MAT 103 Pace University Math Worksheet

Pace UniversityMAT 103/103C Study Guide Test 3
MAT 103A Study Guide Test 4
Mathematics Department
Chapters 1-9
1. Parallel Lines and Perpendicular Lines.
(a) Find the slope of a line parallel to the line that crosses the point (−1, 2) and has a
y-intercept of 2.
(b) Find the slope of a line parallel to the line that crosses the point (−1, 2) and has a
y-intercept of 4.
(c) Find the slope of a line perpendicular to the line that crosses the points (−1, 2) and
(0, 2).
(d) Find the slope of a line perpendicular to the line that crosses the points (−1, 2) and
(2, 4).
2. Solve the following systems of equations by substitution or elimination method.
{
{
2x − 3y = −10
3x + y = −1
(a)
(b)
3x + 4y = 36
y = −2x + 1
3. Simplify the following expressions.
(a) x3 · x4 ÷ x5
(c) (−3x − 5) − (−x + 4)
(b) (2x ) · (−3x )
3 8
(d) (−2×2 − 4x − 3) − (−3×2 + 4x − 5)
2 3
Chapter 10
1. Simplify each of the following radical expressions. Assume all variable bases are positive
integers and all variable exponents are positive real numbers.
√
√
(a) 48
(e) x14 y 11 z 23
√
√
(b) 60
(f)
x36 y 30 z 49
√
√
(c) x4
(g) 3 125a6 y 9
√
√
(d) 8×5
(h) 3 8a16 y 20
2. Combine, if possible.
√
√
(a) 2 6 + 7 6
√
√
(b) 13 − 15
√
√
(c) 3 8 − 2 18
√
√
(d) 2 12 − 2 48
√
√
(e) 2 8 + 50
√
√
(f) 3 20 − 4 45
3. Multiply.
√ √
x· x
√
√
(b) 2x · 3x
√ √
x( x + 3)
√
√
(d) ( x + 1)( x − 2)
(a)
(c)
4. Solve the following radical equations.
(a)
(b)
√
√
x=4
(c)
−x = 4
(d)
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√
√
2x − 1 = 3
3x + 4 = 5
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(e)
(f)
√
√
MAT 103/103C Study Guide Test 3
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x+7=x−5
(g)
2x − 3 = x − 3
(h)
√
3
√
3
Mathematics Department
5x + 4 = 4
4x − 2 = −2
Chapter 11
1. Compute the discriminant to the following quadratic equations, and use it to identify the
number and kind of solutions.
(a) x2 − 2x + 1 = 0
(c) x2 − 2x − 3 = 0
(b) x2 + 2x + 3 = 0
(d) 2×2 + 3x − 4 = 0
2. Solve the following quadratic equations using square root property.
(a) 2×2 = 36
(b) 5×2 − 10 = 70
(x − 5)2
= 32
2
(d) 4 = (4x − 3)2 − 9
(c)
3. Solve the following quadratic equations using completing the square.
(a) x2 − 6x − 6 = 0
(c) 7×2 − 5x + 2 = 0
(b) 2×2 − 8x − 6 = 0
(d) 3×2 − 9x − 12 = 0
4. Solve the following quadratic equations using the quadratic formula.
(a) x2 − 6x − 6 = 0
(c) 7×2 − 5x + 2 = 0
(b) 2×2 − 8x − 6 = 0
(d) 3×2 − 9x − 12 = 0
5. Choose the best method to solve the following quadratic equations.
(a) y 2 − 6y + 9 = 36
(c) x2 − x − 1 = 0
(b) y 2 + 8y + 16 = 25
(d) 3×2 + 14x − 5 = 0
6. Find the vertex, axis of symmetry, x-intercepts (if exist) and y-intercept of the following
quadratic functions. Use what you found to sketch the function.
(a) f (x) = x2 + 4x − 12
(c) f (x) = 2×2 + 3
(b) f (x) = −x2 + 2x + 3
(d) f (x) = −2×2 − 12x − 12
Chapter 12
1. Graph each exponential functions.
(b) y = −3x
(a) y = 3x
( )x
1
2. Let f (x) = 3 and g(x) =
. Evaluate the following.
2
x
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(a) f (3) + g(2)
(c) f (4) · g(3)
(b) f (2) − g(3)
(d) f (3) ÷ g(2)
Mathematics Department
3. Find the inverse function of the following functions.
(a) f (x) = 2x
(c) f (x) = 2x + 3
(b) f (x) = −3x
(d) f (x) = −2x − 3
4. Solve the following exponential equations.
(a) 2x = 16
(c) 33x−2 = 81
(b) 5x = 125
(d) 42x+1 = 83x−1
5. Solve the following logarithmic equations.
(a) log4 x = 3
(e) log2 16 = x
(b) log5 x = 2
(f) log3 27 = x
(c) logx 16 = 4
(g) log4 16 = 2x
(d) logx 64 = 3
(h) log5 125 = 3x
Chapter 13
1. Find the distance between the points. Do not round your answers.
(a) (1, 2) and (3, 4)
(b) (−1, 2) and (3, −4)
2. Find the equation of the circle.
(a) The circle is centered at (1, −2) with a radius of 5.
(b) The circle is centered at (−3, −4) with a radius of 6.
Chapter 14
1. Find the 8th term and the sum of the first 8 terms of the arithmetic sequence: −5, −2, 1, 4, . . . .
2. Find the 10th term and the sum of the first 10 terms of the arithmetic sequence:
2
10 14
, 2, , , . . . .
3
3 3
3. Find the 10th term and the sum of the first 10 terms of the geometric sequence: 16, 8, 4, 2, . . . .
4. Find the 8th term and the sum of the first 8 terms of the geometric sequence:
2 4 8 16
, , , ,….
3 3 3 3
Word Problems
1. One a map, 1 centimeter corresponds to 4 miles. Find the length of a trail represented by
1
a line that is 2 centimeters long on the map.
2
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MAT 103/103C Study Guide Test 3
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Mathematics Department
2. A quality control inspector found 4 defective computer chips in a shipment of 500 computer
chips. At this rate, how many computer chips would be defective in a shipment of 2000
computer chips?
3. Bob and Chris have been assigned to do a job. It is known from before that it takes Bob
4 hours to complete the job alone, and it takes Chris 5 hours to complete the job along.
How long would it take the two employees to do the job together?
4. A ladder has been placed against a house. If the top of this ladder is six feet above the
ground, and the bottom of the ladder is ten feet from the base of the house, how long is
the ladder measured in feet? Do not round your answers.
5. When mixing a 500 ml of water and 800ml of a 30% alcohol solution, what is the alcohol
concentration of the new solution? Round your answer to the nearest percent.
6. A total of $1200 is deposited in two savings accounts for one year, part at 5% and the
remainder at 7%. If $72 was earned in interest, how much was deposited at 5%?
7. An object projected upward with an initial velocity of 48 feet per second will rise and fall
according to the equation s(t) = 48t − 16t2 , where s is the distance above the ground at
time t. At what time will the object be 20 feet above the ground?
8. The total weekly cost for a company to make x high quality watches is given by the formula
C(x) = 10x + 300. If the weekly revenue from selling all x watches is R(x) = 161x − 0.5×2 ,
how many watches must it sell a week to break even?
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