2. Simple Equations & Quadratic Equations

[Pages:16]2. Simple Equations & Quadratic Equations

Quantitative Comparison

1. Quantity A x + y

2.

Quantity A

y

3. Quantity A smaller root

2x + 3y = 29 & 3x + 4y = 41

x =

1 y

1 and y = 1 x

y 4x2 4x 8

Quantity B 12

Quantity B 1 2

Quantity B -2

4. Carlos has 14 coins in his pocket, consisting of quarters and nickels only. The monetary value of these coins is

$2.30.

Quantity A

Quantity B

the number of quarters

the number of nickels

5. Julie is 5 years older than Ravi. Three years ago, Julie was twice as old as Ravi.

Quantity A

Quantity B

Ravi's age now

5

6.

p + q = 16 and 3p + 2q = 44

Quantity A

Quantity B

p

q

7

X2 - 4X - 21=0

Quantity A

Quantity B

sum of the roots

product of the roots

8.

y x2 6x 9

Quantity A minimum y value of the function

Quantity B 9

9. The prom committee orders an arch for the entrance to the dance floor. The arch follows the equation y 2x .1x2

where y is the height of the arch, in feet. Quantity A

maximum arch height

Quantity B 10 feet

Select one Answer Choice

10. At a fruit stand, apples can be purchased for $0.15 each and pears for $0.20 each. At these rates, a bag of apples and

pears was purchased for $3.80. If the bag contained exactly 21 fruits, how many were pears?

(A) 10

(B) 13

(C) 12

(D) 14

(E) 11

11. Jack's age is the addition of his four sons' ages. If his sons were born three years apart, and if Jack was 41 years old

last year, how old was he when his oldest son was born?

(A) 21

(B) 23

(C) 24

(D) 27

(E) 29

y = kx + 3

12. In the equation above, k is a constant. If y = 17 when x = 2, what is the value of y when x = 4 ?

(A) 34

(B) 31

(C) 14

(D) 11

(E) 7

13. Each week, Harry is paid x dollars per hour for the fi rst 30 hours and 1.5x dollars for each additional hour worked that

week. Each week, James is paid x dollars per hour for the fi rst 40 hours and 2x dollars for each additional hour worked

that week. Last week James worked a total of 41 hours. If Harry and James were paid the same amount last week, how

many hours did Harry work last week?

(A) 35

(B) 36

(C) 37

(D) 38

(E) 39

14. A certain store sells all maps at one price and all books at another price. On Monday the store sold 12 maps and 10

books for a total of $38.00, and on Tuesday the store sold 20 maps and 15 books for a total of $60.00. At this store,

how much less does a map sell for than a book?

(A) $ 0.25

(B) $ 0.50

(C) $ 0.75

(D) $ 1.00

(E) $ 1.25

15. Joe and his friend Sue combined their money to spend it together. Originally, Joe had ? of what Sue had.They gave

6% of what Sue had to John and spend one dollar more than ? th of the rest of the money. They lost 10% what Joe

originally had. At the end they are left with $12. How much money did Sue have originally? Approximate your answer

to the nearest cent.

(A) $10.62

(B) $12.62

(C) $14.6

(D) $16.24

(E) $16.42

16. A club collected exactly $599 from its members. If each member contributed at least $12, what is the greatest number

of members the club could have?

(A) 43

(B) 44

(C) 49

(D) 50

(E) 51

17. A positive number x is multiplied by 2, and this product is then divided by 3. If the positive square root of the result of these two operations equals x, what is the value of x ?

9 (A) 4

4 (B) 3

2 (C) 3

3 (D) 2

1 (E) 2

18. If x 12 400 , which of the following could be the value of x ? 5 ?

(A) 15

(B) 14

(C) ?24

(D) ?25

(E) ?26

Select all answer choices

19. Karen collected donations from seven people for the school band. Minimum donation was $7 and the maximum was

$13. Which of the following can be the total amount of money that Karen collected?

