This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 1 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 1 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^2}{y}$ A) $y=\sqrt{\frac{2x^3}{3}-14}$. B) $y=-2e^{\left(-9+\frac{x^3}{3}\right)}$. C) $y=\sqrt{\frac{2x^3}{3}}$. D) $y=-\sqrt{\frac{2x^3}{3}-14}$. Show Answer Correct Answer: D) $y=-\sqrt{\frac{2x^3}{3}-14}$. 2. The general solution of the differential equation $x\frac{\text{d}y}{\text{d}x}+3y=x^2$ A) $y=\frac{x^2}{5}+\frac{c}{x^3}$. B) $y=\frac{x^2}{2}+\frac{c}{x}$. C) $y=\frac{x}{5}+C$. D) $y=x^2+Cx$. Show Answer Correct Answer: A) $y=\frac{x^2}{5}+\frac{c}{x^3}$. 3. If the auxiliary equation has distinct real roots, the complementary function is a combination of: A) Exponentials. B) Trigonometric functions. C) Polynomials. D) None. Show Answer Correct Answer: A) Exponentials. 4. A partial differential equation which is not linear then it is called ..... A) Quasi linear. B) Semi linear. C) Non-linear. D) Collinear. Show Answer Correct Answer: C) Non-linear. 5. $L\left(e^{-ax}\cos bx\right)=\frac{\left(s+a\right)}{\left(s+a\right)^2+b^2}$ A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: B) True. 6. What is the order of the equation $y" + y^3 + y = 0$ A) 1. B) 2. C) 3. D) None of the above. Show Answer Correct Answer: B) 2. 7. The characteristic equation for [-2 2 2 1] is ..... A) $x^2+2x-16$. B) $x^2+4x+6$. C) $x^2+3x-6$. D) $x^2+x-6$. Show Answer Correct Answer: D) $x^2+x-6$. 8. What is the Integrating factor of Mdx+Ndy if it is in form $yf\left(xy\right)dx+xg\left(xy\right)dy=0$ A) $\frac{1}{MX-Ny}$. B) $\frac{1}{MX+NY}$. C) $\frac{1}{M-N}$. D) $\frac{1}{M+N}$. Show Answer Correct Answer: A) $\frac{1}{MX-Ny}$. 9. The ends A and B of a rod of length20cm are at 30$^{0}$C and 80$^{0}$C at end points until steady state prevails. Then the temperature of the rod at ends are changed to 40$^{0}$C and 60$^{0}$C respectively. Final temperature distribution (i.e. in Steady state) is A) 50+x. B) 70+x. C) 40+x. D) 60+x. Show Answer Correct Answer: C) 40+x. 10. Write a differential equation for the statement:"The rate of change of P is proportional to the product of P and 4-P." A) $\frac{dP}{dt}=\frac{kP}{4-P}$. B) $\frac{dP}{dt}=kP$. C) $\frac{dP}{dt}=k\left(4-P\right)$. D) $\frac{dP}{dt}=kP\left(4-P\right)$. Show Answer Correct Answer: D) $\frac{dP}{dt}=kP\left(4-P\right)$. 11. If $|f(x)|$ $f(x)$ A) Odd only. B) Either even or odd. C) Even only. D) Neither even nor odd. Show Answer Correct Answer: B) Either even or odd. 12. Given the differential equation $\frac{dP}{dt}=5P$ $P_{ }$ $P$ $P\left(0\right)=418$ A) P=Ce$^{t}$B) P= 418e$^{t}$. B) P=Ce$^{5t}$B) P= 418e$^{5t}$. C) P=Ce$^{5t}$B) P= 5e$^{418t}$. D) P=Ce$^{10t}$B) P= 418e$^{10t}$. Show Answer Correct Answer: B) P=Ce$^{5t}$B) P= 418e$^{5t}$. 13. Find a general solution of this differential equation $x\frac{dy}{dx}+3y=4x^2-3x$ $x>0$ A) $y=\frac{x^2}{3}-\frac{x}{4}+\frac{c}{2x^3}$. B) $y=\frac{4x^2}{5}-\frac{3x}{4}+\frac{c}{x^3}$. C) $y=\frac{5x^2}{4}-\frac{x^2}{2}+\frac{c}{x^3}$. D) $y=-\frac{4x^2}{3}+x+\frac{c}{x^2}$. Show Answer Correct Answer: B) $y=\frac{4x^2}{5}-\frac{3x}{4}+\frac{c}{x^3}$. 