simplify x 4 x page | 1 math 99 final exam practice section 1: multiple choice questions 1. the...

Simplify x 4 x

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Math 99 Final Exam Practice


Section 1: Multiple Choice Questions

1. The expression 3x2 – 10x + 3 when factored fully is:-

A. (3x – 1)(x – 3) B. (3x – 1)(x + 3) C. (3x + 1)(x – 3) D. (3x + 1)(x + 3)

2. The expression 100y2 – 400x2 when factored fully is:-

A. 100(y2 – 4x2)

C. (2y + 20x)(2y – 20x)

B. 100(y + 2x)(y – 2x)

D. Some other answer

3. The expression 4a2b3 – 8a3b3 when factored fully is:-

C. a2b3(4 – 8a)A. 4a2b3(1 – 2a)

D. 4ab(ab2 – 2a2b2)

B. 4a2b3(– 2a)

E some other answer

4. The expression 3x2 + 10x – 8 when factored fully is:-

C. (3x – 4)(x – 2)A. (3x + 8)(x + 2)

D. (3x + 2)(x + 4)

B. (3x – 2)(x + 4)

E some other answer

4. Simplify4 xy 3

8x 2 y 7

A1

2 xy 4

Bxy 4

4C 2xy4 D 32x3y10 E some other answer

5. Simplify the following expression.

A.16 x 2

z 3B.

z 3

16 x 2C.

8 x 2

z 3D.

x 4 p 4

4 z 5

6.2

x 2 16

Ax

x 4B

x

x 4C

1

4D

x

4E some other answer

x( x 1)

x 1

2 x 1x( x 1)


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7. Add and simplify the following expression 4

x 4

x

x 4

A – 1 B1 C4 x

4 xD

4 x

4 xE some other answer

8. When you add the fractions

A. B.

3 2

x5 5

and simplify the result you get:-

5x 3C. D. some other answer

9. What is the slope of the line that passes through the points A(–5, 3) and B(10, – 2)

A. – 3 B. 3 C.1

3

1

3D.

10. What is the slope and y-intercept of the line with equation 4x + 2y = 20

A. Slope = – 2 and y-intercept = 10 B. Slope = – 4 and y-intercept = 20

C. Slope = 2 and y-intercept = 10 D. Slope = – ½ and y-intercept = 10

11. For the line with equation 4y + 2x = 8 what is the slope of the line.

A. 2 B. 4 C.6 D.½ E. – ½

12. For the line with equation 2x – 4y = 12 where does it cut the y – axis?

A. (0,12) B. (0,6) C. (0,3) D. (0, – 3 ) E. (0, – 4 )

13. Determine which of the following points, if any, satisfy the system of equations.

y =3x – 2y =

4x – 74

A. The point (2,1) B. The point (1, – 3)

C. The point (– 1 , 3) D. None of the points satisfy the system of equations

14. The system of equations x + 2y = 5 are:-102x + 4y =

A. Consistent and have 1 solution B. Inconsistent and have no solutions

C. Dependant and have an infinite solution D. Some other answer


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15. For the pair of equations x + 2y = 52x + 4y = 10

Which is the only coordinates that satisfy both equations?

A. (1,1) B. (1,2) C. (2,1) D. (– 1, – 3 ) E. (– 1, 3 )

16. Simplify the following radical expression 48

A. 2 24 B. 16 3 C. 2 12 D. 4 3 E some other answer

17. Simplify the following radical expression 4x 3 y 7 z 6

A. 2x 2 y 6 z 6 xy B. 2xy 3 z 3 xy C. 2xy 3 z 3

D. 4xy 3 z 3 xy

18. When you simplify the expression

E some other answer

8x 3 y 3 z 5 2x 2 y 3 z into its simplest form you get.-

A. 16x5 y 6 z 6 B. 4x2y3z3 x C. 4x3 y 4 z 2 yz D. Some other answer

19. When you simplify the expression3x 2 y 3

3xby rationalizing the denominator and simplifying the result you get:-

A.9 x3 y 3

3xB.9 x3 y 3 x

3xC. 3xy 3 3x D. Some other answer

20. The solution to the radical equation = 6

A. x = B. x = 4 C. x = D. Some other answer

21. When you express 16 + 9 as a complex number in the form a + bi you get :-

A. 4 – 3i B. 4 + 3i C. 0 + 5i D. Some other answer

22. Express as a complex number a + bi 16 + 9

A 7 B 4 – 3i C 4 + 3i D 0 + 5i E some other answer

V X


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23. The quadratic equation x2 – 4x + 9 = 0 has the following number of real number solutions