(A) $20

(B) $40

(C) $60

(D) $65

(E) $80

20. A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and

bananas from the stand for a total of $26.30, what total number of apples and bananas did the customer purchase?

(A) 39

(B) 43

(C) 47

(D) 51

(E) 55

21. A restaurant meal cost $35.50 and there was no tax.If the tip was more than 10 percent but less than 15 percent of the

cost of the meal, then which of the following could be the total cost of the meal?

(A) $38.825

(B) $39

(C) $39.05

(D) $40.5

(E) $41

Numeric Entry Questions

22. If 3(x - 1) + 2x = 4x + 7, then x - 1 =

23. The addition of Jack's and Jill's ages is 72. When Jack was at Jill's current age, Jill's age was threefourth of Jack's age. What is Jack's current age?

24. A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12. What is the cost per person if 40 people charter the aircraft?

25. A theater sells children's tickets for half the adult ticket price. If 5 adult tickets and 8 children's tickets cost a total of $27, what is the cost of an adult ticket?

14. Functions & Defined Functions

Quantitative Comparison

1 Quantity A

4 4

x y = x y - x,

Quantity B

-4 -4

2.

ab a2b

3

Quantity A

43

Quantity B

3 4

3.

mn mn 1 nm 2

Quantity A

Quantity B

32

21

4.

x

y

x2 y2

y2 x2

Quantity A

3 4

Quantity B

34

5.

m n m2 n2 5

Quantity A

Quantity B

35

53

6.

h h2 9h2 4

Quantity A

Quantity B

2

3

7

x

x2 x2

5x 6 5x 6

Quantity A

Quantity B

3

2

8

n

r

r

!

n! n

r

!

Quantity A

25 8

Quantity B

2517

9.

y x

x y 2

Quantity A

Quantity B

4

3

-1

Select one Answer Choice

10. If g(z) = 1/z + 4, find the value of z for g(z) = 8

3 (A) 4

4 (B) 3

11. The domian of (x 2)(4 x) is

(A) {?2,4}

(B) ( 2,4)

1 (C) 4 (C) {?2, ? 4}

3 (D) 2 (D) [2, 4]

1 (E) 2 (E) [-2, 4]

12. f(x) = 1 if x > 2 ; f(x) = 2 if ?3 < x < 2 ; f(x) = 5 if x < ?3 then f (? 10) =

(A) 3

(B) 4

(C) 2

(D) 1

(E) 5

13. The distance traveled by an object with a constant acceleration is given by the equation: d vt at2 where d is

the distance traveled, v is the initial velocity, a is the acceleration and t is the time. If it takes 5 and 10 seconds for an

object to travel 6 and 15 meters, respectively, what is the acceleration of the object?

(A) 6/25 m/sec.2

(B) 15/100 m/sec.2

(C) 9/10 m/sec.2

(D) 3/50 m/sec.2

(E) 3/2 m/sec.2

14. The height of a certain tree is given by the function, h t a b t , where a and b are constants, h is the height

in feet, t is time in years. When the tree was first planted, it was 2 ' high and after 9 years it became 20' tall. What is

the value of b?

(A) 2

(B) 2.44

(C) 6

(D) 7.5

(E) -6

15. What is the value of x at which the quadratic function f(x) = x2 + 6x + 9 has a minimum?

(A) x = 9

(B) x = 6

(C) x = 3

(D) x = 0

(E) x = -3

16. f x x2 2x 1 and g s s3 . If g(f(x)) = 1, then which of the following can be the value of x?

(A) -3

(B) -2

(C) -1

(D) 0

(E) 1

17. The weight of a new born baby is given by the function w(m) = 7.5 + 2.5m, where m is the age of the baby in months.

At what month the baby's weight is doubled?

(A) 2

(B) 2.5

(C) 3

(D) 3.5

(E) 4

18. If f(x) is a quadratic function with a minimum at point A(2, -1), which of the following has a minimum value at point

O (0, 0)?