14. $\int_0^{\ln2}e^{2x}dx=$ A) $\frac{3}{2}$. B) $2e^2-1$. C) $4$. D) $3$. Show Answer Correct Answer: A) $\frac{3}{2}$. 15. The complementary function of (D$^{3}$-3D$^{2}$ + 3D-1) y = x$^{3}$ is A) Ae$^{x}$ + be$^{-x}$ + ce$^{2x}$. B) E$^{x}$ (a + bx + cx$^{2}$). C) E$^{-x}$ (a cosx + b sinx + c). D) E$^{-x}$ (a + bx + cx$^{2}$). Show Answer Correct Answer: B) E$^{x}$ (a + bx + cx$^{2}$). 16. $\frac{\text{d}y}{\text{d}x}=\tan x+15x^2+e^x+\frac{1}{x}$ A) Y=-ln|sinx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. B) Y=-ln|cosx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. C) Y= secxtanx + 5x$^{3}$ + e$^{x}$ + ln|x| + c. D) Y= sec$^{2}$(x) + 5x$^{3}$ + e$^{x}$ + ln|x| + c. Show Answer Correct Answer: B) Y=-ln|cosx| + 5x$^{3}$ + e$^{x}$ + ln|x| + c. 17. L\{(1/8b$^{4}$)(bt)(sin bt)-(bt)$^{2}$cosbt)\} A) S/(s$^{2}$+b$^{2}$)$^{3}$. B) S/(s$^{2}$+b$^{2}$)$^{4}$. C) S/(s$^{2}$+b$^{2}$). D) S/(s$^{2}$+b$^{2}$)$^{2}$. Show Answer Correct Answer: A) S/(s$^{2}$+b$^{2}$)$^{3}$. 18. $\frac{\text{d}^2y}{\text{d}x^2}-2\frac{dy}{dx}-3y=e^{2x}$ A) $y=e^x\left(A\sin3x+B\cos3x\right)-\frac{1}{3}e^{2x}$. B) $y=Ae^{-x}+Be^{3x}-\frac{1}{3}e^{2x}$. C) $y=Ae^{-2x}+Be^{4x}-3e^{2x}$. D) $y=e^x\left(A\sin3x+B\cos3x\right)-3e^{2x}$. Show Answer Correct Answer: B) $y=Ae^{-x}+Be^{3x}-\frac{1}{3}e^{2x}$. 19. $y\left[\ln\left(\frac{y}{x}\right)+1\right]dx-xdy=0$ A) Separable. B) Bernoulli. C) Linear. D) Homogeneous. Show Answer Correct Answer: D) Homogeneous. 20. A transformation T of functions is said to be linear if $T\left[\alpha f\left(x\right)+\beta g\left(x\right)\right]=$ A) $T\left[\alpha f\left(x\right)\right]-T\left[\beta g\left(x\right)\right]$. B) $T\left[\alpha f\left(x\right)\right]+T\left[\beta g\left(x\right)\right]$. C) $\alpha T\left[f\left(x\right)\right]-\beta T\left[g\left(x\right)\right]$. D) $\alpha T\left[f\left(x\right)\right]+\beta T\left[g\left(x\right)\right]$. Show Answer Correct Answer: D) $\alpha T\left[f\left(x\right)\right]+\beta T\left[g\left(x\right)\right]$. 21. Solve the differential equation $\frac{dy}{dx}+2=2y$ A) $y=1+\sqrt{\frac{4}{-2x-A}}$. B) $y=1+\alpha e^{2x}$. C) $y=\frac{\beta e^x+2}{2}$. D) $y=1+\frac{1}{-4x-A}$. Show Answer Correct Answer: B) $y=1+\alpha e^{2x}$. 22. $\vec{\phi}_1\left(t\right), \ ..... , \\vec{\phi}_n\left(t\right)$ $B\left(t\right)$ A) I don't know. B) Is always invertible. C) Is always not invertible. D) Is invertible only for specific times t. Show Answer Correct Answer: B) Is always invertible. 23. The standard five-point formula assumes that the grid spacing in xand ydirections is: A) Equal. B) Unequal. C) Zero. D) Different. Show Answer Correct Answer: A) Equal. 24. If the roots of the Auxiliary equation are complex conjugates $\alpha\\pm\\beta$ A) $e^{-\alpha x}\left(A\cos\beta x+B\sin\beta x\right)$. B) $e^{\beta x}\left(A\cos\alpha x+B\sin\alpha x\right)$. C) $e^{-\beta x}\left(A\cos\alpha x+B\sin\alpha x\right)$. D) $e^{\alpha x}\left(A\cos\beta x+B\sin\beta x\right)$. Show Answer Correct Answer: D) $e^{\alpha x}\left(A\cos\beta x+B\sin\beta x\right)$. 