This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 9 Differential Equations – Quiz 15 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 9 Differential Equations Quiz 15 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. What is the standard form of a first order linear differential equation? A) Y + P(x)y + Q(x)y = 0. B) Y = f(y). C) Y + y = 0. D) Y + P(x)y = Q(x). Show Answer Correct Answer: D) Y + P(x)y = Q(x). 2. What is the significance of boundary conditions in PDEs? A) Boundary conditions are irrelevant in PDEs. B) Boundary conditions only affect linear equations. C) Boundary conditions are used to simplify PDEs without affecting solutions. D) Boundary conditions determine the uniqueness and behavior of solutions in PDEs. Show Answer Correct Answer: D) Boundary conditions determine the uniqueness and behavior of solutions in PDEs. 3. What would the sign of the square root solution to a differential equation be given the initial condition g(1)=-5? A) Positive. B) Negative. C) All the above. D) None of the above. Show Answer Correct Answer: B) Negative. 4. What is the order of the PDE ( u ..... {xx} + u ..... {yy} = 0 )? A) 1. B) 4. C) 3. D) 2. Show Answer Correct Answer: D) 2. 5. $L^{-1}\left[\frac{1}{s^2}\right]=$ A) $0$. B) $x$. C) $x^2$. D) $1$. Show Answer Correct Answer: B) $x$. 6. What is the antiderivative of sine x in the context of separation of variables? A) Tangent x. B) Cosine x. C) -cosine x. D) Secant x. Show Answer Correct Answer: C) -cosine x. 7. Which of the following is the solution to the differential equation $\frac{dy}{dx}=2e^{3x}$ $y(\ln 2)=5$ A) $y=\frac{2}{3}e^{3x}-\frac{1}{3}$. B) $y=\frac{2}{3}e^{3x}+\frac{1}{3}$. C) $y=2e^{3x}-\frac{1}{3}$. D) $y=\frac{2}{3}e^{x}-\frac{1}{3}$. Show Answer Correct Answer: A) $y=\frac{2}{3}e^{3x}-\frac{1}{3}$. 8. If T is a linear transformation from V to W, which of the following is correct? A) R(T) is a subspace of V. B) R(T) is a subspace of W. C) N(T) is a subspace of W. D) None of the others. Show Answer Correct Answer: B) R(T) is a subspace of W. 9. Identify the type of differential equation:y" -4y = 0 A) Second-order linear homogeneous differential equation. B) Second-order non-linear differential equation. C) Third-order linear homogeneous differential equation. D) First-order non-linear differential equation. Show Answer Correct Answer: A) Second-order linear homogeneous differential equation. 10. Solve $x\left(y-2\right)\frac{\text{d}y}{\text{d}x}-\ln x=0$ $y=10$ $x=1$ A) $\left(y-2\right)^2=2\left(\ln x\right)^2-16$. B) $\frac{\left(y-2\right)^2}{2}=\frac{1}{2}\left(\ln x\right)^2+32$. C) $\frac{\left(\ln x\right)^2}{2}=-\left(y-2\right)^2+8$. D) $\left(\ln x\right)^2=-\frac{1}{2}\left(y-2\right)^2-4$. Show Answer Correct Answer: B) $\frac{\left(y-2\right)^2}{2}=\frac{1}{2}\left(\ln x\right)^2+32$. 11. SOLVE USING LAGRANGE'S LINEAR EQUATON4(p+q)=z A) F(x-y, y/4)=7. B) F(x-y, y/5-logz)=0. C) F(x-y, y/4-logy)=0. D) F(x-y, y/4 logz)=0. Show Answer Correct Answer: D) F(x-y, y/4 logz)=0. 