NumbersandOperations 5. 3?

[Pages:18]SAT Math Hard Practice Quiz

Numbers and Operations

1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red tomatoes in the bag is 4 to 3. When five green tomatoes and five red tomatoes are removed, the ratio becomes 3 to 2. How many red tomatoes were originally in the bag?

5. How many integers between 10 and 500 begin and end in 3?

(A) 12 (B) 15 (C) 18 (D) 24 (E) 30

6. A particular integer N is divisible by two different prime numbers p and q. Which of the following must be true?

I. N is not a prime number. II. N is divisible by pq. III. N is an odd integer.

2. If each digit in an integer is greater than the digit to the left, the integer is said to be "monotonic". For example, 12 is a monotonic integer since 2 > 1. How many positive two-digit monotonic integers are there?

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III

(A) 28 (B) 32 (C) 36 (D) 40 (E) 44

7. A perfect square is an integer that is the square of an integer. Suppose that m and n are positive integers such that mn > 15. If 15mn is a perfect square, what is the least possible value of mn ?

a, 2a - 1, 3a - 2, 4a - 3, . . .

3. For a particular number a, the first term in the sequence above is equal to a, and each term thereafter is 7 greater than the previous term. What is the value of the 16th term in the sequence?

8. M is a set of six consecutive even integers. When the least three integers of set M are summed, the result is x. When the greatest three integers of set M are summed, the result is y. Which of the following is true?

4. If p is a prime number, how many factors does p3 have?

(A) One (B) Two (C) Three (D) Four (E) Five

(A) y = x - 18 (B) y = x + 18 (C) y = 2x (D) y = 2x + 4 (E) y = 2x + 6

tutor

pg. 1

SAT Math Hard Practice Quiz

9. A three-digit number, XYZ , is formed of three different non-zero digits X, Y , and Z. A new number is formed by rearranging the same three digits. What is the greatest possible difference between the two numbers? (For example, 345 could be rearranged into 435, for a difference of 435 - 345 = 90.)

10. An integer is subtracted from its square. The result could be which of the following?

(A) A negative integer. (B) An odd integer. (C) The product of two consecutive even integers. (D) The product of two consecutive odd integers. (E) The product of two consecutive integers.

tutor

pg. 2

SAT Math Hard Practice Quiz

Algebra and Functions

1. Let m be an even integer. How many possible values of m satisfy m + 7 3 ?

(A) One (B) Two (C) Three (D) Four (E) Five

4. Let m and n be positive integers such that one-third of m is n less than one-half of m. Which of the following is a possible value of m ?

(A) 15 (B) 21 (C) 24 (D) 26 (E) 28

2.

Let

x

be defined by

x

=

x+3 x-1

for

any

x

such

that

x = 1. Which of the following is equivalent to x - 1 ?

5. If a and b are numbers such that (a - 4)(b + 6) = 0, then what is the smallest possible value of a2 + b2 ?

(A)

x+2 x-1

(B)

4 x-1

(C)

2x + 4 x-1

(D)

2 x-1

(E)

x+2 x-2

6. Let f (x) = ax2 and g(x) = bx4 for any value of x. If a and b are positive constants, for how many values of x is f (x) = g(x) ?

(A) None (B) One (C) Two (D) Three (E) Four

3. Let a and b be numbers such that a3 = b2. Which of the following is equivalent to b a ? (A) b2/3

7. Let a and b be numbers such that 30 < a < 40 and 50 < b < 70. Which of the following represents all possible values of a - b ?

(B) b4/3 (C) b2

(A) -40 < a - b < -20 (B) -40 < a - b < -10 (C) -30 < a - b < -20 (D) -20 < a - b < -10 (E) -20 < a - b < 30

(D) b3

(E) b4

tutor

pg. 3

SAT Math Hard Practice Quiz

x 3

+

y 12

=

z

8. In the equation shown above, x, y, and z are positive

integers. All of the following could be a possible value of

y EXCEPT

11. Amy is two years older than Bill. The square of Amy's age in years is 36 greater than the square of Bill's age in years. What is the sum of Amy's age and Bill's age in years?

(A) 4 (B) 6 (C) 8 (D) 12 (E) 20

72 + 72 = m n

9. In the equation above, m and n are integers such that m > n. Which of the following is the value of m ?

y

1 1

y = f (x) x

(A) 6 (B) 12 (C) 16 (D) 24 (E) 48

12. The function f is graphed in its entirety above. If the function g is defined so that g(x) = f (-x), then for what value of x does g attain its maximum value?

t

0

1

2

N (t)

128

16

2

(A) -3 (B) -2 (C) 0 (D) 2 (E) 3

10. The table above shows some values for the function N . If N (t) = k ? 2-at for positive constants k and a, what is the value of a ?

