Free Printable MATH122 - Engineering Mathematics II

1.

Degree of Differential equation is

a)

Power of highest order derivative

b)

Highest power

c)

Highest order derivative in DE

d)

None of above

2.

I.F. of $\frac{\text{d}x}{\text{d}y}+Px=Q$ is given by:

a)

$e^{\int Pdx_{ }}$

b)

$e^{\int Pdy_{ }}$

c)

$e^{\int Qdy_{ }}$

d)

$e^{\int Qdx_{ }}$

3.

What is homogeneous

a)

same degree

b)

same order

c)

different degree

d)

different order

4.

State the order and degree of this following Differential Equations

$\left(\frac{\text{d}^3x}{\text{d}y^3}\right)^5+xy=1-x$

a)

Order=5

Degree=3

b)

Order=1

Degree=1

c)

Order=1

Degree=0

d)

Order=3

Degree=5

5.

State TWO types of solutions of Differential Equations

a)

General Solutions Only

b)

General and Particular Solutions

c)

Particular Solutions Only

d)

No solutions

6.

Solve the differential equation

\frac{dy}{dx}+2=2y

.

a)

$y=1+\sqrt{\frac{4}{-2x-A}}$

b)

$y=\frac{\beta e^x+2}{2}$

c)

$y=1+\alpha e^{2x}$

d)

$y=1+\frac{1}{-4x-A}$

7.

What is the general solution to the DE $\frac{\text{d}^2x}{\text{d}y^2}+7\frac{\text{d}x}{\text{d}y}-8y=0$ ?

a)

$y=Ce^{-x}+De^{8x}$

b)

$y=Ce^x+De^{-8x}$

c)

$y=Ce^x+De^{7x}$

d)

$y=Ce^{-x}+De^{7x}$

8.

The roots of the auxiliary equation
$\frac{\text{d}x}{\text{d}y}+16y=0$ is

a)

$y=A\cos16x+B\sin16x$

b)

$y=A+Be^{-16x}$

c)

$y=A\cos4x+B\sin4x$

d)

$y=Ae^{4x}+Be^{-4x}$

9.

$\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}-6y=0$

Which of the following options are TRUE about the above DE?

a)

The roots of the auxiliary equations are two complex roots.

b)

The auxiliary equation has two different roots.

c)

The auxiliary equation has two equal roots.

d)

The equation is homogeneous.

10.

$x^2\frac{\text{d}^2x}{\text{d}y^2}-5y=x+1$ is a non homogeneous second order differential equations with constant coefficients.

a)

TRUE

b)

FALSE

11.

The complete solution(or complete integral ) of Linear Differential equations involves

a)

complete function + particular integral

b)

complementary function + particular integral

c)

complementary function + definite integral

d)

complete function + indefinite integral

12.

The particular integral of the equation $\left(D-1\right)y=e^{3x}$ is

a)

$\frac{e^{3x}}{2}$

b)

$\frac{e^{-3x}}{2}$

c)

$\frac{e^{3x}}{4}$

d)

$\frac{e^x}{2}$

13.

The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ is

a)

$c_1e^{-3x}+c_2e^{-x}$

b)

$c_1e^{3x}+c_2e^{3x}$

c)

$c_1e^{3x}+c_2e^{-x}$

d)

$c_1e^{3x}+c_2e^{-3x}$

14.

The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ is

a)

$c_1e^{-3x}+c_2e^{-x}$

b)

$c_1e^{3x}+c_2e^{3x}$

c)

$c_1e^{3x}+c_2e^{-x}$

d)

$c_1e^{3x}+c_2e^{-3x}$

15.

Number of arbitrary constant in the general solution of a differential equation of degree 3 and order 4 is

a)

3

b)

4

c)

0

d)

4³

16.

$\frac{\text{d}^2x}{\text{d}y^2}+6\frac{\text{d}x}{\text{d}y}-5y=0\$

The general solution to the DE is,

a)

$y=Ae^x+Be^{5x}$

b)

$y=Ae^{-x}+Be^{-5x}$

c)

$y=Ae^{\left(-3+\sqrt{14}\right)x}+Be^{\left(-3-\sqrt{14}\right)x}$

d)

$y=A\cos\left(-3+\sqrt{14}\right)x+B\sin\left(-3-\sqrt{14}\right)x$

17.

