1 Algebra / Trigonometry Quiz
Rationale: This math quiz is intended to test the skills you ought to
have in preparation for an introductory physics class. It emphasizes
word problems, and emphasizes the ability to think clearly and
carefully. It requires you to know some algebra and trigonometry, but
not assume you already know any calculus or physics.
Some of the questions that look hard may be easy, if you think about
them the right way. Some of the problems that look easy may be
slightly harder than they look.
Instructions: In all cases, show your work. Even if you can solve the
problem in your head, briefly explain how you solved it. Closed
book. No computers or electronic calculators; just pencil and paper.
- 1.
What is the remainder when 1,000,000,001 is divided by
9?
- 2.
As indicated in figure 1, we have two
squares, partially overlapping. The side of one is equal to the
diagonal of the other. We wish to compare the area of the red square
to the area of the blue square. The ratio of areas is
________.
- 3.
In general, the power dissipated in a resistor can be
calculated using the formula P = I2 R. Consider a situation where
the power is P = 50 watts and the resistance is R = 1 ohm. What
is I? Please answer in numerical form, accurate to 1% or better.
- 4.
You are given a blank square piece of plywood, 4 ft on a
side. The plan is to cut off the corners so as to make a regular
octagon, which you can then paint to make a giant stop sign, as shown
in figure 2. What is the length of each side of the
resulting octagon? Also, what is the length of the legs of the
triangles that get cut off? Please answer in numerical form, accurate
to 1% or better.
- 5.
We are given a cube 1 inch on a side. How long is
the diagonal of the cube? An algebraic / symbolic answer suffices.
- 6.
Which is bigger, the sine of one degree, or
the tangent of one degree?
- 7.
Find x such that x = √(6 + √(6 + √(6 +
⋯))) where “⋯” means “and so on, forever”.
- 8.
How many different ways are there of making change
for a quarter? You may use pennies, nickels, and dimes. (When
deciding what is a “different way” we do not care about the order in
which the coins are dispensed, just how many coins of each type are
used.)
Note for those who are unfamiliar with US currency: “quarter” = 25
cents; “dime” = 10 cents; “nickel” = 5 cents; “penny” = 1 cent.
- 9.
Let’s analyze a coin-tossing experiment. The coin is
lopsided such that it comes up “heads” 60% of the time. Each toss
is statistically independent of the others. We will toss the coin six
times. What is the probability that it will come up “heads” exactly
four times out of six?
- 10.
Given the parallelogram shown in
figure 3, find a general expression for the
length of the diagonal D in terms of the sides A and B and the angle
θ.