25. If the roots of A. E are imaginary then C. F is y=e$^{x}$(Acosx+Bsinx) A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 26. Find general solution of by using integrating factor $x^3\frac{\text{d}y}{\text{d}x}+3x^2y=e^x$ A) $x^2y=e^x+c$. B) $x^3y=\frac{e^x}{x^3}+c$. C) $y=e^xx^3+c$. D) $y=\frac{e^x+c}{x^3}$. Show Answer Correct Answer: D) $y=\frac{e^x+c}{x^3}$. 27. What is a general solution of a differential equation? A) A solution containing arbitrary constants. B) A solution that is always true. C) A solution with no arbitrary constants. D) A solution that cannot be derived. Show Answer Correct Answer: A) A solution containing arbitrary constants. 28. What is not Differential Equation? A) Y' = 2y + 2. B) Dy/dx = 2x. C) Y" = g(x) + f(x, y). D) 2y = 2. Show Answer Correct Answer: D) 2y = 2. 29. Determine the general solution of $4y"-4y'+y=0$ A) $y=C_1e^{\frac{x}{2}}+C_2xe^{-\frac{x}{2}}$. B) $y=C_1e^{\frac{x}{2}}+C_2xe^{\frac{x}{2}}$. C) $y=e^{-\frac{x}{3}}\left(C_1\cos x+C_2\sin x\right)$. D) $y=C_1e^{\frac{x}{2}}+C_2e^{\frac{x}{2}}$. Show Answer Correct Answer: B) $y=C_1e^{\frac{x}{2}}+C_2xe^{\frac{x}{2}}$. 30. Identify the equilibrium solution of the differential equation $\frac{dy}{dt} = y(1-y)$ A) $y = 0$. B) $y = 1$. C) $y = 2$. D) Both A and B. Show Answer Correct Answer: D) Both A and B. 31. The complete solution of partial differential equation z=px-qy+6pq is ..... A) Z=ax-by+6xy. B) Z=px-qy+6ab. C) Z=ax-by-6ab. D) Z=ax-by+6ab. Show Answer Correct Answer: D) Z=ax-by+6ab. 32. What is Bessel's equation? A) A first order linear differential equation. B) A second order linear differential equation. C) A non-linear differential equation. D) A polynomial equation. Show Answer Correct Answer: B) A second order linear differential equation. 33. The Clairaut's equation is of the form ..... A) Y=px+f(p). B) X=qy+f(p). C) X=py+f(q). D) Y=px+f(q). Show Answer Correct Answer: A) Y=px+f(p). 34. The following Differential Equation is $\frac{\text{d}y}{\text{d}x}=y^2+xy^2$ A) Separable. B) Non Separable. C) All the above. D) None of the above. Show Answer Correct Answer: A) Separable. 35. Find the nature of the one-dimensional wave equation. A) Hyperbolic. B) Parabolic. C) Elliptic. D) None of These. Show Answer Correct Answer: A) Hyperbolic. 36. L$^{-1 }$\{(s-a)/((s-a)$^{2}$+b$^{2}$)\} A) E$^{at }$cos bt. B) Cos bt. C) T$^{n }$e$^{at}$. D) E$^{at }$sin bt. E) Sin bt. Show Answer Correct Answer: A) E$^{at }$cos bt. 37. The original value of a painting is $ 1400, and the value increases by 9% each year. Write an exponential growth function to model this situation. A) Y=1.09(1400)$^{x}$. B) Y=1.09$^{x}$. C) Y=1400(1.09)$^{x}$. D) Y=1400(.91)$^{x}$. Show Answer Correct Answer: C) Y=1400(1.09)$^{x}$. 38. Which differential equation is separable? A) $y" +y=0$. B) $\dfrac{dy}{dx}+y=e^x$. C) $\dfrac{dy}{dx}=xy$. D) $\dfrac{dy}{dx}=x+y$. Show Answer Correct Answer: C) $\dfrac{dy}{dx}=xy$. 