12. The complementary function of the equation $2\frac{d^2y}{dt^2}+5\frac{\text{d}y}{\text{d}t}-12y=0$ A) $c_1e^{\frac{3}{2}t}+c_2e^{-4t}$. B) $c_1e^{\frac{3}{2}x}+c_2e^{-4x}$. C) $c_1e^{-\frac{3}{2}t}+c_2e^{4t}$. D) $c_1e^{-\frac{3}{2}x}+c_2e^{4x}$. Show Answer Correct Answer: A) $c_1e^{\frac{3}{2}t}+c_2e^{-4t}$. 13. If the PDE is of the type f(x, p)=g(y, q), then we will take ..... A) P=q. B) F(x, p)=a=g(y, q). C) P=a. D) P=aq. Show Answer Correct Answer: B) F(x, p)=a=g(y, q). 14. The particular integral of (D$^{2}$+1)y=x is A) 0. B) X$^{2}$. C) 1. D) X. Show Answer Correct Answer: D) X. 15. What is the characteristic equation / auxiliary equation for $\frac{\text{d}^2y}{\text{d}x^2}-2y=0$ A) $\lambda^2+2=0$. B) $\lambda^2-2\lambda=0$. C) $\lambda^2-2=0$. D) None of the above. Show Answer Correct Answer: C) $\lambda^2-2=0$. 16. If the PDE is of the type f(p, q)=0, then we will take ..... A) F(y, q)=a. B) F(x, p)=a. C) P=a. D) P=q. Show Answer Correct Answer: C) P=a. 17. The complete integral of $F_1\left(x, p\right)=F_2\left(y, q\right)$ A) $z=\int_{ }^{ }qdx+\int_{ }^{ }pdy$. B) $z=\int_{ }^{ }pdx-\int_{ }^{ }qdy$. C) $z=\int_{ }^{ }pdx+\int_{ }^{ }qdy$. D) None of the above. Show Answer Correct Answer: C) $z=\int_{ }^{ }pdx+\int_{ }^{ }qdy$. 18. In R$^{2}$, define (x, y)+(a, b) = (x+a-1, y+b-1) and $\alpha$(x, y)=($\alpha$x, $\alpha$y), which is of the following axiom of the vector space is fails: A) ($\alpha$ + $\beta$)u = $\alpha$u + $\beta$u. B) Identity. C) Associative. D) Commutative. Show Answer Correct Answer: A) ($\alpha$ + $\beta$)u = $\alpha$u + $\beta$u. 19. The roots of the auxiliary equation $m^2-7m+12=0$ A) -3, -4. B) 3, 4. C) -3, 4. D) 3, -4. Show Answer Correct Answer: B) 3, 4. 20. $1 + x + x^2 + x^3 + ..... = i$ A) $(1-x)^{-2}$. B) $(1-x)^{-1}$. C) $(1-x)^{-1}$. D) None of these. Show Answer Correct Answer: B) $(1-x)^{-1}$. 21. The singular solution of Clairaut's equation represents: A) The envelope of the family of lines. B) A particular integral. C) A complementary function. D) None. Show Answer Correct Answer: A) The envelope of the family of lines. 22. What mathematical operation is used to eliminate the reciprocal in an equation? A) Differentiation. B) Integration. C) Taking the reciprocal of both sides. D) Multiplying by the derivative. Show Answer Correct Answer: C) Taking the reciprocal of both sides. 23. The solution of linear differential equation $x\frac{\text{d}y}{\text{d}x}+2y=x^2$ A) $y=\frac{x^4+c}{4x^2}$. B) $y=\frac{x^2}{4}+c$. C) $y=\frac{x^2+c}{4x^2}$. D) $y=\frac{x^4+c}{x^2}$. Show Answer Correct Answer: A) $y=\frac{x^4+c}{4x^2}$. 24. A differential equation is an equation involving ..... of a function. A) Cube roots. B) Derivatives. C) Ranges. D) Logarithms. Show Answer Correct Answer: B) Derivatives. 25. Solve $(D^2 + 1)y = 0$ A) Y = Acosx + Bsinx. B) Y = Acosx-iBsinx. C) $e^{x}i$. D) None of these. Show Answer Correct Answer: A) Y = Acosx + Bsinx. 