Sample Test

1. Suppose that f is a function with the property that, for all x and y, f(x+y)=f(x)+f(y)+1, and that f(1)=2. What it the value f(3)?

A. 4
B. 5
C. 6
D. 7
E. 8

2. If the width of a rectangle is doubled and the length is increased by 5, the area is tripled. What is the length of the rectangle?

A. 5
B. 10
C. 15
D. 20
E. 30

3. What is the coefficient of x⁷ in the expansion of:

A. -8
B. 12
C. 6
D. -12
E. 8

4. Suppose that an operation * is defined by a*b=a³+b². What is the value of 5*(3*4)?

A. 1849
B. 1857
C. 1974
D. 1999
E. 2441

5. Two vertical poles of different heights stand on level ground. Straight lines from the top of each pole to the base of the other cross 24 feet above the ground. The short pole is 40 feet. What is the height of the long pole?

A. 48 feet
B. 58 feet
C. 60 feet
D. 62 feet
E. 70 feet

6. Suppose f(x) is a polynomial with integer coefficients, and 3 and -3 are roots. Which of the following is a possible value of f(8)?

A. 9
B. 27
C. 44
D. 60
E. 110

7. If sin(x) + cos(x) = 1/2, find the value of sin³(x)+cos³(x).

A. 1/4
B. 5/16
C. 9/16
D. 11/16
E. 11/8

8. What is the area of an equilateral triangle if the circumscribed circle has radius 10?

A. 503
B. 75
C. 753
D. 1003
E. 140

9. What is the largest prime divisor of 2¹⁶-16?

A. 7
B. 11
C. 13
D. 17
E. 19

10. The lengths of the sides of a triangle are 8, 10, 12. Find the perimeter of a similar triangle whose area is 9 times that of the given triangle.

A. 45
B. 60
C. 90
D. 180
E. 270

11. How many pairs of integers (m, n) satisfy the equation m+n=mn?

A. 1
B. 2
C. 3
D. 4
E. More than 4

12. The average age of a group of doctors and lawyers is 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the number of doctors to the number of lawyers is

A. 3:2
B. 3:1
C. 2:3
D. 2:1
E. 1:2

13. The latitude of a point of the surface of the earth is specified by the angle in the picture. Virginia Beach is 3652' North. Find the distance from Virginia Beach to the equator. The radius of the earth is approximately 3960 miles.

A. 1550
B. 2050
C. 2550
D. 3050
E. None of these.

14. Evaluate the sum 1 - 2 + 3 - 4 + 5 - 6 + ... + 97 - 98 + 99 - 100

A. 101
B. 99
C. -50
D. 50
E. 101

15. A ball is dropped from a height of 6 feet. On each bounce the ball rebounds one third of its previous height. Find the total distance traveled by the ball.

A. 27
B. 30
C. 33
D. 36
E. None of these.

16. The set of all points P such that the sum of the distances from P to two fixed points A and B equals twice the distance between A and B is

A. the line segment from A to B.
B. the line passing through A and B.
C. the perpendicular bisector of the line from A to B.
D. an ellipse.
E. a parabola.

17. Each of the three circles in the figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is

A. 36 + 92
B. 36 + 93
C. 36 + 63
D. 18 + 183
E. 45

18. Point E is on side AB of the square ABCD. If EB has length 1 and EC has length 2, then the area of the square is

A. 13
B. 25
C. 3
D. 23
E. 5

19. How many of the first 100 numbers are divisible by 2, 3, 4, and 5?

A. 0
B. 1
C. 2
D. 3
E. 4

20. The measure of the interior angles of a convex polygon are in arithmetic progression. If the smallest angle is 100 and the largest 140, then the number of sides the polygon has is

A. 6
B. 8
C. 10
D. 11
E. 12