The P.I. of the equation $\left(D^3+D\right)y=\cos x\$ is

a)

$-\frac{x\cos x}{2}$

b)

$\frac{x\cos x}{2}$

c)

$-\frac{\cos x}{2}$

d)

$-\frac{x\cos x}{5}$

18.

$x^2\frac{\text{d}^2x}{\text{d}y^2}-5y=x+1$ is a non homogeneous second order differential equations with constant coefficients.

a)

TRUE

b)

FALSE

19.

$\frac{d^2y}{dx^2}+ay=\operatorname{cosec}x\$ can be solved by finding

a)

C.F and P.I

b)

singular solution

c)

General solution by variation of parameter method

d)

total differential equation steps

20.

Inverse Laplace transform, transform

a)

$f\left(t\right)\ to\ F\left(s\right)$

b)

$F\left(s\right)\ to\ f\left(t\right)$

c)

$f'\left(t\ \right)\ to\ f\left(t\right)$

d)

$f\left(t\right)\ to\ f'\left(t\right)$

21.

$L^{-1\ }\left(\frac{4s}{s^2+25}+\frac{2}{3\left(s^2+1\right)}\right)$

a)

$4\ \cos\ 25t+\frac{2}{3}\sin\ t$

b)

$\cos\ 5t+\ \sin\ t$

c)

$4\ \cos\ 5t\ +\frac{2}{3}\sin\ t$

22.

$L\left(t^{\left(4\right)}\right)$ ?

a)

$\frac{24}{s^5}$

b)

$\frac{4}{s^5}$

c)

$\frac{1}{s^5}$

23.

$L^{-1}\left(F\left(s-5\right)\right)=e^{5t}f\left(t\right)$ Based on first shifting property for Inverse Laplace transform, what is $a$ value?

a)

4

b)

5

c)

3

24.

Find the Inverse Laplace for $\frac{\left(2s-8\right)}{\left(s-2\right)\left(s-3\right)}$

a)

$4e^{-2t}-2e^{-3t}$

b)

$2e^{-2t}-4e^{-3t}$

c)

$e^{-2t}-e^{-3t}$

25.

By convolution theorem, find F(s) and G(s) $X\left(s\right)=\frac{1}{s\left(s^2+4\right)}$

a)

$F\left(s\right)=\frac{1}{s};G\left(s\right)=\frac{1}{s^2+4}$

b)

$F\left(s\right)=\frac{1}{s^{ }};G\left(s\right)=\frac{1}{s^3+4s}$

c)

$F\left(s\right)=\frac{1}{s^2+4};G\left(s\right)=\frac{1}{s^{ }}$

26.

Find: $L\left\{e^{3t}t\right\}$

a)

$\frac{1}{s^3}$

b)

$\frac{1}{\left(s-3\right)^2}$

c)

$\frac{1}{\left(s-1\right)^3}$

d)

$\frac{3}{s-1}$

27.

Find: $L\left\{\sinh\ t\right\}$

a)

$\frac{s}{s^2+1}$

b)

$\frac{s}{s^2-1}$

c)

$\frac{1}{s^2+1}$

d)

$\frac{1}{s^2-1}$

28.

$L\left[e^{-2t}\sin3t\right]$

a)

$L\left[\sin3t\right],\ s\rightarrow s+2$

b)

$L\left[\sin3t\right],\ s\rightarrow s-2$

c)

$L\left[e^{-2t}\right],\ s\rightarrow s-3$

d)

$L\left[e^{-2t}\right],\ s\rightarrow s+3$

29.

Define this property: $L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$

a)

Linearity Property

b)

First Shifting Property

c)

Convolution Theorem

d)

Second Shifting Property

30.

A complex number is denoted by______.

a)

C

b)

Z

c)

R

d)

z

31.

The argument of the product of two complex numbers is the _______ of the arguments of the complex numbers

a)

sum

b)

product

c)

inverse

d)

conjugate

32.

In polar coordinates, r is the ________.

a)

argument

b)

modulus

c)

diameter

d)

angle

33.

A function which is analytic everywhere in the finite plane is called _______.

a)

entire function

b)

analytic function

c)

differentiation

d)

continuous function

34.

Say TRUE or FALSE.