26. Differential equation of y=mx+c, where m, c are arbitrary constant is A) Y$^{"}$=0. B) Y$^{" '}$=0. C) Y$^{'}$+xy"=0. D) Y$^{'}$=0. Show Answer Correct Answer: A) Y$^{"}$=0. 27. The Wronskian of two solutions of a second-order linear differential equation is: A) Always constant. B) Always zero. C) May be zero or nonzero function of x. D) Either zero or nonzero constant. Show Answer Correct Answer: D) Either zero or nonzero constant. 28. Apa akar dari persamaan karakteristik untuk y" + 2y' + y = 0? A) R =-1 (akar ganda). B) R =-2 (akar tunggal). C) R = 1 (akar ganda). D) R = 0 (akar tunggal). Show Answer Correct Answer: A) R =-1 (akar ganda). 29. Determine whether the functions $y_{1} = e^{-2x}$ $y_{2} = e^{4x-5x}$ A) No, the functions are linearly dependent. B) The functions are identical. C) Yes, the functions are linearly independent. D) The functions are not defined. Show Answer Correct Answer: A) No, the functions are linearly dependent. 30. The complementary function is the solution of ..... A) Non linear equation. B) Homogeneous ordinary differential equation. C) Partial differential equation. D) Linear equation. Show Answer Correct Answer: B) Homogeneous ordinary differential equation. 31. Find the composite function of (D$^{3}$-4D$^{2}$D'+DD'$^{2}$)z=0 A) CF=f 1 (y)+f 2 (y+t). B) CF=f 1 (x)+f 2 (y+x)+f 3 (y+3x). C) CF=f 1 (y)+f 2 (y+x)+f 3 (y+4x). D) CF=f 1 (y)+f 2 (y+x)+f 3 (y+3x). Show Answer Correct Answer: D) CF=f 1 (y)+f 2 (y+x)+f 3 (y+3x). 32. If $x^{\alpha}y^{\beta}$ $\left(2x^2y-3y^4\right)dx-\left(3x^3+2xy^3\right)dy=0$ $\alpha$ $\beta$ A) $\alpha=-\frac{49}{13}, \\beta=-\frac{28}{13}$. B) $\alpha=\frac{49}{18}, \beta=-\frac{28}{18}$. C) $\alpha=\frac{49}{18}, \\beta=\frac{28}{18}$. D) $\alpha=-\frac{49}{13}, \\beta=\frac{28}{13}$. Show Answer Correct Answer: A) $\alpha=-\frac{49}{13}, \\beta=-\frac{28}{13}$. 33. Which method is suitable for solving a system of first-order linear ODEs? A) Matrix exponentials. B) Cramer's Rule. C) Taylor series only. D) Gaussian Elimination. Show Answer Correct Answer: A) Matrix exponentials. 34. Newton's Law of Cooling states that the rate at which an object cools is proportional to what? A) Difference in temperature of the object and the temperature of its surroundings. B) Temperature of surroundings. C) Temperature of the object. D) Object's mass. Show Answer Correct Answer: A) Difference in temperature of the object and the temperature of its surroundings. 35. If the roots of A. E are real and equal then C. F is C$_{1}$e$^{m1x}$+C$_{2}$e$^{m2x}$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 36. Differential equation representing the curve y=cx+c$^{2}$ where c being arbitrary constant is A) Y=xy+(y$^{'}$)$^{2}$. B) Y=xy$^{'}$+(y$^{' }$)$^{2}$. C) Y=xy$^{'}$+(y)$^{2}$. D) Y=xy$^{'}$ +y$^{2}$. Show Answer Correct Answer: B) Y=xy$^{'}$+(y$^{' }$)$^{2}$. 37. A differential equation is said to be ..... if it can be expressed as $f(x)dx=g(y)dy$ $f$ $g$ $x$ $y, $ A) Solved. B) Separable. C) Integrable. D) Differentiable. Show Answer Correct Answer: B) Separable. 