A 0 solutions B 1 solution C 2 solutions D 3 solutions

24. The solution to the quadratic equation 2x2 + 9 = 59 is:-

A x = 5 and x = – 5 B. x = 625 C. x = 25 D x = 12.5 and x = – 12.5

25. The solution to the quadratic equation (x – 5)2 = – 36 is

A x = 1301 B. x = C. x = 11i D. some other answer

26. The coordinates of the vertex of the quadratic equation y = x2 + 4x + 6 is:

A (– 2,2) B (– 2,0) C (– 2, – 6) D (2, –2) E some other answer

27. When the formula V = 2ab 2 is expressed in terms of b we get.

Ab =1

2aV 2 Bb= aV

1

2Cb=

2V

aDb=

V

2aE some other answer

28. If you take the formulaV = x 4r 2 and change the subject of the formula to r you get:-

A.

C.

r=

r=

V 2 x

4

42

B.

D.

r=

r=

V 2 x

4

x V 2

4

2

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Section 21. Simplify each of the following expressions, if an expression cannot be factored then say so.

(a)x 2 7 x

3x 2 20 x 7

(b)a 2 2a 1

a 2ab b 2

a 1

a b

2. Add or subtract the following and simplify the answer where possible.

(a)3

p

6

p 3(b)

2

x 2

x

x 2


3. Solve the following equations.

(a) x 5

2

x 7

3(b)

5

3

x

2

x

4

4. Vecca can paddle her kayak 4 mph in still water. It takes her as long to paddle 6 miles upstream as ittakes her to paddle 18 miles downstream. Find the speed of the water current.

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5. (a) Draw on the grid the line with

yx + 2y

==

2x – 18

equation y = 2x – 1 .

x

(b) Draw on the grid the line withEquation x + 2y = 8

(c) Use your diagram to find the solutionto the system of equations.

y


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6.(a) Solve the pair of equations by the substitution method.2x + y =3x – 2y =

101

6.(b) Solve the system of equations x – 3y = – 10 by using the addition method.3x + 2y = 25


7.

8.

If 2 hamburgers and 3 fries cost $3.50 and 4 hamburgers and 3 fries cost $6.50.What are the cost of a single hamburger and a single portion of fries?

A pet store owner wants to make 600 pounds of mixed bird seed, the sunflower seeds cost $2.30 perpound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixturewas $1200. How much of each seed was used?

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9. Solve the following Radical Equations.

(a) x = 4

(b) 2x 17 = 9

10. Solve the quadratic equation x2 + 8x – 20 = 0 by using “Completing the square” method.



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12. Solve the quadratic equation 5x2 + 2x – 7 = 0 by using the quadratic formula x = b b2 4ac

2a

13. Solve the quadraticequation. x2+ 10x + 16 = 0 by using the quadratic formula x = b b2 4ac

2a


14. For the quadratic y = x2 – 2x – 6

15. The height h, in feet of a ball after t seconds is given by the model h = 12t – 2t2

(a) What is the height of the ball after 5 seconds?

(b) How long does it take the ball to reach a height of 3.6 feet?

(c) How long does it take the ball to reach its maximum height?

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(a) What is the shape of the quadratic?

(b) What is the y-intercept?

x

(c) What are the x-intercepts?

(d) What are the coordinates of the vertex?

(e) Sketch the quadratic – use appropriate scales.

y


16. V = x y 2 change the subject of the formula to y.

17. H = r2h change the subject of the formula to r.

18. John traveled from his own home to his mother’s home 200 miles away and then he returned back home thatsame day. During these journey’s he noticed that his speed was 10 mph faster coming homethan it was when he traveled to his mothers home. The total time spent on both parts of the journey was 9hours. What were his speeds during this journey?

