Free Printable Engineering. Maths int 2

1.

What is an antiderivative?

a)

The opposite of a derivative

b)

The same as a derivative

c)

A second derivative

d)

It always represents velocity.

2.

What does C represent in an antiderivative?

a)

A variable

b)

A constant

c)

None of these

d)

Unknown

3.

These are methods for solving 1st order differential equations, except

a)

Separating the variables

b)

Homogeneous

c)

Nonlinear

d)

Exact

4.

A differential equation is called separable if it can be written in the form of g(y) dy = h(x) dx. True or false?

a)

True

b)

False

5.

A differential equation dy/dx = f(x, y) is called homogeneous if f(λx, λy) = λf(x, y). True or false?

a)

True

b)

False

6.

There are many applications of 1st order differential equations such as

a)

Newton's law of cooling

b)

Population growth

c)

Mechanical motions

d)

Free oscillation

7.

For undetermined coefficient method, we can solve the above equation with g(x) = ?

a)

constant

b)

cubic function

c)

exponential function

d)

logarithmic function

8.

The nonhomogeneous 2nd order differential equation can be solved by

a)

separating the variables

b)

undetermined coefficient method

c)

variation of parameters method

d)

wronskian method

9.

Identify the type of solution.

a)

Infinite solutions

b)

One solution

c)

No solution

10.

What is the definition of one solution?

a)

no matter what constant is substituted in for the variable, the equation is ALWAYS true

b)

no matter what constant is substituted in for the variable, the equation will NEVER be true

c)

only this ONE solution will make the equation true

11.

3r - 5 = 2r

a)

No solution

b)

Infinitely many solutions

c)

r = 5

12.

Using substitution, which of the following equations are solutions to the partial differential equation?

a)

b)

c)

d)

13.

The partial differential equation

is classified as

a)

elliptic

b)

parabolic

c)

hyperbolic

d)

none of the above

14.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=x^2y+x$

a)

Separable.

b)

Non Separable.

15.

dy/dx = 4x/y. Suppose y(0)=1

The particular solution is

a)

B

b)

C

c)

D

d)

E

16.

Is this a Linear Differential equation

a)

Yes

b)

No

c)

Maybe

d)

red

17.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=e^x$

a)

$1=e^x+C$

b)

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

c)

$y=e^x+C$

d)

$0=e^x+C$

18.

The degree of a differential equation is define by a

a)

Positive real number

b)

Positive rational number

c)

Positive integer

d)

All the above

19.

The complete solution 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

20.

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)

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

21.

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)

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

22.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

$y=\ln\left|15t\right|$

b)

$y=16t^3$

c)

$y\ =\ \sqrt{2t^3-16}$

d)

$y=2t^3-16$

23.

If $\int_{-5}^3g\left(x\right)dx=-2\$ and $\int_{-5}^{15}g\left(x\right)dx=9$

Find the value of $\int_{15}^3\ \ g\left(x\right)dx$

a)

7

b)

11

c)

- 11

d)

- 7

24.

$\int3x^2\ +2x^{3^{ }}\ dx\ =$

a)

$6\ x+6\ x^2\ +C$

b)

$x^{3^{ }}\ +\frac{1}{2}\ x^2\ +C$

c)

$2x^{3\ }\ \ +\ 3x\ +\ C$

d)

$\frac{1}{2}x^3\ +\ 3x^2\ +C$

25.

$\int\sin\ 3x\ +\ \cos\ 2x\ dx$

a)

$\frac{1}{3}\cos\ 3x\ -\ \frac{1}{2}\ \sin\ 2x\ +C$

b)

$-\ \frac{1}{3}\sin\ 3x\ +\ \cos\ 2x\ +\ C$

c)

$-\ \cos\ 3x\ \ +\ \sin\ 2x\ +C$

d)

$-\frac{1}{3}\ \cos\ 3x\ +\ \frac{1}{2}\ \sin\ 2x\ +\ C$

26.

$\int\left(\sin\ 6x\ \cdot\cos\ 3x\right)\ dx$

a)

$-\frac{1}{18}\cos\ 9x\ -\frac{1}{6}\cos\ 3x\ +\ C$

b)

$-\frac{1}{18}\sin\ 9x\ -\ \frac{1}{6}\sin3x\ +\ C$

c)

$\frac{1}{9}\cos\ 9x\ +\frac{1}{3}\cos\ 3x\ +C$

d)

$\frac{1}{6}\sin\ 6x\ +\ \frac{1}{3}\sin\ 3x\ +\ C$

27.

$\int2x\ \left(x^2\ +\ 4\right)^4\ dx$

a)

$\frac{1}{10}\left(x^2\ +\ 4\right)^{5\ \ \ }+\ C$

b)

$\frac{2}{5}x\ \left(x^2\ +\ 4\right)^5\ \ +\ C$

c)

$\frac{1}{5}\left(x^2\ +\ 4\right)^5\ +\ C$

d)

$\frac{1}{3}\left(x^2\ +4\right)^4\ \ +C$

28.

$\int_{ }^{ }x\ \sin\ x\ dx$

a)

$-\ x\ \cos\ x\ +\sin\ x+\ C$

b)

$-x\ \sin\ x\ -\ \sin\ x\ +C$

c)

$x\ \cos\ x\ +\ \sin\ x\ +C$

d)

$x\ \sin\ x\ +\ \cos\ x\ \ +\ C$

29.

$\int2x\ \left(x-6\right)^4\ dx$

a)

$\frac{1}{5}x^2\ \ \left(x-6\right)^2\ \ +\ \frac{2}{3}\left(x-5\right)^3\ \ +C$

b)

$\frac{1}{5}x^{2\ }\left(x-6\right)^5\ -\frac{2}{3}\ \left(x-6\right)^{3\ \ \ }+C$

c)

$\frac{2}{5}x\ \left(x-6\right)^{4\ \ }-\frac{1}{5}\ \left(x-6\right)\ ^5\ \ +C$

d)

$\frac{2}{5}x\ \left(x-6\right)^{5\ }-\frac{1}{3}\left(x-6\right)^6\ \ +C$

30.