39. What is a differential equation? A) An equation involving only derivatives of a function. B) An equation involving only integrals of a function. C) An equation that relates a function and its derivatives. D) An equation that relates a function and its integrals. Show Answer Correct Answer: C) An equation that relates a function and its derivatives. 40. What are roots of unity in complex numbers? A) Numbers that result in 1 when raised to a certain power. B) Numbers that result in a complex number when raised to a certain power. C) Numbers that result in-1 when raised to a certain power. D) Numbers that result in 0 when raised to a certain power. Show Answer Correct Answer: A) Numbers that result in 1 when raised to a certain power. 41. Find the order and degree of the differential equation are respectively: A) 3, 7. B) 3, 2. C) 2, 3. D) 7, 2. Show Answer Correct Answer: C) 2, 3. 42. The z-transform of a signal X(n) whose definition is given by X(z)= $\sum_{n=0}^{\infty}x\left(n\right)z^{-n}$ A) Unilateral Z-transform. B) Bilateral Z-transform. C) Rational Z-transform. D) None of the above. Show Answer Correct Answer: A) Unilateral Z-transform. 43. F(s) = L\{(t$^{n}$)\} (s) A) S/(s$^{2}$+b$^{2}$). B) N!/(s-a)$^{(n+1)}$. C) N!/s$^{(n+1)}$. D) 1/s. E) 1/(s-a). Show Answer Correct Answer: C) N!/s$^{(n+1)}$. 44. The general solution of $y" +y'=0$ A) $y=c_1-c_2e^x$. B) $e^{-x}$. C) $y=c_1+c_2e^{-x}$. D) $y=c_1+c_2e^x$. Show Answer Correct Answer: C) $y=c_1+c_2e^{-x}$. 45. For a continuous function, the mixed partials satisfy: A) Always zero. B) F xy = f yx. C) F xy =-f yx. D) Only defined for polynomials. Show Answer Correct Answer: B) F xy = f yx. 46. Separate the variables for the equation $\frac{dy}{dx}=\frac{(x^2-1)}{(y^2+1)}$ A) $\frac{dy}{(y+1)}=\frac{dx}{(x-1)}$. B) $(y^2+1)dy=(x^2-1)dx$. C) $(y^2-1)dy=(x^2+1)dx$. D) $(y^2-1)dy=\frac{1}{(x^2+1)}dx$. Show Answer Correct Answer: B) $(y^2+1)dy=(x^2-1)dx$. 47. Given the differential equation dP/Dt=5PA) Find the general solution for P to the differential equationB)Find the particular solution for P to the differential equation given P(0)= 418 A) P=P$_{0}$e$^{5t}$B) P= 418e$^{5t}$. B) P=P$_{0}$e$^{10t}$B) P= 418e$^{10t}$. C) P=P$_{0}$e$^{5t}$B) P= 5e$^{418t}$. D) P=P$_{0}$e$^{t}$B) P= 418e$^{t}$. Show Answer Correct Answer: A) P=P$_{0}$e$^{5t}$B) P= 418e$^{5t}$. 48. The dimension theorem states that A) Rank(T)=Dim(V)-Nullity(T). B) Rank(T)=Dim(V)+Nullity(T). C) Rank(T)+Dim(V)=Nullity(T). D) None. Show Answer Correct Answer: A) Rank(T)=Dim(V)-Nullity(T). 49. Solve $\frac{\partial^3z}{\partial x^3}-2\frac{\partial^3z}{\partial x^2\partial y}=0$ A) $Z=f_1\left(x\right)+yf_2\left(x\right)+f_3\left(y+2x\right)$. B) $Z=f_1\left(y\right)+f_2\left(y+x\right)+f_3\left(y+2x\right)$. C) $Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y+2x\right)$. D) $Z=f_1\left(y\right)+f_2\left(y-x\right)+f_3\left(y+2x\right)$. Show Answer Correct Answer: C) $Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y+2x\right)$. 50. $.