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,  Dn,  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  25b  represents a 2 digit number to the base  b.  If the number  52b  is twice  25b  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