WORCESTER POLYTECHNIC INSTITUTE

TWENTY-FIRST ANNUAL INVITATIONAL MATH MEET

OCTOBER 21, 2008

INDIVIDUAL EXAM QUESTION SHEET WITH ANSWERS

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

Ans: (-3,0 and (0,4) 1 point

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?

Ans: 2n 1’s 1 point

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?

Ans: b = 8 1 point

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

Ans: 10 1 point

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?

Ans: x-2 = -1/(4*2) (y-5)² 2 points

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?

Ans: 2.5 : 1 ratio 2 points

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

Ans: 1 2 points

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

Ans: 20 2 points

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?

Ans: b = 4 3 points

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?

Ans: p(E) = 7/12 3 points

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?

Ans: k = 4/3 3 points