FINAL EXAM
REVIEW
1. Complete each ordered pair, so that it
satisfies the equation.
a)
( 0 , __ ) , ( 6 , __ ) , ( __ , 9 ) , ( __ , -2 )
b) ( 5 ,
__ ) , ( -5 , __ ) , ( , __ ) , ( __ , 0 )
2. Solve by graphing. Check each solution
a) b) c)
3. Without graphing, determine whether
each system has one solution, no solution, or infinitely many
solutions.
Explain your thinking.
a) b) c)
4. Solve each system of equations by
substitution.
a) b)
5. Simplify each system, and solve by substitution. Check your solution.
a)
6. Solve each system of equations by
elimination. Check your solution.
a) b)
c) d)
7. A supermarket sells 2-kg and 4-kg bags
of sugar. A shipment of 1100 bags of
sugar has a total mass of
2900 kg. How
many 2-kg bags and 4-kg bags are in the shipment?
8. The school car wash charged $5 for a car and $6 for a
van. A total of 86 cars and vans were
washed on
Saturday, and the amount earned was $475. How many vans were washed on Saturday?
9. A lab technician needs to combine some 30% alcohol solution
and 35% alcohol solution to make 5 L of 33% alcohol solution. How many litres of the 30% and of the 35%
solution will be used?
10. A plane makes a trip of 5040 km in 7 h,
flying with the wind. Returning against
the wind, the plane
makes the trip in 9 h. What is the speed of the wind?
11. Determine the length of the line segment
joining each pair of points. Express
each length as an exact
solution and as an approximate solution, to the
nearest tenth.
a) (
3 , 7 ) and ( -1 , -5 ) b) ( 0 , 5 ) and ( 6 , 10 )
12. Write the equation for a circle with
centre O( 0 , 0 ) and through the point ( 3 , 4 ).
13. The equation for a circle with centre O(
0 , 0 ) is x2 + y2 = 361.
What is the radius?
14. Determine the midpoint of each line
segment with the given endpoints.
a) (
-6 , 2 ) and ( 4 , 8 ) b) ( -200 , -100 ) and ( 350 , 600 )
15. One endpoint of a line segment is D( 5 , -7 ). The midpoint of the line segment is M( 3.5 ,
1.5 ). Explain how to find the
coordinates of the other endpoint, E, of the line segment.
16. The vertices of a quadrilateral are S( 1 , 2 ) , T( 3 , 5 ) ,
U( 6 , 7 ) , and V( 4 , 4 ). Verify each
of the following:
a) STUV
is a parallelogram b) The diagonals of STUV bisect each other
17. DABC has vertices A( 3 , 5 )
, B( 2 , 3 ) , and C( 5 , 2 ). Find the
equation of the altitude from A to BC.
18. Find the equation of the median from
vertex A in DABC, if the coordinates of
the vertices are
A( -3 , -1 ) , B( 3 , 5 ) , and C( 7 , -3 ).
19. Find the equation of the perpendicular
bisector of the line segment joining P( -1 , 4 ) to Q( 3 , -2 ).
20. Find the shortest distance from the given
point to the given line. Round to the
nearest tenth, if
necessary.
a) (
0 , 0 ) and b) ( 6 , 5 ) and 7x + y + 23 = 0
POLYNOMIALS
21. Classify each polynomial by degree and by
number of terms.
a) 3x2 - 2x b) 4a2b3 c) 8 + 2y4 + 3y3 d) 4x5
- 2x3 + x2 + 4
22. Evaluate the expression for the given
values of the variables.
a) 2x2
- 4xy - 5y2 for x = -3 , y = 2
23. Simplify:
a) (6y
- 2) + (2y + 8) b) (8 + 6x) - (9 + x)
c) (3x2
+ 2x - 6) + (2x2 - 4x + 7) d) (5a2b + 2ab - 3b2)
- (6a2b - 3ab + b2)
e) (3ab)(-2ab2)(2a3) f) (-6x2yz)(-5y3z)
g) h)
24. Expand and Simplify
a) 4m(m2
- mn - n2) - 2n(6m2 + mn + 4n2)
25. Expand and Simplify
a) 2(m
- 3)(m + 8) b) 3(6x - 2y)(2x - 3y)
c) (y - 4)(y - 3) - (y - 2)(y - 5) d) 6(m - 2)(m + 3) - 3(3m - 4)
26. Expand
a) (x + 4)2 b) (y
- 7)2 c) (x - 5)(x + 5) d) (5m +
2n)(5m - 2n)
27. Expand and Simplify
a) 3(2b
- 1)2 - 2(4b - 5)2 b) 4x2 - (2 - 3x)2 +
6(2x - 1)(2x + 1)
28. Factor
a) 2ax + 10ay - 8az b) 3x3y2
- 12x2y3 + 18x2y + 15xy2
c) 3x(y - z) - 2(y - z) d) 4t(r + 6) - (r + 6)
29. Factor by grouping
a) 3x2y
- 6x2 - 2y + y2 b) 4ab2 - 12a2b -
3bc + 9ac
30. Factor completely
a) x2
- 5x + 6 b) a2 + 6a + 5 c) x2 - 5xy -66y2
d) m2
+ 12mn + 32n2 e) 4x2 - 16x - 48 f) 2x2
- 16x - 66
31. Factor completely
a) 3y2 + y - 4 b) 20x2 - 7x - 6 c) 18y2 + 15y - 18
d) 8m2 + 6m - 20 e) 15x2
- 13xy + 2y2 f) 9x2 + 3xy - 20y2
32. Factor completely
a) x2
- 25 b) 49 - 64m2 c) 81x2 - 121p2
d) 16a4
+ 40a + 25 e) 4x2 - 36 f) 36x2 - 81y2
33. Sketch each parabola and state the
direction of the opening, the coordinates of the vertex, the equation of
the axis of symmetry, the domain and range, and the
maximum or minimum value.
