This test covers mathematical topics which you (technically) have seen the various mathematics courses prior to your enrollment in ECE 202. This test has been designed to help you and your instructor assess your skill level in the various mathematical operations that are necessary for your successful completion of ECE 202.

Please answer in the blanks below. Staple any additional sheets of paper which contain extra work.

  1) For the following simultaneous equation set:

(x-120) + (x-y)/20 + (x-z)/20 = 0

(y-x)/20 + y/2 +(y-z)/40 = 0

(z-x)/20 + (z-y)/40 + (z+120) = 0

transform these equations to the standard form:

a₁₁x+a₁₂y+a₁₃z = b₁

a₂₁x+a₂₂y+a₂₃z = b₂

a₃₁x+a₃₂y+a₃₃z = b₃

and solve for the numerical values of x,y, and z. Fill in the values for the coefficients and variables in the blanks below.

a₁₁ = ___22___ a₁₂ = ___-1__ a₁₃ = __-1________ b₁ = __2400_________

a₂₁ = ___-2___ a₂₂ = ____23__ a₂₃ = ___-1_______ b₂ = ____0__________

a₃₁ = ___-2__ a₃₂ = ___-1___ a₃₃ = ____43______ b₃ = ____4800_______

x = ___104.44_______ y = __4.44________ z = __-106.67______________

  2.) In the blank below, put the functional form (with all constants evalueated) of the solution of the following first order differential equation subject to the initail condition given:

dy/dt + 100y = 1000

y(0) = 0.

y(t) = ____10(1- e^-100t )_________________

  3.) Show that the function below

x(t) = A₁e^at+A₂e^bt

is the form of the solution to the second order differential equation

[(d²x)/(d²t)]+500[dx/dt]+40,000x = 0

with the initial conditions

x(0) = 200;[[dx/dt]]_{t = 0} = -17,000.

Find the numerical value of the constants in the solution above and provide them in the blanks below:

A₁ = ____80_____ A₂ = ___-80_____ a = ____-500________ b = ______0_________

4. Evaluate the definite integral shown below

z = ò¹⁰₀e^[(-x)/10]dx = _____ 10(1-e^-1) = 6.32 __________

  5. Evaluate the definite integral shown below

w(t) = ò^t₀xe^[(-x)/20]dx = _____ 20(t-20)e^-t/20+20 _______________

  6. Evaluate the following functions:

a) w(t) = ò^t₀20cos[([(2p)/10])x]dx = (100/p)sin([(2p)/10]x)

b) u(t) = [(d{5cos[(2000p)x]})/dx] = ___ -10,000psin(2000px) ___________________

7. The Euler identity is defined by

e^jq = cos(q)+jsin(q)

where j = [Ö(-1)]

and qis an angle, which may be specified in either degrees or radians.

a) For q= 45 degrees = [(p)/4] rad, find:

       z = 15e^-jq = ____ 15[(Ö2)/2] ___________ + j__ -15[(Ö2)/2] _________________.

b) For q= 30 deg = [(p)/6] rad, find:

        y = 10e^-jq = ____8.66_______ + j_____-5____________.

  8. Given the complex numbers:

 a=10+j6b=5 -j8, 

Find:

a) c = a + b = ____15____ + j____-2________

b) d = a -b = ____5______ + j____14_____

c) e = a * b = ___98_____ + j__-50______

e) f = a / b = ____2/89_______ + j___170/89________

9. Given the sinusoidal functions

       x(t) = 20sin(1000pt) = 20cos(1000pt+p/2) = 20Ð90deg = (0+j20)

y(t) = 50cos(1000pt+p/6) = 50Ð30deg = (43.3+j25)

a) Find u(t) = x(t) + y(t)

                   u(t)=___ 43.3+j45 = 62.5^Ð46.102 = 62.5(cos1000pt+46.102^°) __

b) Find v(t) = x(t) - y(t)

                    v(t)=_____ (-43.3-j5) = 43.59^Ð-173.41 = 43.59cos(1000pt-173.41) _______

  10. Plot the following functions:

a) y₁ =10x + 50

b) y₂ = 2x²

c) y₃ = 100cos[(2p/10)x]

d) y₄ = 150sin[(2p/20)x]