(A) -3

(B) -2

(C)

1 3

13. If (x + 1)2 = 4 and (x - 1)2 = 16, what is the value of x?

(A) -3 (B) -1 (C) 1 (D) 3 (E) 5

(D) 2

(E) 3

tutor

pg. 4

SAT Math Hard Practice Quiz

x

x

12

8

14. On the number line above, the tick marks correspond to consecutive integers. What is the value of x ?

16. Two cars are racing at a constant speed around a circular racetrack. Car A requires 15 seconds to travel once around the racetrack, and car B requires 25 seconds to travel once around the racetrack. If car A passes car B, how many seconds will elapse before car A once again passes car B ?

15. The value of y increased by 12 is directly proportional to the value of x decreased by 6. If y = 2 when x = 8, what is the value of x when y = 16 ?

(A) 8 (B) 10 (C) 16 (D) 20 (E) 28

tutor

pg. 5

Geometry

SAT Math Hard Practice Quiz

a

c

y x2

y= 2

b Note: Figure not drawn to scale.

(a, b)

x y=

2

3. In the figure above, 3 < a < 5 and 6 < b < 8. Which of the following represents all possible values of c ?

O

x

(A) 0 < c < 3

(B) 1 < c < 3

(C) 0 < c < 13

(D) 1 < c < 13

(E) 3 < c < 13

1. The curve y = x2/2 and the line y = x/2 intersect at the origin and at the point (a, b), as shown in the figure above. What is the value of b ?

(A)

1 8

(B)

1 4

(C)

1 2

(D) 1

(E) 2

4. Line l goes through points P and Q, whose coordinates

are (0, 1) and (b, 0), respectively. For which of the fol-

lowing

values

of

b

is

the

slope

of

line

l

greater

than

1 -2

?

(A)

1 2

(B) 1

B

60

6

8

A

C

2. In the figure above, AB = 6 and BC = 8. What is the area of triangle ABC ?

(C)

3 2

(D)

5 3

(E)

5 2

(A) 122 (B) 123 (C) 242 (D) 243 (E) 36 3

tutor

pg. 6

SAT Math Hard Practice Quiz

C

D

B

A

5. In the figure above, AB = 6 and BC = 8. What is the length of segment BD ?

(A) 2

(B)

12 5

(C) 4

(D)

24 5

(E) 6

8. In the figure above, a square is inscribed in a circle. If the area of the square is 36, what is the perimeter of the shaded region?

(A)

6

+

3 2

2

(B) 6 + 3

(C) 6 + 3 2

(D) 36 + 6 2

(E)

9 2

-

9

6. If four distinct lines lie in a plane, and exactly two of

them are parallel, what is the least possible number of

points of intersection of the lines?

B

(A) Two (B) Three (C) Four (D) Five (E) More than five

A

7

C

Note: Figure not drawn to scale.

7. The perimeter of a particular equilateral triangle is numerically equal to the area of the triangle. What is the perimeter of the triangle?

9. In the figure above, AC = 7 and AB = BC. What is the smallest possible integer value of AB ?

(A) 3 (B) 4 (C) 43 (D) 123 (E) 18 3

tutor

pg. 7

y (2, a)

SAT Math Hard Practice Quiz y

y = x+c

x

P

O

(10, 0)

10. In the figure above, two line segments in the x-y plane form a right triangle with the x-axis. What is the value of a ?

(A) 2 2 (B) 4 (C) 5 (D) 42 (E) 5 2

y = 2x - 1

x O

Note: Figure not drawn to scale.

11. The perimeter of square ABCD is x, and the perimeter of isosceles triangle EF G is y. If AB = EF = F G, which of the following must be true?

(A)

0

<

y

<

x 4

(B)

x 4

<

y

<

x 2

(C)

x 2

<

y

<

x

12. In the x-y plane, the lines y = 2x - 1 and y = x + c intersect at point P , where c is a positive number. Portions of these lines are shown in the figure above. If the value of c is between 1 and 2, what is one possible value of the x-coordinate of P ?

(D) x < y < 2x

(E) 2x < y < 4x

tutor

pg. 8

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download