38. Part B. Find the particular solution under the given conditions:13. y" + y'-6y = 0, y(0) = 2, y'(0) =-1 A) $y = e^{2x} + e^{-3x}$. B) Y = $2e^{2x}-e^{-3x}$. C) Y = $-e^{2x} + 2e^{-3x}$. D) $y = e^{2x}-2e^{-3x}$. Show Answer Correct Answer: A) $y = e^{2x} + e^{-3x}$. 39. What is the role of constants in linear combinations of functions? A) They are always zero. B) They must be integers. C) They determine the order of the equation. D) They can be any real numbers. Show Answer Correct Answer: D) They can be any real numbers. 40. A particle has acceleration given by a(t)=12t-4 with initial velocity-10. What is the velocity equation? A) 2t$^{3}$-2t$^{2}$. B) 12t-14. C) 6t$^{2}$-4t. D) 6t$^{2}$-4t-10. Show Answer Correct Answer: D) 6t$^{2}$-4t-10. 41. The Auxiliary equation of x$^{2}$y" -xy'+4y=cos(logx)+xsin(logx) is ..... A) M$^{2}$+2m+4=0. B) M$^{2}$+2m-4=0. C) M$^{2-}$2m+4=0. D) M$^{2}$+2m+2=0. Show Answer Correct Answer: C) M$^{2-}$2m+4=0. 42. Greenium has a half-life of 23 years and decays following the Law of Exponential Change. After how many years will 80% of an original amount have decayed? A) T = 74.0341 years. B) T = 7.4143 years. C) T = 10.2646. D) T = 53.4043 years. Show Answer Correct Answer: D) T = 53.4043 years. 43. It is an equation that contains derivatives A) Linear Equation. B) Differential Calculus. C) Polynomial Equation. D) Differential Equation. Show Answer Correct Answer: D) Differential Equation. 44. The solution of $\frac{dy}{dx}-y = e^{2x}$ A) Y = $Ce^{-x} + \frac{1}{3}e^{2x}$. B) Y = $Ce^{x} + \frac{1}{3}e^{2x}$. C) $y = Ce^{x} + e^{2x}$. D) $y = Ce^{x}-e^{2x}$. Show Answer Correct Answer: B) Y = $Ce^{x} + \frac{1}{3}e^{2x}$. 45. How do differential equations apply to CPS? A) Differential equations are used to model dynamic behaviors and interactions in Cyber-Physical Systems. B) Differential equations are primarily for static systems analysis. C) Differential equations are only used in financial modeling. D) Differential equations have no relevance to system dynamics. Show Answer Correct Answer: A) Differential equations are used to model dynamic behaviors and interactions in Cyber-Physical Systems. 46. Find the particular solution of the differential equation $\frac{dy}{dx}=12x^3y$ $y\left(0\right)=3$ A) $y=3e^{3x^4}$. B) $y=e^{x^4}$. C) $y=2e^{3x^4}$. D) $y=e^{2x^4}$. Show Answer Correct Answer: A) $y=3e^{3x^4}$. 47. $\left(1+x^2\right)\frac{\text{d}y}{\text{d}x}+xy=0$ A) $\ln\left|y\right|=\frac{1}{2}\ln\left|1+x^2\right|+c$. B) $-\frac{1}{y^2}=2\ln\left|1+x\right|+c$. C) $-\frac{1}{y^2}=\frac{1}{2}\ln\left|1+x\right|+c$. D) $\ln\left|y\right|=-\frac{1}{2}\ln\left|1+x^2\right|+c$. Show Answer Correct Answer: D) $\ln\left|y\right|=-\frac{1}{2}\ln\left|1+x^2\right|+c$. 48. A function $y=f\left(x\right)$ $y\frac{\text{d}y}{\text{d}x}=x$ $f\left(0\right)=1$ A) $f\left(x\right)=e^x$. B) $f\left(x\right)=x$. C) $f\left(x\right)=\sqrt{x^2+1}$. D) $f\left(x\right)=\sqrt{1-x^2}$. Show Answer Correct Answer: C) $f\left(x\right)=\sqrt{x^2+1}$. 