Download Math 20 * 1 Prerequisite Skills

Math 20 – 1 Prerequisite Skills 2014
The following are some questions that relate to the knowledge and skills you need to be fluent with going into
Math 20 – 1.
LINEAR RELATIONS
1. Determine whether each relation is linear or non-linear. Justify each answer.
a. A = πr 2
c. (0, 0), (1, 1), (4, 2), (9, 3), (16, 4)
b. y = 5x – 3
d. (2, 5), (4, 10), (6, 15), (8, 20), (10, 25)
2. Julian creates a number pattern that starts with the number 4. Each subsequent term is 5 less than the previous
term.
a. Create a table of values for the first five numbers in the pattern.
b. What equation can be used to represent the pattern? Verify your answer by substituting a known value
into your equation.
c. What is the value of the 49th term?
d. Which term has a value of 89?
3. Create a graph and a linear equation to represent the table of values.
𝑥
−𝟑
−𝟐
−𝟏
𝟎
𝟏
𝟐
𝟑
𝒚
−𝟖
−𝟓
−𝟐
𝟏
𝟒
𝟕
𝟏𝟎
4. Express each equation in slope-intercept form.
a. 2x + y = 6
d. 6x – y = 4
b. 3x + y + 9 = 0
e. 7x – y + 9 = 0
c. 5x + 6y = 8
f.
8x – 4y = 3
5. What are the slope and y-intercept of each line?
a. y  3x + 4
b.
1
y 
2
1
x 
5
3
Mrs. Connor
c. 3x  2y  7
d. 4.2  2y  3.6x
Math 20 – 1 Prerequisite Skills 2014
6. Write the equation of each line, using the given information.
a. passing through (3, 1) with slope, m  2
b. passing through (3  4) and perpendicular to y 
3
2
x7
7. Write a system of linear equations to model each situation.
a. The sum of two numbers is 752 and their difference is 174.
b. The total number of adult and youth tickets for a play is 256. Adult tickets cost $5 each, and youth tickets
cost $3 each. The total sales for one performance were $767.
c. A newspaper box contains quarters and loonies. The total number of coins is 73. The total value of the
coins is $37.
d. The membership fee at one dance club is $75 for the first year, plus $15 per month. The fee at another
dance club is $35
per month.
8. Predict the number of solutions for each system of linear equations. Explain how you made your prediction.
a. y  2x  3
y  2x  1
b. y  3x  10
2y  6x  20
c. 2x  3y  6  0
14x  21y  42  0
d. 2x  y  10  0
4x  y  30  0
9. Solve each system of linear equations by graphing. Express answers to the nearest tenth.
a. y  2x  6
y  2x  8
b. x  y  1
5x  4y  12
c.
2
x7
5
5
y x2
8
y 
d. 6x  5y  45
2x  5y  40
10. Solve each linear system by substitution.
a. y  3x  1
x  y  11
b. 2  y  3x
6x  5y  8
2
Mrs. Connor
c. 0.1y  0.3x  1.5
x  0.2y  5.6
d. 2x  6y  9
y  2x  4
Math 20 – 1 Prerequisite Skills 2014
11. Solve each linear system using the elimination method.
a. x  y  17
x  y  9
b.
y
 2x  3
2
3x  2 y 
9
2
c. 3x  2y  10
2x  y  4
d. x  7  y
2x  y  8
12. What are the domain and range of each function shown? State your answers using set notation.
a)
b)
3
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
13. What are the slope and y-intercept of each line?
a. y  6x  2
b.
1
y   x 3
2
14. Write the equation of each line in slopeintercept form.
a)
b)
c)
d)
4
Mrs. Connor
c. y  1.2  0.75x
d. 5x  3  2y
Math 20 – 1 Prerequisite Skills 2014
15. Write an equation for a line that passes through each pair of points.
a. (5, 1) and (3, 7)
c. (3, 6) and (0, 0)
b. (5, 8) and (1, 4)
d. (8, 3) and (4, 6)
GEOMETRY
1. Use the diagram to help answer the questions.
a. Name the angle at vertex A in two different ways.
b. What are two different ways to express the hypotenuse in ABC?
c. Express the Pythagorean relationship for ABC in two ways.
d. What is an expression for sin A?
e. Write the equation for tan C.
f.
If cos C =
a
, what is an expression for a?
b
2. Determine the length of the unknown side in each right triangle. Give the answer to the nearest tenth of a
centimetre.
a)
b)
c)
d)
5
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
3. Determine the numerical value for each expression, to two decimal places.
a. 4.5 tan 60°
b. 4.5 cos 60°
c.
d.
sin 56

