WORCESTER POLYTECHNIC INSTITUTE

TWENTY-FIRST ANNUAL INVITATIONAL MATH MEET

OCTOBER 21, 2008

INDIVIDUAL EXAM QUESTION SHEET

1. Consider the conic x²/9 +y²/16 = 1 and the line 3y – 4x = 12. At what points, if any, do they meet?

2. A Double Mersenne Number, D_n, is of the form 4ⁿ – 1 where n is a positive integer. What is the binary form (base 2) for such a number?

3. The symbol 25_b represents a 2 digit number to the base b. If the number 52_b is twice 25_b then what is the value of b?

4. What is the number of digits in the number 2¹² *5⁸ ?

5. A parabola is known to have its vertex at (2, 5) and its focus 2 units to the left of the vertex. What is its equation?

6. Two concentric circles are formed. What must be the ratio of the larger to smaller radii so that the area in between them is 84% of the area of the larger circle?

7. Express as single complex number: 1 + i + i² + i³ + . . . + i¹⁰⁰ where i² = - 1

8. How many different 5 digit numbers can be constructed using the digits 1,1,1,4,7?

9. The parabola y = ax² + bx + c has vertex (p,p) and y intercept at (0, -p) where p ≠ 0. What must the value of b equal in order for this to happen?

10. A fair six-sided die is tossed three times and the resulting sequence of numbers is recorded. What is the probability of the event E that either all of the numbers are equal or none of them is a 4?

11. Consider the solutions to the equation 3x² – 4x + k = 0. The value of k for which the

product of the roots is a maximum is what?