49. $\frac{\partial x}{\partial y}$ A) Yes. B) No. C) Either. D) Neither. Show Answer Correct Answer: A) Yes. 50. How does feedback influence CPS behavior? A) Feedback decreases the adaptability of CPS behavior. B) Feedback has no impact on CPS behavior. C) Feedback enhances the adaptability and efficiency of CPS behavior. D) Feedback complicates the efficiency of CPS behavior. Show Answer Correct Answer: C) Feedback enhances the adaptability and efficiency of CPS behavior. 51. The general solution of (D$^{2}$-4)y=0 is ..... A) C$_{1}$e$^{x}$+c$_{2}$e$^{2x}$. B) C$_{1}$e$^{-2x}$+c$_{2}$e$^{-2x}$. C) C$_{1}$e$^{-2x}$+c$_{2}$e$^{2x}$. D) C$_{1}$e$^{3x}$+c$_{2}$e$^{2x}$. Show Answer Correct Answer: C) C$_{1}$e$^{-2x}$+c$_{2}$e$^{2x}$. 52. Find the particular solution of the differential equation $\frac{dy}{dx} = 6x$ $y(0) = 5$ A) $y = 3x^2 + C$. B) $y = 3x^2 + 5$. C) $y = 6x^2 + 5$. D) $y = 6x^2 + C$. Show Answer Correct Answer: B) $y = 3x^2 + 5$. 53. The formula for u'1 in Variation of Parameters is: A) U'1=-y1 f(x)/W(y1, y2). B) U'1=y'2 f(x)/W(y1, y2). C) U'1=y" 2 f(x)/W(y1, y2). D) U'1=-y2 f(x)/W(y1, y2). Show Answer Correct Answer: D) U'1=-y2 f(x)/W(y1, y2). 54. The order and degree of the differential equation $dy/dx + x^2 d^2y/dx^2 + xy = sin(x)$ A) 1, 1. B) 2, 1. C) 2, not defined. D) 3, 2. Show Answer Correct Answer: B) 2, 1. 55. How do you sketch a slope field for a given differential equation? A) Use a protractor to measure the angles of the lines on the coordinate plane. B) Use a compass to draw circles on the coordinate plane. C) Write out the equation in words instead of sketching it on the coordinate plane. D) Identify the equation and use small line segments to represent the slope at various points on the coordinate plane. Show Answer Correct Answer: D) Identify the equation and use small line segments to represent the slope at various points on the coordinate plane. 56. Differential equation of the curve y= Acos2x+B sin2x, where A & B are constant is A) Y$^{"}$-4y=0. B) Y$^{"}$+4y$^{'}$=0. C) Y$^{"}$+4y=0. D) Y$^{"}$-4y$^{'}$=0. Show Answer Correct Answer: C) Y$^{"}$+4y=0. 57. The solution z = ax + g(a)y +c is getting from A) F(x, p)=g(y, q). B) F(z, p, q)=0. C) F(p, q)=0. D) F(x, p, q)=0. Show Answer Correct Answer: C) F(p, q)=0. 58. What is the general form of a separable differential equation? A) Dy/dx = g(x)-h(y). B) Dy/dx = g(x) + h(y). C) Dy/dx = g(x) * h(y). D) Dy/dx = g(x) / h(y). Show Answer Correct Answer: C) Dy/dx = g(x) * h(y). 59. CHOOSE THE CORRECT FORM OF PARTIAL DIFFERENTIAL EQUATION BY ELIMINATING ARBITARY CONSTANTS z=ax+by+ab A) Z=py-rx+pq. B) Z=ax+by+ab. C) Z=py+q+p. D) Z=px+qy+pq. E) Z=px+qy-q. Show Answer Correct Answer: D) Z=px+qy+pq. 60. F(s) = L{(1)} (s) A) N!/s$^{(n+1)}$. B) N!/(s-a)$^{(n+1)}$. C) 1/(s-a). D) S/(s$^{2}$+b$^{2}$). E) 1/s. Show Answer Correct Answer: E) 1/s. ← 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