If a function is continuous at a point, then the function is differentiable at a point.

a)

TRUE

b)

FALSE

35.

Suppose f(z) is analytic in a domain, if its real part is a constant then f(z) is

a)

a continuous function

b)

a harmonic function

c)

an analytic function

d)

a constant function

36.

The value of $\ \ \int_c^{ }\frac{\left(3z^{2\ }+7z+1\right)}{z+1}dz$ where C is l Z l = 1/2 is

a)

$2\pi i$

b)

0

c)

$\pi i$

d)

$\frac{\pi i}{2}$

37.

$The\ value\ of\int_c^{ }\frac{\left(3z+4\right)}{z\left(2z+1\right)}dz\ \ \ \$ where C is the circle l Z l =1 is

a)

$2\pi i$

b)

$3\pi i$

c)

$4\pi i$

d)

$5\pi i$

38.

$The\ value\ of\int_c^{ }\frac{z}{\sin z}dz$ where C is l Z l = 4 is

a)

$2\pi i$

b)

0

c)

$-2\pi i$

d)

$4\pi i$

39.

$The\ poles\ of\frac{\left(z^3-1\right)}{z^3+1}$

a)

1, $\frac{1}{2}\left(-1\pm\sqrt{3}\right)$

b)

-1, $\frac{1}{2}\left(-1\pm\sqrt{3}\right)$

c)

1, $\frac{1}{2}\left(1\pm\sqrt{3}\right)$

d)

-1, $\frac{1}{2}\left(1\pm\sqrt{3}\right)$

40.

$The\ value\ of\int_c^{ }\frac{\left(4z^2+z+5\right)}{z-4}dz$ where C is $9x^2+4y^2=36$ is-----

a)

0

b)

1

c)

2

d)

3

41.

$Residue\ of\ f\left(z\right)=z\cos\left(\frac{1}{z}\right)\ at\ z=0$

a)

$\frac{1}{2}$

b)

$-\frac{1}{2}$

c)

1

d)

-1

42.

$Poles\ off\left(z\right)=\frac{z^2}{z^4+1}\ in\ \left|z\right|=2$

a)

$\pm\frac{1}{2}\pm\frac{i}{2}$

b)

$\pm\frac{1}{2}\pm\frac{i}{\sqrt{2}}$

c)

$\pm\frac{1}{\sqrt{2}}\pm\frac{i}{\sqrt{2}}$

d)

$\pm\frac{1}{\sqrt{2}}\pm\frac{i}{2}$

43.

$The\ zeroes\ and\ \sin gularities\ off\left(z\right)=\frac{\left(z^2+1\right)}{1-z^2}$

a)

$\pm1,\pm i$

b)

$\pm2,\pm2i$

c)

$\pm i,\pm1$

d)

$\pm3,\pm3i$

44.

The non-isolated singularities of f(z)=cot( $\frac{\pi}{z}$ )

a)

$\pm1,\pm2,--$

b)

$\pm1,\pm\frac{1}{2},---$

c)

1,2,--

d)

0

45.

$The\ function\ f\left(z\right)=\log z\ is$

a)

differentiable everywhere

b)

not differentiable everywhere

c)

not differentiable at z=0

d)

none of these

46.

$The\ \ function\ f\left(z\right)=\frac{1}{z-1}\ is$

a)

continuous everywhere

b)

not continuous at z=1

c)

not continuous everywhere

d)

none of these

47.

$The\ function\ f\left(z\right)=\frac{1}{z^2+1}is$

a)

not analytic at z=1

b)

not analytic at z= i and -i

c)

not analytic at z= 1 and -1

d)

none

48.

If f(z)= u+iv is analytic then

a)

both u and v are harmonic functions

b)

both u and v are harmonic functions and one is conjugate to other

c)

u and v are analytic

d)

none

49.

The value of $\ \ \ \ \ \ \ \ \ \int_c^{ }4x^3dx+3y^2z^2dy+2y^3zdz$ where C is any path joining A(-1,1,0) to B(1,2,1)is

a)

0

b)

1

c)

-8

d)

8

50.

What is the conjugate of -2+5i?

a)

-5+2i

b)

-2-5i

c)

2+5i

d)

5-2i

MATH122 - Engineering Mathematics II

Total questions: 50

Worksheet time: 40mins

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