Math 0250 Exam 1Name:

February 9, 1998

F. Beatrous

1.

(a)

Express the complex number -2+2i in polar form $r e^{i\theta}$.

(b)

  Express the function $\cos t + 2 \sin t$ as the real part of a complex exponential. I.e., find complex numbers A and $\lambda$ so that the given function is the real part of $A e^{\lambda t}$.

(c)

The function given in part 1b oscillates with period $2\pi$. Use your answer to part 1b to find the amplitude of the oscillation.

2.

Find all solutions, if any,to the system

$$\left[\begin{array}{rrr} 2&1&5\\ 1&1&3\\ 1&-1&1 \end{array}\right] X = \left[\begin{array}{r}4\\ 3\\ -1\end{array}\right]$$

3.

Calculate the determinant of

$$\left[\begin{array}{rrr} 2&1&5\\ 1&1&3\\ 1&-1&1 \end{array}\right]$$

4.

Decide whether the matrix below is invertible, and if it is, calculate its inverse.

$$\left[\begin{array}{rrr} 2&1&-1\\ 0&-1&2\\ 0&0&1 \end{array}\right]$$

5.

Find all solutions to the differential equation

$$t \frac{dy}{dt} - 3 y = t, \qquad t>0$$

6.

A cup of coffee at 100 degrees centigrade is placed outdoors on a cold day, when the outdoor temperature is 0 degrees. In one minute, the coffee cools to 85 degrees. Assuming that it cools at a rate proportional to the difference between the coffee temperature and the air temperature (Newton's Law of Cooling), how long will it take for the temperature of the coffee to drop to 50 degrees?