$\int\frac{x+1}{1-x^2}dx\ =$

a)

$\ln\left(1-x\right)\ +c$

b)

$-\ln\left(1-x\right)\ +c$

c)

$\frac{1}{\left(1-x\right)^2}+c$

d)

$-\frac{1}{\left(1-x\right)^2}+\ c$

31.

$\int\frac{1}{x^2}\ dx\ =$

a)

$x-2+c$

b)

$-\frac{2}{x^3}+c$

c)

$-\ \frac{1}{x}+c$

d)

$\frac{2}{x}+c$

32.

$\int_0^2\left(\frac{4x}{1+x^2}\right)dx=$

a)

$4\ln4$

b)

$4\ln5$

c)

$2\ln4$

d)

$2\ln5$

33.

$\int\frac{\ln x}{x}dx=$

a)

$\frac{1}{2}\ln^2x\ +\ c$

b)

$\frac{1}{x^2}+c$

c)

$\ln x\ +\ c$

d)

$\ln x\ +\ \frac{1}{x^2}\ +c$

34.

$\int f'\left(x\right)g\left(x\right)\ dx\ =$

a)

$\int f\left(x\right)g'\left(x\right)\ dx$

b)

$\int f\left(x\right)g\left(x\right)\ dx$

c)

$f\left(x\right)g\left(x\right)\ -\ \int f\left(x\right)g'\left(x\right)\ dx$

d)

$f\left(x\right)\cdot\int g\left(x\right)\ dx$

35.

What is the domain of the function

f\left(x,y\right)=\sin x\ln y

?

a)

All points of $\left(x,y\right)$

b)

$y\ne0$

c)

$y>1$

d)

$y>0$

36.

What is the range of the function

z=f\left(x,y\right)=\sin x\ln y

?

a)

$\left[-1,1\right]$

b)

$\left[-1,\infty\right)$

c)

$\left(-\infty,\infty\right)$

d)

$\left(-\infty,1\right]$

37.

What is the domain of the function

f\left(x,y,z\right)=e^z\ln\left(x^2+y^2\right)

?

a)

$\left(x,y\right)\ne\left(0,0\right)$

b)

All points of $\left(x,y,z\right)$

c)

$z\ge0$

d)

$z>0$

38.

Which of the following is the level curve of the function

f\left(x,y\right)=x\cos y

at

\left(1,\pi\right)

?

a)

$x\cos y=\pi$

b)

$x\cos y=1$

c)

$x\cos y=0$

d)

$x\cos y=-1$

39.

$If\ x\ =r\cos\ \theta\ and\ y\ =\ r\ \sin\ \theta\ ,\ then\ \frac{\partial\left(x,y\right)}{\partial\left(r,\theta\right)}\ =$

a)

$0$

b)

$-1$

c)

$r$

d)

$2$

40.

$If\ u,\ v\ ,\ w\ are\ functionally\ dependent\ functions\ of\ three\ independent\ variables\ x,\ \ y,\ \ z\ then\ \ \frac{\partial\left(u,v,w\right)}{\partial\left(x,y,z\right)}=$

a)

$1$

b)

$0$

c)

$-1$

d)

$3$

41.

If u is a homogeneous function of degree n in x and y then $x\ \frac{\partial u}{\partial x}+\ y\ \frac{\partial u}{\partial y}=\$

a)

$nu$

b)

$-nu$

c)

$n+u$

d)

$n-u$

42.

$If\ u\ =e\ ^x\ \sin\ y\ where\ x=st^2\ y=s^2t\ ,\ then\ \frac{\partial u}{\partial s}\$

a)

$2st\ e^x\ \sin y\ +\ s^2\ e\ ^x\cos y$

b)

$2st\ e^x\ \sin y\ -\ s^2\ e\ ^x\cos y$

c)

$\ e^x\ \sin y\ \ t^2+\ 2st\ \ e\ ^x\cos y$

d)

$\ e^x\ \sin y\ -\ s^{ }\ e\ ^x\cos y$

43.

If $\int_{-5}^3g\left(x\right)dx=-2\$ and $\int_{-5}^{15}g\left(x\right)dx=9$

Find the value of $\int_{-5}^3\ \ \frac{1}{4}g\left(x\right)dx$

a)

- $\frac{1}{2}$

b)

$\frac{1}{2}$

c)

- $\frac{9}{2}$

d)

$\frac{9}{2}$

44.

The methods of solving ODE that based on Taylor series expansion are

a)

Euler method

b)

second order Taylor series method

c)

Runge-Kutta method

d)

finite difference method

45.

Find the lower limit of z in the triple integral

\int_{ }^{ }\int_{ }^{ }\int_D^{ }\ \ f\left(x,y,z\right)dV

if D is the solid bounded by

z=4-y,\ z=y-6

and

y=x^2,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

46.

Find the upper limit of y in the triple integral

\int_{ }^{ }\int_{ }^{ }\int_D^{ }\ \ f\left(x,y,z\right)dV

if D is the solid bounded by

z=4-y,\ z=y-6

and

y=x^2,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

47.

1

a)

5

b)

1

c)

3

d)

7

48.

5

a)

1/3

b)

1/2

c)

3/4

d)

5/8

49.

7

a)

1/8

b)

1/9

c)

1/10

d)

1/2

50.

a)

0

b)

1

c)

2

d)

$\infty$

Engineering. Maths int 2

Total questions: 50

Worksheet time: 1hrs 18mins

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