}$+(z-c)$^{2}$=r$^{2}$ A) Px = qy. B) Py+qx=0. C) Py = qx. D) Px+qy=0. Show Answer Correct Answer: C) Py = qx. 46. Which of the following is a solution to the differential equation y" + y = 0? A) $y = e^x$. B) Y = sin(x). C) Y = x. D) Y = ln(x). Show Answer Correct Answer: B) Y = sin(x). 47. If $\frac{dP}{dt} = P-3$ A) $e^{t-4} + 1$. B) $e^{t-4}-1$. C) $-e^{t-4} + 3$. D) $-e^{t-4}-3$. Show Answer Correct Answer: C) $-e^{t-4} + 3$. 48. Which of the following represents a condition for the existence of a unique solution in simultaneous equations? A) Determinant of coefficients is zero. B) Determinant of coefficients is non-zero. C) Sum of ratios equals one. D) All variables are equal. Show Answer Correct Answer: B) Determinant of coefficients is non-zero. 49. Find the degree of the differential equation:$\left(\frac{d^4y}{dx^4}\right)^3 + 2\left(\frac{d^2y}{dx^2}\right)^2-y = 0$ A) 3. B) 2. C) 4. D) 1. Show Answer Correct Answer: A) 3. 50. The Wronskian of two solutions of a 2x2 linear systems is? A) Always zero. B) Nonzero if solutions are linearly independent. C) Always constant. D) Equal to determinant of A. Show Answer Correct Answer: B) Nonzero if solutions are linearly independent. 51. Find particular integral of (D$^{2}$+D'$^{2}$)z=x$^{2}$+y$^{2}$ A) PI=x$^{2}$y$^{2}$/4. B) PI=x$^{2}$y$^{2}$/2. C) PI=x$^{2}$/4. D) PI=y$^{2}$/2. Show Answer Correct Answer: B) PI=x$^{2}$y$^{2}$/2. 52. Identify the order of the differential equation:$\sin\left(\frac{d^2y}{dx^2}\right) +\cos\left(\frac{dy}{dx}\right) = 0$ A) 0. B) 2. C) 3. D) 1. Show Answer Correct Answer: B) 2. 53. If $\frac{dy}{dx} = 4y$ $y(0) = 2$ A) $y = 4e^{2x}$. B) $y = 4e^{x}$. C) $y = 2e^{4x}$. D) $y = 2e^{x}$. Show Answer Correct Answer: C) $y = 2e^{4x}$. 54. Find the particular solution W = W(t) to the differential equation $\frac{dW}{dt}=\frac{1}{25}\left(W-300\right)$ A) $W\left(t\right)=300+\frac{1}{25}e^{1100t}$. B) $W\left(t\right)=300+1100e^{\frac{1}{25}t}$. C) $W\left(t\right)=300t+1100e^{\frac{1}{25}}$. D) $W\left(t\right)=1100+300e^{\frac{1}{25}t}$. Show Answer Correct Answer: B) $W\left(t\right)=300+1100e^{\frac{1}{25}t}$. 55. A population of fish starts at 8, 000 and decreases by 6% per year. What is the population of fish after 10 years? A) 839. B) 4309. C) 7680. D) 14327. Show Answer Correct Answer: B) 4309. 56. In which quadrant(s) is the differential equation $\frac{dy}{dx} = \frac{xy^2}{e^y}$ A) Quadrants I and II only. B) Quadrants II and III only. C) Quadrants I and III only. D) Quadrants II, III, and IV only. Show Answer Correct Answer: B) Quadrants II and III only. 57. In the method of undetermined coefficients, what is the first step to find the particular solution? A) Substitute the complementary function into the original equation. B) Guess a form for the particular solution based on f(t). C) Differentiate the non-homogeneous term. D) Solve the characteristic equation. Show Answer Correct Answer: B) Guess a form for the particular solution based on f(t). 58. Name the method used to solve first order linear ODEs? A) Laplace transform. B) Separation of variables only. C) Variation of parameters. D) Integrating factor method. Show Answer Correct Answer: D) Integrating factor method. 59. The matrix exponential is defined as ..... A) $e^A = A^t$. B) $e^A = \Sigma_{n=0}^{\infty} \left(\frac{A^n}{n!}\right)$. C) $e^A = I + A$. D) $e^A = det(|A|)$. Show Answer Correct Answer: B) $e^A = \Sigma_{n=0}^{\infty} \left(\frac{A^n}{n!}\right)$. 60. If $\overrightarrow{f}=\left(x^2+y^2\right)\overrightarrow{i}+\left(x^2-y^2\right)\overrightarrow{j}$ $\int_Cf.d\overrightarrow{r}$ $y=x^2$ A) 7/10. B) 9/10. C) 2. D) 0. Show Answer Correct Answer: A) 7/10. ← 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 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