MATH 131:100 Quiz 1
Diagnostic Test: Algebra
1) Evaluate each expression without using a calculator
523
a) (−3)4
b) −34
c) 3−4
d) 21
5
e)
−2
2
3
f) 16−3/4
2) Simplify each expression. Write your answer without negative exponents.
2/3 3 −2
√
√
3x y
3 3
2 2
a) 200 − 32
b) (3a b )(4ab )
c)
x2 y −1/2
3) Expand and simplify.
a) 3(x + 6) + 4(2x − 5)
b) (x + 3)(4x − 5)
d) (2x + 3)2
c)
√
a+
√ √
√ b
a− b
e) (x + 2)3
4) Factor each expression.
a) 4x2 − 25
b) 2x2 + 5x − 12
e) 3x3/2 − 9x1/2 + 6x−1/2
d) x4 + 27x
f) x3 y − 4xy
5) Simplify the rational expression.
a)
x2 + 3x + 2
x2 − x − 2
b)
x2
x+1
c) 2
−
x −4 x+2
2x2 − x − 1 x + 3
x2 − 9 2x + 1
d)
y
x
1
y
−
−
x
y
1
x
6) Rationalize the expression and simplify.
√
10
√
a)
5−2
√
b)
4+h−2
h
7) Rewrite by completing the square.
a) x2 + x + 1
b) 2x2 − 12x + 11
8) Solve the equation. (Find only the real solutions)
b)
2x
2x − 1
=
x+1
x
c) x2 − x − 12 = 0
d) 2x2 + 4x + 1 = 0
f) 3|x − 4| = 10
9) Solve each inequality. Write your answer using interval notation.
a) −4 < 5 − 3x ≤ 17
c) x(x − 1)(x + 2) > 0
1
d) |x − 4| < 3
10) State whether each equation is true or false. If it is false, provide a counterexample.
NOTE: This is probably the most important problem of this entire exercise because most
of these are the common mistakes that students keep making. Remember these so that you
don’t fall into the same trap!
a) (p + q)2 = p2 + q 2 .
b)
c)
√
√
ab =
√ √
a b.
a2 + b2 = a + b. We could just as well state this as
√
a+b=
√
a+
√
b.
d)
1 + TC
=1+T
C
e)
1
1
1
= − .
x−y
x y
f)
1/x
1
=
.
a/x − b/x
a−b
Note: In general, if you are unsure if something is true or not, try plugging in numbers to
see if it works. But you should typically avoid using 0 or 1 to test your equations, because
this can lead you to some wrong conclusions.
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Diagnostic Test: Analytic Geometry
1) Find an equation for the line that passes through the point (2, −5) and
a) has slope −3
b) is parallel to the x-axis
c) is parallel to the y-axis
d) is parallel to the line 2x − 4y = 3
2) Find an equation for the circle that has center (−1, 4) and passes through the point
(3, −2).
4) Let A = (−7, 4) and B = (5, −12) be two points in the plane.
a) Find the slope of the line that contains A and B.
b) Find an equation of the line that passes through A and B. What are the intercepts?
d) Find the length of the segment AB.
Diagnostic Test: Functions
1) You will have to look at the textbook for this one.
2) If f (x) = x2 , evaluate the difference quotient
f (2 + h) − f (2)
h
and simplify your answer.
NOTE: I changed problem 2; the textbook usese f (x) = x3 .
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