tan 70
cos 7°
sin 84
– 0.56
cos 30
e. 4.9 +
f.
 3.1 tan 45
cos 78
4. Determine the measure of each angle, to the nearest degree.
a. tan C = 0.75
e. sin  =
b. tan B = 2.5108
c. cos  = 0.6779
d. cos A =
f.
1
2
sin  = 0.873
3
8
5. Ben and Sophie are skiing at Holiday Mountain Ski Resort located at La Riviere, MB. The beginner’s slope is
inclined an average of 12.1° from the horizontal. One advanced run has an average angle of elevation of 27.3°.
Ben skis 1000 m down the beginner’s slope and Sophie skis 1000 m down the advanced run.
a. Draw a diagram to represent the situation.
b. Determine the difference in the vertical distances the two friends ski. Give your answer to the nearest
metre.
ALGEBRAIC EXPRESSIONS
1. Perform the indicated operations. Simplify each answer.
a. 7x2 – 3x + x2 – x
c. (2x –5)2
b. (4x – 3)(x + 7)
d. (x – 1)2 – (2x + 3)(x – 4)
2. A linear function is expressed as g(x) = 3x – 8.
a. If you were to draw a graph of function g, how should you label the axes?
b. What is the value of g(–2)?
c. Is the point A(5, 7) on the graph of function g? Explain how you know.
d. What is the domain of function g?
3. What is the degree of each polynomial?
a. 6x – 3y + 1
b. 2x2 – 3x2y – 7y
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Mrs. Connor
c. 5x2 – 10 + 3y2
Math 20 – 1 Prerequisite Skills 2014
4. An ecologist investigating the effect of air pollution on plant life finds the percent p(x) of diseased plants at a
distance x kilometres from an industrial site is defined by the function p(x) = 40 –
3x
for 50 ≤ x ≤ 200.
50
a. Sketch a graph of function p. Title the axes and give the graph a title.
b. What is the value of each of the following: p(50), p(150), and p(200)?
c. What is the range of function p?
5. Multiply and combine like terms.
a. 3(2x – 7) – 4(x – 1)
c. (4x – 3)(2x + 5)
b. 5x(3x – 2)
d. (5x – 4)2
6. Factor fully.
a. 3xy – 8x2y
d. 4a2 – 9y2
b. 3p – 9p2
e. 8r2 + 20r + 8
c. x2 – 13x + 12
f.
2x2 – 0.08y2
7. Solve for x.
a. 7x – 3 = 2x – 5
c. (4x + 3)(x – 1) = (2x – 1)(2x + 1)
b. 19 – 2(x + 3) = 1
POWERS AND ROOTS
1. Which of the following numbers are perfect squares, perfect cubes, or both?
a. 144
c. 16
b. 2197
d. 64
2. Evaluate.
a.
49
c.
 81
b.
38
d.
196
961
3. Determine an integer that is equal to each expression.
7
a.
 3 27
c.
b.
144
d.
Mrs. Connor
3 17 576
Math 20 – 1 Prerequisite Skills 2014
4. Simplify each expression. State the answer using positive exponents.
a. (x3)(x–5)
b.
y 4
y 2
c.
t3
t7
d.
 g 1 
 g0 


3
5. Simplify each expression. Express the answer to four decimal places, when possible.
a.
 0.5 
b.
  2 3 
 3  
  
c.
 5  53 


2 3
 64 
 64 
 
e.
8
 3
8 
f.
 3 4  3  2 
 4    4  
  
 
3
 

d.
1

1
6. Simplify each expression by restating it using positive exponents only.
a.
 2
x  x5 
 
 
 
3
3
8
c.
 
d.
  x 2  


3
  xy  
d4


b.

