AMATYC SML Training Session – October 1, 2009

As always, these will be posted, with solution, to my website:  www.montgomerycollege.edu/~rpenn

Additional problems are available at the AMATYC website, http://www.amatyc.org

 

 

  1. What is the slope of a line parallel to the line with equation 2x – 5y = 10?

A.  2/5             B.  -2/5                        C.  5/2              D. -5/2             E. -2

 

Solving for y, y=2/5 x -2, so its slope is 2/5, and that would also be the slope of any line parallel to it.

 

  1. If f(x) = 3x-2, find f(f(f(3)))

A. 19               B.  55               C.  75               D. 107             E.  163

 

f(3)=7, f(7)=19, f(19)=55

 

  1. If h(x) = 2x-8, find

A.  -4               B.  ¼                C.  7                 D.  11              E. 20

 

2x-8 = 6 -> x=7

 

  1. Suppose f(x) = ax+b, and g(x)=bx+a (a, b are integers).  If f(1)=8 and f(g(50))-g(f(50))=28, find the product

A.  5                B.  12               C.  48               D.  182                        E.  210

 

f(1)=8 -> a+b=8.

f(g(50))-g(f(50))= 28 -> (50ab+a2 + b)-(50ab+b2+a) = 28 -> a2-b2 + b-a = 28 -> (a+b)(a-b)-(a-b) = (a+b-1)(a-b)=28

Since a+b=8, a+b-1=7, so 7(a-b)=28, or a-b=4.  This then tells us that a=6 and b=2, so ab=12

 

 

  1. If L has equation ax+by=c, M is its reflection across the y-axis, and N is its reflection across the x-axis,

which of the following must be true about M and N for all nonzero choices of a, b, and c?

A.  The x-intercepts are equal                          B.  The y-intercepts are equal

C.  The slopes are equal                                   D.  The slopes are opposite     

E. The slopes are reciprocals

 

The slope of each reflection is the opposite of the slope of L, so M and N have the same slope – C

(an easy way to see that C is the answer is to draw a “typical” line L, and then visually draw M and N)

 

  1. Which of the following is NOT a factor of ?

A. x-2              B.  x+2             C.  x-1             D.  x+1            E. x-3

 

(x-a) is a factor if and only if a is a root, which is also the same as saying that the graph of the polynomial hits the x-axis at a.

Graphing this, we see it crosses the x-axis at 2, -2, 1 and 3 but not -1, so D is the answer.

(btw – if you are using a TI-89, you can just ask it to factor the polynomial for you!)

 

  1. What is the remainder when  x3 - 2x2 +4 is divided by x+2?

A.  -12             B.  0                 C.  4                 D.  6                E.  12

 

This is the same as asking for the value of the polynomial when x=-2.  This is -12.

 

  1. A newspaper advertises that it sells the Sunday paper for one-third the price of the rest of the week’s papers. 

If a weekly subscription costs between $2.20 and $2.30, what is the cost of one Sunday paper and one daily paper?

A.  56¢             B.  81¢             C.  84¢             D.  87¢             E. $1.12

 

S=2D. 

1 week is 1S+6D = 8D, and the only number in the given range divisible by 8 is $2.24, which corresponds to $.28 / day.

So 1S and 1D costs $.84

 

  1.  What is the coefficient of in the expansion of

A.  -2               B.  -1               C. 2                 D.  7                E.  9

 

When you distribute the product, the quadratic terms come from (x2)(-1), (3x)(3x), and (-1)(x2), so the coefficient is 7.

 

  1.   Which of the following is a factor of ?

A.      B.     C.     D.     E.

 

For simplicity, let x=102009.  (x+1)2+(x+2)2-x2 = x2 +2x+1+ x2 +4x+4- x2 = x2 +6x+5 = (x+1)(x+5), so D.

 

 

 

  1. When the polynomial P(x) is divided by (x-2)2 , the remainder is 3x-3.

What is the remainder when (x-1)P(x) is divided by (x-1) (x-2)2 ?

A. 3x-3            B. 3x2  – 6x + 3           C. 3                  D. x-1              E. x-2

 

B.  The easiest way to see this:  Let P(x) = 3x-3.  Then P(x)/(x-2)2 = 0 with a remainder of x-3, and

(x-1)P(x)/ ((x-1)(x-2)2) = 0 with a remainder of (x-3)(x-1) = B