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(a 1) (a b)(a b) (a b)

3x 20 x 7

2

3 ( p 3)

2 x ( x 2)2 ( x 2)

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Math 99 Final Exam Practice – Solutions.Section I

9. D19. D

10. A20.B

1. A11. E21. D

2.B12. D22. C

3. A13. A23. A

4. A14. C24. A

5. A15. B25. B

6. B16. D26. A

7. B17. D27.D

8. C18. B28. A

Section 21. Factor each of the following expressions, if an expression cannot be factored then say so.

(a)x 2 7 x

2=

x( x 7)

(3x 1)( x 7)=

x

(3x 1)

(b)a 2 2a 1

a 2ab b 2

a 1

a b=

a 2 2ab b 2

a 2 2a 1

(a 1)

(a b)= =

(a b) (a 1)(a 1) (a 1)

2. Add or subtract the following and simplify the answer where possible.

(a)3

p

6

p 3=

p ( p 3)

p

p

6

( p 3)=

3( p 3)

p( p 3)

6 p

p( p 3)=

9 p 9

p( p 3)

(b) x

x 2

( x 2) ( x 2)=

x 2 ( x 2) ( x 2)=

2( x 2)

( x 2)( x 2)

x( x 2)

( x 2)( x 2)

=

=

x 2 2 x 2 x 4

( x 2)( x 2)

x 2 4 x 4

( x 2)( x 2)3. Solve the following equations.

(a) x 5

2

x 7

3(b)

5

3

x

2

x

4

2(x+ 7) =

2x + 14 =

3(x – 5)

3x – 15

x

2x

2

3x 20

12 123x 20

1229 = x 6x + 40 =

40 =12x

6x20

3

40

6so x

Distance Speed Time

Upstream 6 4 – x

Downstream 18 4+x

x y0 42 34 2

x y0 – 11 12 3


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4. Vecca can paddle her kayak 4 mph in still water. It takes her as long to paddle 6 miles upstream as ittakes her to paddle 18 miles downstream. Find the speed of the water current.

and Speed (Downstream) = 4 + xLet x = speed of current then

Since times are the same

Speed (upstream) = 4 – x

=

18(4 – x)

6(4 + x) =

72 – 18x

24 + 6x =

24x =

48

x = 2

5. (a) Draw on the grid the line withequation y = 2x – 1 .

(b) Draw on the grid the line withEquation x + 2y = 8

(c) Use your diagram to find the solution to the system of equations. yx + 2y

==

2x – 18

6.(a) Solve the pair of equations 2x + y = 10 by the substitution method.3x – 2y = 1

3x – 2y3x – 2(10 – 2x)3x – 20 + 4x

7x – 207xx

======

1111213

Rearrange equation 1 to get y = 10 – 2x and substitute it into equation 2

So y = 10 – 2x = 10 – 2(3) = 10 – 6 = 4

So solution is (3,4)

x

So the Speed of current = 2 mphy

y = 2x - 1

X + 2y = 8


6.(b) Solve the system of equations x – 3y = – 10 by using the addition method.3x + 2y = 25

x – 3y = – 10 Equation 1 x – 3 = – 3x +9y = 303x + 2y = 25 Equation 2 = 3x + 2y = 25

Add the equations 11yy

==

555

Put y = 5 into 3x + 2y3x + 2(5)

3x + 103x

====

25252515

x = 5 So solution is(5,5)

7. If 2 hamburgers and 3 fries cost $3.50 and 4 hamburgers and 3 fries cost $6.50.What are the cost of a single hamburger and a single portion of fries?

Let x = cost of hamburger and y = cost of fries

2 hamburgers and 3 fries cost $3.50 gives us the equation4 hamburgers and 2 fries cost $6.50. gives us the equation

2x + 3y =4x + 3y =

3.56.5

2x + 3y =4x + 3y =

3.5 Equation 1 x – 16.5 Equation 2

– 2x – 3y =4x + 3y =

– 3.56.5

2xx

==

3.1.50

Put x = 1.50 into

Add the equations

2x + 3y =

3.5

2x + 3(1.50) =

3.5

2x + 3

= 3.5

2x

= 0.5

x

= 0.25

So cost of hamburger = $1.50 and the cost of fries = $0.25

8. A pet store owner wants to make 600 pounds of mixed bird seed, the sunflower seeds cost $2.30 perpound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixturewas $1200. How much of each seed was used?