2
m 2 3

 12 
m 


5
7. Write each expression as a power with a single rational exponent. Then, evaluate. Express the answer to four
decimal places, when possible.
a.
b.
 5
3  36 
 
 
 
2
c.
  3  3
(8)  4 4  
   
d.
 23  5
 82 
 
c.
3
s3
t5
5
5
y3
2
 52 
1

125 3 


4
8. Express each radical as a power.
a.
b.
12 p 3
3 4 x3
d.
8
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
9. Evaluate each expression. State the result to four decimal places.
a.
4 17
b.
(65) 3
c.
1
0.3(22) 2
36
7
d.
2
FACTORS
1. Determine each product.
a.
(3)  1 
d.
 3  2 
  
 8  3 
b.
(6)  
2
1
 
e.
 3  1 
  
 7  6 
c.
(2)  
5
6
 
f.
 3  4 
  
 4  9 
2
2. Express each product as an improper fraction.
a.
 1  3 
1  
 3  4 
c.
 3  2 
 2 1 
 7  3 
b.
 1  1 
 2  2 
 3  3 
d.
 3  4 
  2 
 5  13 
1
1

4
3
3. Determine each quotient.
a.
1
2
5
d.
b.
1
2
4
e. 1 
c.
2
6
3
f.
1
a.
3  1  2 
   
4  2  3 
c.
2  3 
5
   
3  7 
6
b.
2
3
5
9
10
5
3
2
12
4
4. Simplify.
9
1 4
1
  
5 5
4
Mrs. Connor
2
1
2
 1
d. 1 
 2   (1)  2  
3
 5  2 
 8
Math 20 – 1 Prerequisite Skills 2014
5. Multiply using the distributive property.
a. (x  5)(x  2)
c. (x  3)2
b. (c  d )(c  d )
d. (4j  2k)(6j  3k)
6. Multiply. Then, combine like terms.
a. (4n  2)  (2n  3)(3n  2)
c. (b  2d )  (5b  3d )  (b  d )(4b  d )
b. ( f  7)(2f  4)  (3f  1)2
d. (4x  2)(3x  5)  2(7x  5)(2x  6)
7. Identify the least common multiple of each pair of numbers.
a. 12 and 15
c. 20 and 25
b. 18 and 32
d. 49 and 3
8. Identify the least common multiple of each pair of terms.
a. 2x and 3x2
c. rs2 and s3t
b. 3y and 4xy
3
x2
and
4
2
9. Factor the following polynomials.
a. 3y ( y  2)  4 ( y  2)
c. 3x2  9x  8x  24
b. 5a (a  4)  2(4  a)
d. 2y4  y3  10 y  5
10. Factor, if possible.
a. x2  7x  10
d. 2m2  3m  9
b. 2r 2  14rs  24 s2
e. 12 q2  17q  6
c. 4x2  11x  6
f.
a2  11ab  24b2
11. Factor completely.
a. b2  121
d. 18 x3  24x2  8x
b. 4t 2  100
e. x4  16
c. 10 x3y  90xy
f.
10
Mrs. Connor
x4  18x2  81
Math 20 – 1 Prerequisite Skills 2014
ANSWERS
LINEAR RELATIONS
1. a) Non-linear. Each increase in the value of r increases the value of A by a different amount
b) Linear. Each increase in the value of x increases the value of y by the same amount, 5.
c) Non-linear. Each increase in the value of the first coordinate increases the value of the second coordinate by a different amount.
d) Linear. The same increase in the value of the first coordinate (2) increases the value of the second coordinate by the same amount,
2. a)
Term Number
Value
1
4
2
9
3
14
4
19
5
24
b) v = 5t + 1
Substitute t = 3. The result should be 14.
v = 5(3) + 1
v = 15 + 1
v = 14
c) 244 d) t = 18
3.
y = 3x + 1
4
3
5
4. a) y = 2x + 6 b) y = 3x  9 c) y     x 
d) y = 6x  4 e) y = 7x + 9 f) y  2 x 
3
4
6
5. a) m   3, y-intercept  4
2
5
1
3
3
2
7
2
b) m  , y-intercept  
c) m  , y-intercept  
d) m  1.8, y-intercept  2.1
11
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
6. a) y  2x  5 b) y  
2
x2
3
7. a) x  y  752 b) a  c  256
x  y  174
c) q  l  73
5a  3c  767
d) 75  15m  C
0.25q  l  37
35m  C
8. a) None. The lines have the same slope, but different y-intercepts, so the lines are parallel.
b) Infinite. The equations are multiples of each other, so the lines are congruent.
c) Infinite. The equations are multiples of each other, so the lines are congruent.
d) One. These are linear equations with different slopes and y-intercepts, so the lines intersect.
9. a) (3.5, 1.0)
b) (8.0, 7.0)
c) (8.8, 3.5)
d) (21.3, 16.5)
12
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
10. a) (3, 8) b)
 92 ,  43 
c) (6.5, 4.5) d) (1.5, 1)
11. a) (4, 13) b) (1.5, 0) c)
 187 , 78 
d) (5, 2)
12 a) domain: {x  x  R}; range: {y  y  R}
b) domain: {y  y  R}; range: {y  y  18, y  R}
1
2
5
3
c) m  0.75, b  1.2 d) m   , b 
2
2
13a) m  6, b  2 b) m   , b 3
14a) y  2x  3 b) y  x  5
2
3
c) y  3 d) y   x  1
15 a) y  4x  19 b) y  3x  7
3
4
c) y  2x d) y   x  3
GEOMETRY
1. a) A or BAC or CAB b) b or AC or CA
c) b2 = a2 + c2 or (AC)2 = (BC)2 + (AB)2
BC
or sin A =
AC
AB
e) tan C =
or tan C =
BC
d) sin A =
a
b
c
f) a = b cos C
a
2. a) 3.3 cm b) 3.3 cm c) 6.7 cm d) 8.3 cm
3. a) 7.79 b) 2.25 c) 0.21 d) 14.91
e) 7.67 f) 0.59
4. a) 37° b) 68° c) 47° d) 68° e) 30° f) 61°
5. a)
b) 249 m
13
Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
ALGEBRAIC EXPRESSIONS
1. a) 8x2 – 4x b) 4x2 + 25x – 21
c) 4x2 – 20x + 25 d) –x2 + 3x + 13
2. a) Label the horizontal axis x and the vertical axis g(x). b) –14
c) Yes. If you substitute x = 5, you get g(5) = 7.
d) g is the set of all real numbers
3. a) 1 b) 3 c) 2
4. a)
b) p(50) = 37; p(150) = 31; p(200) = 28
c) {p(x) | 28 ≤ p(x) ≤ 37}
5. a) 2x – 17 b) 15x2 – 10x
c) 8x2 + 14x – 15 d) 25x2 – 40x + 16
6. a) xy(3 – 8x) b) 3p(1 – 3p) c) (x – 1)(x – 12)
d) (2a – 3y)(2a + 3y) e) 4(2r + 1)(r + 2)
f) 2(x – 0.2y)(x + 0.2y)
2
Check:
5
2
2
7    3  2    5
 5 
 5 
7. a) x 
 29   29 