$ $\left(D^2-5DD'+6D'^2\right)z=e^{\left(x+y\right)}$ A) $\frac{1}{3}e^{\left(x+y\right)}$. B) $e^{\left(x+y\right)}$. C) $\frac{1}{4}e^{\left(x+y\right)}$. D) $\frac{1}{2}e^{\left(x+y\right)}$. Show Answer Correct Answer: A) $\frac{1}{3}e^{\left(x+y\right)}$. 51. The following Differential Equation is $\frac{\text{d}y}{\text{d}x}=\frac{e^y}{e^x}$ A) Separable. B) Non Separable. C) All the above. D) None of the above. Show Answer Correct Answer: A) Separable. 52. The population $P(t)$ $\frac{dP}{dt}=3P\Big(4-\frac{P}{6000}\Big)$ A) $24000$. B) $\tfrac13$. C) $4$. D) $6000$. Show Answer Correct Answer: A) $24000$. 53. $2\frac{\text{d}^2y}{\text{d}x^2}+\frac{\text{d}y}{\text{d}x}-y=2x^2+1$ A) $y=Ae^x+Be^{-x/2}+2x^2+4x-13$. B) $y=Ae^x+Be^{-x/2}-2x^2-4x-13$. C) $y=Ae^{-x}+Be^{x/2}-2x^2-4x-13$. D) $y=Ae^{-x}+Be^{x/2}+2x^2+4x-13$. Show Answer Correct Answer: C) $y=Ae^{-x}+Be^{x/2}-2x^2-4x-13$. 54. The particular solution of the differential equation $\frac{\text{d}y}{\text{d}x}+2y=x$ A) $y=\frac{1}{2}x^2e^{-2x}$. B) $y=\frac{e^{-2x}+2x-1}{4}$. C) $y=\frac{1-e^{-2x}}{2}$. D) $y=1-e^{-2x}$. Show Answer Correct Answer: B) $y=\frac{e^{-2x}+2x-1}{4}$. 55. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year.Find the general solution. A) P=1000e$^{ln(1.05)t}$. B) P=lne$^{1000t}$. C) P=.05e$^{1000t}$. D) P=1000e$^{ln(5)t}$. Show Answer Correct Answer: A) P=1000e$^{ln(1.05)t}$. 56. 83% of a certain airline's flights depart on time. Of these flights, 90% also arrive on time. 30% of flights from this airline that depart late manage to make up time in the air to still arrive on time.You see a flight from this airline arriving on time. What is the probability that it departed on time? A) 0.9361. B) 0.8532. C) 0.9721. D) 0.9608. Show Answer Correct Answer: A) 0.9361. 57. Which of the following is an application of PDEs in engineering? A) Calculating stress in structural beams. B) Analyzing electrical circuits in devices. C) Modeling heat conduction in materials. D) Simulating fluid flow in pipes. Show Answer Correct Answer: C) Modeling heat conduction in materials. 58. Find the general solution of the differential equation y" +6y'-16y=10x-32 A) $y=Ae^{2x}+Be^{-8x}-x+2$. B) $y=\left(e^{2x}+e^{-8x}\right)\left(A+Bx\right)$. C) $y=Ae^2+Be^{-8}-x+2$. D) $y=Ae^{2x}+Be^{-8x}$. Show Answer Correct Answer: A) $y=Ae^{2x}+Be^{-8x}-x+2$. 59. The nature of PDE u$_{xx }$+4 u$_{xy }$+3 u$_{yy}$, =0 A) Hyperbolic. B) Circular. C) Parabolic. D) Elliptic. Show Answer Correct Answer: A) Hyperbolic. 60. What is the method of reduction of order used for? A) To solve non-linear equations. B) To convert a linear differential equation to a lower order. C) To find the Wronskian. D) To determine linear independence. Show Answer Correct Answer: B) To convert a linear differential equation to a lower order. Next →Related QuizzesScience QuizzesClass 12 QuizzesClass 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 8Class 12 Mathematics Chapter 9 Differential Equations Quiz 9 🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books