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 K, to the nearest degree.

 

a) tan K = 2.750 b) sin K = 0.208 c) cos K = 0.174

 

53. Find Q, to the nearest degree.

 

a) tan Q = b) sin Q = c) cos 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 , A = 50 , a = 9 m , and b = 8 m. Find B?

 

 

 
63. Solve the triangles.

 

 

 

 

 

 

 

 

 

 

 


65. In D

 
KLM , k = 54.2 cm , l = 45.7 cm , and m = 36.9 cm. Find K?

 

66. Solve D

 
WXY , w = 120 m , x = 77 m , and y = 115 m