(A) f(x) + 1

(B) f(x - 2) + 1

(C) f(x + 2) + 1

(D) f(x) - 2

(E) f(2 - x) + 1

Select all answer choices

19. x @ z means xz and x & z means x/z.If a @ b = a & b, then which of the following pairs could be the values of

a and b?

(A) (1, 1)

(B) (2, 1)

(C) (2, -2)

(D) (1, -1)

(E) (1, 2)

20. f x x x 2 16 a bx 2 , where a and b are constants. Which of the following value of x is not the x-intercept

of f(x)?

(A) -4

(B) 0

(C) 4

(D) a/b

(E) b/a

21.

For what values of x, the function

x2 2x 1 4x3 x

is not defined

1 (A)

4

(B) 1 2

1

(C) 0

(D) 2

(E)

2

Numeric Entry Questions

22. If g x x 2 2 and f(x) = 2x - 1, then f(g(3)) = 23. If g x 2x 1 , then g(5) =

24. f x x2 2x 1 and g(x) = 2f(3x) + 10 What is g(-2)?

x2 2x 1 25. For what value of x, the function x 2 is not defined

11. Indices &Surds

Quantitative Comparison

1 Quantity A

a is a positive integer.

1 a7

2.

a = -921

Quantity A

a 7 + a 15

3.

10x = 125y

Quantity A x

4.

Quantity A

107 106 9

5.

Quantity A

1 5

1

2

5

1

2

5

6.

k2 =9

Quantity A

2k 3

Quantity B 1 a8

Quantity B a 8 + a 14

Quantity B y

Quantity B

1003

Quantity B 1

Quantity B 3 k

7.

Quantity A

104

8

Quantity A

46 44 45 43

9.

Quantity A

13 + 12

Select one Answer Choice

10. (2 10 - 2 9 )(2 8 - 2 7 ) =

(A) 2

(B) 22

11.

Find x, if

55x+5

=

8x 25

-

3

(A) 0

(B) 3

Quantity B 56

Quantity B

1

Quantity B 5

(C) 24

(C) 2

(D) 216

(D) 1

(E) 28

(E) 5

12. If x > 0, which of the following statements may possibly be true?

I.. x

1 0 x

(A) I only (E) I and II and III

II.

x

1 0 x

(B) II only

III.

x

1 0 x

(C) III only

(D) I and III

13. What is the minimum value of x in the product 0.0000123678 x10x such that the product is more than 99?

(A) 7

(B) 3

(C) 2

(D) 6

(E) 4

14. Which of the following is the largest among the following?

(A) 2 222

(B) 2222

(C) 2222

15. If x = a5 and y = a6, a ? 0, which of the following is equivalent to a13?

(A) xy

(B) x2y

x3 (C) y

16.

3

a2 2

4

a4 4

1

=

(A) 7

(B) 3

3

4

1

17. If 4 a 2a 3 23 ,then a = ?

(C) 2

1

(A) 4 2

(B) 4 2

1

(C) 4 2

(D) 2 2 2 22

x4

(D) y

(E) 222 y3

(E) x

(D) 6

(D)

1 4

2

(E) 4

(E) 4 2

18. Which of the following statements is true for all positive values of x?

(A) x 2 x 0 (E) x 2 x 1 2

(B) x 2x 0

(C) x 2x 0

(D) x 2 x 1 2

Select all answer choices

19. Which of the following is correct?

(A)

2 83

4

(E)

3

32 5

8

(B) (64)0.5 8

1

(C) (0 .0 0 0 1) 3

1

1000

20. Which of the following is incorrect?

(A)

1

52

1

104

1

43

111

(B) 10 4 5 2 4 3

11

1

(C) 43 52 10 4

(E)

10

1 3

1

52

2

43

21.

Which of the following expressions below is equal to

a

3 2

?

2

(A) 3 a

3

(B) 2 a

1 3

(C)

a

2

1

(D)

3

3 8

3

2 3

(D)

1

43

1

104

1

52

1

(D) a3 2

1 (E) a3

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