Let x = amount of sunflower seed and y = amount of peanut seed

to make 600 pounds of mixed bird seed gives us the equation x+y = 600The total cost of all the seed used in the mixture was $1200 gives us the equation 2.3x + 1.4y = 1200

Rearranging x + y = 600 to y = 600 – x and substituting it into the equation 2.3x + 1.4y = 12002.3x +1.4(600 – x) = 1200

2.3x + 840 – 1.4x = 12000.9x

x= 360= 400

Put x = 400 into x + y = 600 to get y = 200, solution is 400 of sunflower and 200 pounds peanuts

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9. Solve the following Radical Equations.

(a) (b)x

x ==

416

2x 172x + 17

==

981

2xx

==

6432


So solutions are x + 4x

==

62

x2 + 8x – 20x2 + 8x(x + 4)2

===

02020 + 16

(x + 4)2 = 36 and x + 4 = – 6x+4 = x = – 10x+4 =


So solutions arex2 + 3x – 10x2 + 3x

==

010

x + 1.5x

==

3.52

10 + 1.52

12.25(x + 1.5)2

(x + 1.5)2

x + 1.5

===

and x + 1.5x==

– 3.5– 5

x + 1.5 =

12. Solve the quadratic equation 5x2 + 2x – 7 = 0 by using the quadratic formula x = b b2 4 ac

2a

a=5 b=2 c= – 7 b2 – 4ac = (2)2 – 4(5)(– 7) = 4 + 140 = 144

x = = = so x =

x=

13. Solve the quadraticequation. x2+ 10x + 16 = 0 by using the quadratic formula x = b b2 4 ac

2a

a=1 b = 10 c = 16 b2 – 4ac = (10)2 – 4(1)(16) = 100 – 64 = 36

x = = = so x =

x=

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b b2 4 ac

2a

b b2 4 ac

2a

12t – 2t = 12(3) – 2(3)2 = 36 – 18 = 18 ft


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14. For the quadratic y = x2 – 2x – 6

x= =

x = 3.65 and x = – 1.65

(d) What are the coordinates of the vertex?

x = = = 1

y = (1)2 – 2(1) – 6 = 1 – 2 – 6 = – 7 Coordinate of vertex is (1,– 7)

(f) Sketch the quadratic – use appropriate scales.

h = 12t – 2t215. The height h, in feet of a ball after t seconds is given by the model

(a) What is the height of the ball after 5 seconds?

h = 12t – 2t2 = 12(5) – 2(5)2 = 60 – 2(25) = 60 – 50 = 10 ft

(b) How long does it take the ball to reach a height of 3.6 feet?

h = 12t – 2t2

3.6 = 12t – 2t2

– 2t2 + 12t – 3.6 = 0 Use quadratic formula to get t = =

t = 5.7sec and t = 0.3 sec

(c) How long does it take the ball to reach its maximum height?

Maximum height will occur at vertex. t = = = 3 sec

h = 2

16. V =x y 2 change the subject of the formula to y.

V2 =

x + y2

V2 – x =

y2

=

y

y

=

(a) What is the shape of the quadratic?Concave up

(b) What is the y-intercept?y= – 6

x

(c) What are the x-intercepts?Use quadratic formula to get

y


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17. H = r2hchange the subject of the formula to r.

H

= r2h

=

r2

=

r

r

=

18. John traveled from his own home to his mother’s home 200 miles away and then he returned back home thatsame day .During these journey’s he noticed that his speed was 10 mph faster coming home than it was whenhe traveled to his mothers home. The total time spent on both parts of the journey was 9 hours. What were hisspeeds during this journey?

= 9

=

=

=

9(x2 + 10x)9x2 + 90x

==

400x + 2000400x + 2000

9x2 + 90x – 400x – 2000 = 0= 09x2 – 310x – 2000

Use quadratic formula to get x =

=

x = = = 40 So the speeds are x = 40 mph and a + 10 = 50 mph

x = = = – 5.55 not a usable solution

Distance Speed Time

Going to mother

Returning Home

Since total time was 12 hours

200

200

x

x + 10

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