 5   5 
b) x = 6 c) x = –2
POWERS AND ROOTS
1. a) perfect square b) perfect cube
c) perfect square d) both
2. a) 7 b) 2 c) 9 d) 14
3. a) 3 b) 12 c) 31 d) 26
1
1
1
1
4. a) 2 b) 2 c) 4 d) 3
x
t
y
g
5. a) 64 b) 38.4434 c) 0.0016
d) 1 e) 16 777 216 f ) 0.1780
17
1
6. a) x 5 b)
23
m6
c)
1
d
d)
3
2
1
15
x2
9
y2
17
7. a) 3 6  22.4824 b) 51  5
12

5
c) 23  8 d) 2
8. a) 12 p 
3
2
 0.1895
3
b) 3x 4 c) st

5
3
1
d) y 3
9. a) 16.4924 b) 16.1662 c) 1.4071 d) 2.2678
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Mrs. Connor
Math 20 – 1 Prerequisite Skills 2014
FACTORS
12
2
3
1
b) 3 c)
or 2
d)
5
5
2
4
49
18
1
85
2. a)  1 b)
c)
d)
9
13
1
21
1
1
1
16
3
3. a)
b)
c)
d)
e)
10
8
9
9
4
12
5
9
4. a)
b) 4 c)
d)
5
12
14
1. a)
e)
1
14
f)
17
33
f)
1
3
5. a) x2  3x  10 b) c2  d 2
c) x2  6x  9 d) 24j 2  6k2
6. a) 6n2  9n  8 b) f 2  4f  29
c) 4b2  5bd  6b  5d  d 2 d) 40x2  90x  50
7. a) 60 b) 288 c) 100 d) 147
3
8
8. a) 6x2 b) 12xy c) x 2 d) rs3t
9. a) (3y  4)(y  2) b) (5a  2)(a  4)
c) (x  3)(3x  8) d) (y3  5)(2y  1)
10. a) (x  2)(x  5) b) 2(r  4s)(r  3s)
c) (4x  3)(x  2) d) (2m  3)(m  3)
e) (3q  2)(4q  3) f ) (a  3b)(a  8b)
11. a) (b  11)(b  11) b) 4(t  5)(t  5)
c) 10xy(x  3)(x  3) d) 2x (3x  2)(3x  2)
e) (x2  4)(x  2)(x  2) f ) (x  3)(x  3)(x  3) (x  3)
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Mrs. Connor