MATH 1010 Sec. 4 Solution Midterm 1
October 21, 2010
(Total: 100 points + extra credits: 5 points)
You have to show all the work on the space given. Please be neat! If you need more
space, you can use extra sheets of paper provided. (Please put your name on all the
sheets and label question numbers.) Not showing your work gets 0 point. Where
appropriate, clearly indicate your final answer by circling it. You are not allowed to
get help from your textbook, class notes, calculators, other students, or any other form
of outside aid. Do not forget to turn off your cell phone. If you have any question,
please ask the instructor. Good luck!
1. (Total: 10 points) Solve the following equations
(a) (5 points) 5x − 4 + 2(2x − 3) = 12x − 3(5x + 1)
(b) (5 points) |5x − 9| = 3x + 1
(a)
5x − 4 + 2(2x − 3) = 12x − 3(5x + 1)
5x − 4 + 4x − 6 = 12x − 15x − 3
9x − 10 = −3x − 3
9x + 3x = 10 − 3
12x = 7
7
x=
12
(b)
|5x − 9| = 3x + 1
We have two possibilities: 5x − 9 = 3x + 1 or 5x − 9 = −3x − 1 In the first case we
get x = 5. In the second case we get x = 1. Therefore, the solutions are x = 5 and
x = 1.
1
2. (10 points) You are driving from Rome to Paris, a trip of 1000 kilometers. Your average
speed is 50 kilometers per hour. How long does it take you to reach your destination?
Let time = t. Since
speed =
we have
distance
time
1000
t
1000
= 20
t=
50
It takes 20 hours to go from Rome to Paris.
50 =
2
3. (Total: 10 points) Solve the following linear inequalities
(a) (5 points) 3x − 4 > 2
(b) (5 points) 4x + 5 ≤ x + 2
(a) 3x − 4 > 2
3x > 6
x>2
The set of solutions is {x | x > 2}.
(b) 4x + 5 ≤ x + 2
4x − x ≤ 2 − 5
3x ≤ −3
x ≤ −1
The set of solutions is {x | x ≤ −1}.
3
4. (5 points) The number 6 is 30% of what number?
Let n be a number
6 = 30% × n
6
6
n=
=
= 20
30%
0.3
5. (10 points) Determine whether the set of points {A = (0, 4), B = (4, 1), C = (3, 0)} is
collinear. Explain why.
The distance between A and B is
p
√
√
d1 = (4 − 0)2 + (1 − 4)2 = 16 + 9 = 25 = 5
The distance between A and C is
p
√
√
d2 = (3 − 0)2 + (0 − 4)2 = 9 + 16 = 25 = 5
The distance between B and C is
p
√
√
d3 = (4 − 3)2 + (1 − 0)2 = 1 + 1 = 2
Since d1 6= d2 + d3 , the points are not collinear.
4
6. (5 points) Rewrite 6, 300, 000 into scientific notation.
6, 300, 000 = 6.3 × 106
7. (5 points) Simplify
6x2 y
3x5 y −3
2y 1+3
2y 4
6x2 y
=
=
3x5 y −3
x5−2
x3
5
8. (10 points) Sketch the graph of the line given by 8x + 4y = 3. Then find the slope of the
line, the x-intercept, and the y-intercept.
The line is y = −2x + 43 .
Slope: −2.
x-intercept: ( 38 , 0).
y-intercept: (0, 43 ).
y
x
6
9. (Total: 10 points) Let f (x) = x2 − 3x − 1. Find each value of the function.
(a) f (−2) = (−2)2 − 3(−2) − 1 = 4 + 6 − 1 = 9
(b) f (3) + f (0) = 32 − 3(3) − 1 + (−1) = 9 − 9 − 1 − 1 = −2
10. (10 points) Solve the following system of equations
−x + 2y = 5
2x − 3y = 1
From the first equation x = 2y − 5. Substituting it in the second equation we have
2(2y − 5) − 3y = 1
4y − 10 − 3y = 1
Therefore, y = 11. Substituting the value of y in x = 2y − 5 we get
x = 2(11) − 5 = 22 − 5 = 17
The solution is (17, 11).
7
11. (10 points) Solve the following system of linear equations

 x − 3y + z = 6
3y + 2z = 5

3y − z = 2
Do R3 − R2 in the third row and get

 x − 3y + z = 6
3y + 2z = 5

−3z = −3
Divide by −3 the third row

 x − 3y + z = 6
3y + 2z = 5

z = 1
Now substituting in the second row we get y = 1. Substituting y = 1, z = 1 in the first
equation we find x = 8. The solution is (8, 1, 1).
8
12. (10 points) Sketch the graph of the following

 x−y
x

y
The lines are y = x − 5, x = 3, and y = 0.
y
y=0
x=3
x
y=x−5
9
system of linear inequalities
< 5
> 3
≤ 0
13. (Extra problem: 5 points) An object is dropped from the top of the Tower of Pisa, which
is 180 feet high. The initial velocity is 0 feet per second. Use the position function
h(t) = −16t2 + v0 t + s0
to find the time when the object hits the ground. (You can leave the solution as a fraction).
We have
h(t) = −16t2 + 180 = 0
16t2 = 180
180
t2 =
16
r
180
t=
16
The object hits the ground in
q
180
16
seconds.
10