MTH401 Midterm MCQs 2024 Set 2 by Miss Mehwish

MTH401 (Live Quiz) · MID TERM · Past Papers for Mids./Finals are also Available

1. The annihilator operator for the function, g(x) = 2e^x, is

D+5 D+1 ✔ D D²

2. Consider the non-homogeneous linear differential equation,

the input function, g(x)=x2e5xg(x) = x^2 e^{5x}, then the assumed particular solution ypy_p could be (Ax + B)e^{5x} ✔ (Ax² + Bx + C)e^{5x} Ae^{5x} cos 5x + Be^{5x} sin 5x Ae^{5x}

3. If y=c1eαx+c2e4xy = c_1 e^{\alpha x} + c_2 e^{4x} is the complementary solution of

✔ y=c1eαx+c2e4xy = c_1 e^{\alpha x} + c_2 e^{4x}

y=c1eαx+c2e−αxy = c_1 e^{\alpha x} + c_2 e^{-\alpha x}

4. If the auxiliary equation am2+bm+c=0a m^2 + b m + c = 0 has the complex roots, m1=α+iβm_1 = \alpha + i\beta and m2=α−iβm_2 = \alpha - i\beta, where α\alpha and β>0\beta > 0, then the solution of the differential equation, ay′′+by′+cy=0ay'' + by' + cy = 0, could be y=c1e(α+iβ)xy = c_1 e^{(\alpha+i\beta)x}

y=c2e(α−iβ)xy = c_2 e^{(\alpha-i\beta)x}

Q5. If y=2+xy = 2 + x, then which of the following is true for it?

Its annihilator operator is D Its annihilator operator is D³ ✔ Its annihilator operator is D² Its annihilator operator is D + 1

6. Consider the non-homogeneous linear differential equation

the input function, g(x)=sin⁡(4x)g(x) = \sin(4x), then the assumed particular solution ypy_p could be Ae4xA e^{4x} (A+B)(cos⁡4x+sin⁡4x)(A + B)(\cos 4x + \sin 4x) Ax2+Bx+CA x^2 + B x + C ✔ Acos⁡4x+Bsin⁡4xA \cos 4x + B \sin 4x

7. An annihilator operator LL for the function f(x)=e3xf(x) = e^{3x}, is

✔ D−3D - 3 3e3x3 e^{3x} e3xe^{3x} D−e3xD - e^{3x}

8. The polynomial linear differential operators with constant coefficients

✔ distributive identical inverse commute

9. The solution of the auxiliary equation determines the

order ✔ solutions linearity degree

10. For non-homogeneous differential equations, the form of the particular solution, yp=yp1+yp2+⋯+ypny_p = y_{p1} + y_{p2} + \dots + y_{pn}, is a linear combination of all the linearly functions generated by repeated differentiation of the input function g(x)g(x). ✔ independent dependent
11. For the non-exact differential equation M(x,y)dx+N(x,y)dy=0,M(x, y)dx + N(x, y)dy = 0,M(x,y)dx+N(x,y)dy=0, if ∂N∂x−∂M∂y\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} ∂x∂N −∂y∂M is a function of yyy, then the integrating factor is a: ❌ multi-variable function of both x and y ✔ function of y ❌ function of x ❌ function of dx
12. The differential equation: dydx+5(ddx)3y−3y=esin⁡x\frac{dy}{dx} + 5 \left( \frac{d}{dx} \right)^3 y - 3y = e^{\sin x} dxdy+5(dxd)3y−3y=esinx is an example of: ❌ linear ❌ Bernoulli ✔ non-exact ❌ exact