COMALGE

FINAL EXAM (SET B)

September 5, 2013

I. MULTIPLE CHOICE
Write the letter corresponding to the correct answer. Write your answers on the first page of your test booklet.
(2 points each)
1. The multiplicative inverse of the additive inverse of
−a
is b−c a b−c A.
C.
c−b a a c−b B.
D.
b−c a 2. Which of the following is TRUE?

8. Factor x2 − 2xy + y 2 − xw + yw completely.
A. (x − y − w)(x − y)

C. xy(x − 2)(y + w)

B. (x − y + w)(x − y)

D. w(x − y)2

9. Each of the following is a factor of u6 − 64w6 EXCEPT
A. u − 2w

A. Rational numbers are integers.

B. u + 2w

B. Integers are whole numbers.

C. u2 − 2uw + 4w2

C. Whole numbers are rational numbers.

D. (u + 2w)2

D. There is a number that is both rational and irra- 10. Find the quotient: (x3 + 2x2 − 1) ÷ (x + 1) tional. A. x2 − x + 1
3. Which of the following is NOT TRUE?
B. x2 + x − 1
A. a2 (a3 ) = a5
2

3

6

B. (3a )(2a ) = 6a

4.

3x−1 y −1
5x0 y 3
6
A.
10x2 y 8
9
B.
25x2 y 16

C. 2a(a2 ) = 2a3
1
D. − 4a2 = −2a
2

2

is equal to
9
25x2 y 8
9
D.
25y 8
C.

5. Which of the following is TRUE?
I. Every integer has a multiplicative inverse.
II. The sum of two rational numbers is rational.
A. I only

C. Both I and II

B. II only

D. Neither I not II

6. 2(x + y)−1 is equal to
1
2x + 2y
2
B. x+y A.

C. x2 + 3x + 3
D. x2 − 3x − 3 s−1 2 is equal to
− 2 s + 2 s + 2s s−1 s2 + 5s + 2
C. 2
A.
2 + 2s) s + 2s
(s + 2)(s
3−s
s+1
D.
B. 2
(s + 2)(s2 + 2s) s + 2s
√
√
12. What is the reciprocal of ( 5 + 3) ?
√
√
√
√
5− 3
5+ 3
C.
A.
2
2
2
2
√
√
B. √
D. √
5− 3
5+ 3
11. The expression

13. If 6x2 − xyz − y 2 z 2 = 100 and 3x + yz = 10 , then what is the value of 2x − yz ?
A. 10

1
1
+
2x 2y
2
2
D.
+ x y
C.

7. Perform the indicated operations: x + 2(x − 2)2
A. 2x2 − 7x + 8

C. x3 − 2x2 − 4x + 8

B. 9x2 − 24x + 16

D. x4 − 8x2 + 16

B. 50

C. 200

D. 1000

x2 + 2xy + y 2 x2 − y 2
6x + 6y
÷
·
(x − y)3 x+y 4x − 4y when simplified is equal to

14. The expression

4(x + y)2
6(x − y)2
2(x + y)2
B.
3(x − y)2

C.

4(x + y)
6(x − y)

D.

A.

2(x + y)
3(x − y)

15. What is the sum of the roots of x2 − 4x + 1 = 0 ?
A. 2

B. −2

C. 4

D. −4

16. The LCD of

5 − y 3y + 1
7y 3 is + 2
−
y + 2 y − 4 (y − 2)2

22. When simplified,

A. (y + 2)(y − 2)

A. 4 − 2i

20 + 10i is equal to
3 + 4i

B. 4 + 2i

C. 2 − 4i

D. 2 + 4i

2

B. (y − 2) (y + 2)
C. (y − 2)2 (y + 2)2
2

23. The solution to the inequality ax+b ≤ 0 where a < 0 and b > 0 is

2

D. (y − 4)(y − 2)

17. In the quadratic equation ax2 + bx + c = 0 , which of the following gives the product of the roots?
A.

b a B.

c a 18. Simplify the expression

C. −

b a D. −

c a 4x2 y 3/4 z 1/6
32x−1 y 0 z −5/6

b a b
B. x ≤ a b a b
D. x ≤ − a A. x ≥

C. x ≥ −

24. The solution to

xy 3/4
C. 8x3 y 3/4 z
2/3
8x x3 y 3/4 z
8y 3/4
B.
D.
8
xz
√
√
√
√
3
4
19. Simplify 8a3 b3 + 3 ab − 8a4 b4 − 4a2 b2
√
√
A. (1 − 2ab)( 3 ab − 2ab)
√
√
B. ( ab + 3 ab)2ab
√
√
C. (2ab − 1) ab + (1 − 2ab) 3 ab
√
√
D. (1 − 2ab) ab + (2ab − 1) 3 ab
A.

20. The roots of the equation x2 + 2x + 2 = 0 are

A.

31
7

B. 8

25. Solving for L in M =

C. 3

L
P

PM
25 + f
25 + f
B. L =
PM

A. L =

D. −

25
+ 1 , we get f MPf − f
25
MPf
D. L =
25 + f
C. L =

B. −1

C. 0

D.

B. irrational
C. complex

5
3

27. If a, b, c ∈ R and a > 0 , b < 0 and c > 0 , which of the following is TRUE?

D. real
21. The expression (i15 − i7 )3 is equal to
A. 1

3
11

26. The solution to 2[x − (4 + 2x) + 3] = 2x + 2 is
A. 1

A. rational

x
5
5
3
+ =
+ is x+1 8
2(x + 1) 4

B. −1

C. 0

A. a + c < 0
D. i

C. a − b + c > 0

B. b − c > 0

D. b − a − c > 0

II. Solve the following problems. Show complete solution. Each item is worth 7 points.
1. Solve for x : 6(2x − 1)−2 + (2x − 1)−1 = 2
√
√
2. Solve for x in the equation: 3 + x = 15 + x

6. Solve for the variable z 
ONLY in the system below
=
2
 x+y−z
2x − y + 3z = −5 using Cramer’s rule.

x
+ z = −2.
3. The product of two consecutive negative even integers is 24. Find the numbers.
7. Solve the following system using the matrix inversion method: 4. A total of Php6000 is invested into two simple inter2x + 7y =
6
est accounts. The annual simple interest rate on one x + 5y = −3. account is 9%; on the second account, the annual simple interest rate is 6%. How much should be invested 8. Draw a sketch of the region defined by the following system of inequalities: in each account so that both accounts earn the same

amount of annual interest?
 y ≥ 2x − 5
2x ≤ 3y − 10
5. Find the solution to the system below using Gauss

x ≥ −2.
= −1
 x + 3y + 3z
2x − y + z
= −3
Jordan.

3x + 5y + 7z = −1.