a) y = (x + 3)2 - 2 b) y = - (x - 4)2 - 3
c) y = 2(x - 1)2 + 1 d)
34. Write an equation for a parabola with
vertex ( 3 , -1 ) and a = -2
35. Without graphing, state whether each
function has a maximum or a minimum.
Then, write each
function in the form y = a(x - h)2 + k
and find the minimum or maximum value and the value of x for which it occurs.
a) y = 3x2 - 18x + 1 b) y = -4x2 - 32x - 11
c) y = -7x2 + 84x + 19 d) y = 4x2 - 20x + 7
36. A ball is thrown upward with an initial
velocity of 18 m/s. Its height, h metres
after t seconds, is given
by the equation
h = -5t2 + 18t + 1.8 where 1.8 represents the height at which
the ball is released by the thrower.
a) What
is the maximum height the ball will reach?
b) How
much time elapses before the ball reaches the maximum height?
c) How
long is the ball in the air, to the nearest tenth of a second?
37. Phil wants to make the largest possible
rectangular vegetable garden using 18 m of fencing. The garden
is right behind the back of his house, so he has to
fence it on only three sides. Determine
the dimensions that maximize the area of the garden.
38. A pizza company's research shows that a
$0.25 increase in the price of a pizza results in 50 fewer pizzas
being sold.
The usual price of $15 for a three-item pizza results in sales of 1000
pizzas. Write the algebraic expression
that models the maximum revenue for this situation.
39. The length of a rectangle is 2 m more
that the width. The area is 48 m2. Find the dimensions of the
rectangle.
40. The sum of the squares of three
consecutive integers is 77. Find the
integers.
41. The hypotenuse of a right triangle is 15
cm. The other two sides have a total
length of 21 cm. Find the
lengths of the two unknown sides.
42. State the roots of each equation
a) (x
- 2)(x + 7) = 0 b) (3x + 1)(2x - 3) = 0
c) 7x(x
- 5) = 0 d) (2x + 5)(2x + 5) = 0
43. Solve the following equations
a) x2
- 6x + 8 = 0 b) 6t2 = t + 35
c) d) (3x - 1)2 = 25
44. Sketch the graphs of the following
quadratic functions by locating the x-intercepts, and then finding the
coordinates of the vertex.
a) y
= (x - 3)(x - 5) b) y = x2 - 7x + 12
45. Solving using the quadratic formula. Round to the nearest hundreth, if necessary.
a) x2
- 8x + 12 = 0 b) 20x2 + 27x = 14
c) 3x2
- 6x - 8 = 0 d) 4x(x + 8) = 3
46. The sum of the squares of three
consecutive odd integers is 875. Find
the integers.
47. The length and width of a rectangle are 6
m and 4 m. When each dimension is
increased by the same
amount, the area of the new rectangle is 50 m2. Find the dimensions of the new rectangle, to
the nearest tenth of a metre.
48. A rectangular skating rink measures 40 m
by 20 m.
It is to be doubled in area by extending each side
by the same amount.
Determine how much each side should be extended, to the nearest tenth of
a metre.
49. The triangles in each pair are
similar. Find the unknown side lengths.
·
a)
x
b)
50. DPQR ~ DKLM. PQ = 4 cm and KL = 6 cm. The area of DPQR is 12 cm2. Find the area of DKLM.
51. Use a calculator to find each angle, to
the nearest thousandth.
a) tan
84° b) sin 21° c) cos
43°
52. Find Ð K, to the nearest degree.
a) tan
Ð K = 2.750 b) sin Ð K = 0.208 c) cos
Ð K = 0.174
53. Find Ð Q, to the nearest degree.
a) tan Ð Q = b) sin Ð Q = c) cos Ð Q =
54. Calculate x, to the nearest tenth of a
metre.
a) b) c)
55. Solve each triangle.
Round each side length to the nearest tenth of a unit, and each angle,
to the nearest degree.
a) b)
57. Find XY, to the nearest tenth of a
centimetre.
56. Find BC, to the nearest centimetre.
58. From the window of one building, Sam finds the angle of
elevation of the top of a second building is 41° and the angle of depression
of the bottom is 54°. The buildings are 56 m apart. Find, to the nearest metre, the height of the
second building.
59. Perce Rock is a popular tourist
attraction on the shore of the Gaspe Peninsula.
To find its height,
measurements were taken at low tide, as shown in the
diagram. What is the height of Perce
Rock, to the nearest metre?
60. Find all unknowns.
62. In D
ABC ,
Ð A = 50° , a = 9 m , and b = 8
m. Find Ð B?
63. Solve the triangles.
65. In D
KLM ,
k = 54.2 cm , l = 45.7 cm , and m = 36.9 cm.
Find Ð K?
66. Solve D
WXY ,
w = 120 m , x = 77 m , and y = 115 m