[Math Center]

1. Ashley and Beatrice received the same score on a physical fitness test. The scores for this test, [latex]t[latex], are determined by the formula [latex]t=3ps-25m[latex] where [latex]s[latex] and [latex]p[latex] are the numbers of sit-ups and push-ups the athlete can do in one minute and [latex]m[latex] is the number of minutes she takes to run a mile. Ashley did [latex]10[latex] sit-ups and [latex]10[latex] push-ups and ran an [latex]8[latex]-minute mile. Beatrice did half as many sit-ups and twice as many push-ups. If both girls received the same overall score, how many minutes did it take Beatrice to run the mile?

(A) [latex]4[latex]
(B) [latex]8[latex]
(C) [latex]10[latex]
(D) [latex]16[latex]
(E) [latex]20[latex]


2. Define [latex]x*[latex] by the equation [latex]{ x }^{ * }=\frac{ \pi }{ x }[latex].  Then [latex]{\left( {\left( \pi \right) }^{\ast }\right) }^{\ast }=[latex]

(A) [latex]-\frac{ 1 }{ \pi }[latex]
(B) [latex]-\frac{ 1 }{ 2 }[latex]
(C) -π
(D) [latex]\frac{ 1 }{ \pi }[latex]
(E) π


3. The functions [latex]f[latex] and [latex]g[latex] are defined as [latex]f(x, y) = 2x + y[latex] and [latex]g(x, y) = x + 2y[latex]. What is the value of [latex]f(3, 4)[latex] ?

(A) [latex]f(4, 3)[latex]
(B) [latex]f(3, 7)[latex]
(C) [latex]f(7, 4)[latex]
(D) [latex]g(3, 4)[latex]
(E) [latex]g(4, 3)[latex]


4. A function [latex]f(x)[latex] is defined for all real numbers by the expression [latex](x-\frac{ 1 }{ 5 })(x-\frac{ 2 }{ 5 })(x-\frac{ 3 }{ 5 })(x-\frac{ 4 }{ 5 })[latex]. For which one of the following values of [latex]x[latex], represented on the number line, is [latex]f(x)[latex] negative?


(A) Point [latex]A[latex]
(B) Point [latex]B[latex]
(C) Point [latex]C[latex]
(D) Point [latex]D[latex]
(E) Point [latex]E[latex]


5. The functions [latex]f[latex] and [latex]g[latex] are defined as [latex]f(x, y) = 2x + y[latex] and [latex]g(x, y) = x + 2y[latex].

Quantity A Quantity B
[latex]f(3, 4) + g(3, 4)[latex] [latex]f(4, 3) + g(4, 3)[latex]


6. pikachu(1)In the function above, if [latex]f(k) = 2[latex], then which one of the following could be a value of [latex]k[latex] ?

(A) [latex]-1[latex]
(B) [latex]0[latex]
(C) [latex]0.5[latex]
(D) [latex]2.5[latex]
(E) [latex]4[latex]


7. The function [latex]f(x, y)[latex] is defined as the geometric mean of [latex]x[latex] and [latex]y[latex] (geometric mean of [latex]x[latex] and [latex]y[latex] equals [latex]\sqrt{ xy })[latex], and the function [latex]g(x, y)[latex] is defined as the least common multiple of [latex]x[latex] and [latex]y[latex]. [latex]a[latex] and [latex]b[latex] are two different prime numbers.

Quantity A Quantity B
[latex]f(a, b)[latex] [latex]g(a, b)[latex]


8. If [latex]f(x) = { x }^{ 2 }[latex], what is [latex]f(m + n) + f(m-n)[latex]?

(A) [latex]{ m }^{ 2 }+{ n }^{ 2 }[latex]
(B) [latex]{ m }^{ 2 }-{ n }^{ 2 }[latex]
(C) [latex]{ 2m }^{ 2 }+{ 2n }^{ 2 }[latex]
(D) [latex]{ 2m }^{ 2 }-{ 2n }^{ 2 }[latex]
(E) [latex]{ m }^{ 2 }{ n }^{ 2 }[latex]


9. A pottery store owner determines that the revenue for sales of a particular item can be modeled by the function [latex]r(x) = 50\sqrt{ x }-40[latex], where [latex]x[latex] is the number of the items sold. How many of the items must be sold to generate $[latex]110[latex] in revenue?

(A) [latex]5[latex]
(B) [latex]6[latex]
(C) [latex]7[latex]
(D) [latex]8[latex]
(E) [latex]9[latex]

10. At time [latex]t = 0[latex], a projectile was fired upward from an initial height of [latex]10[latex] feet. Its height after [latex]t[latex] seconds is given by the function [latex]h(t) = p-10{ (q-t) }^{ 2 }[latex] , where [latex]p[latex] and [latex]q[latex] are positive constants. If the projectile reached a maximum height of [latex]100[latex] feet when [latex]t = 3[latex], then what was the height, in feet, of the projectile when [latex]t = 4[latex] ?

(A) [latex]62[latex]
(B) [latex]70[latex]
(C) [latex]85[latex]
(D) [latex]89[latex]
(E) [latex]90[latex]

11.  Define the symbol [latex]*[latex] by the following equation: [latex]{ x }^{ * } = 1-x[latex], for all non-negative [latex]x[latex].  If [latex]{ ({ (1-x) }^{ * }) }^{ * } ={ (1-x) }^{ * }[latex], then [latex]x =[latex]

(A) [latex]1/2[latex]
(B) [latex]3/4[latex]
(C) [latex]1[latex]
(D) [latex]2[latex]
(E) [latex]3[latex]

12. The figure above shows the graph of the function [latex]f[latex] defined by for all numbers [latex]x[latex]. For which of the following functions [latex]g,f (x)[latex]=⎪[latex]2x[latex]⎪[latex]+4[latex] defined for all numbers [latex]x[latex], does the graph of [latex]g[latex] intersect the graph of [latex]f[latex] ? 

(A) [latex]g(x)=x-2[latex] 
(B) [latex]g(x)=x+3[latex] 
(C) [latex]g(x)=2x-2[latex] 
(D) [latex]g(x)=2x+3[latex] 
(E) [latex]g(x)=3x-2[latex]

13. For each of the following functions, give the domain and a description of the graph in the [latex]xy[latex]-plane, including its shape, and the [latex]x[latex]- and [latex]y[latex]-intercepts. [latex]y=f(x)[latex]

(a) [latex]f(x)=-4[latex]
(b) [latex]f(x)=100-900x[latex]
(c) [latex]2 f(x)=5-{ (x+20) }^{ 2 }[latex]
(d) [latex]f(x)=\sqrt{ x+2 }[latex] 
(e) [latex]f(x)=x+[latex]⎪[latex]x[latex]⎪


14. The operation [latex]\bigotimes[latex] is defined for all integers [latex]x[latex] and [latex]y[latex] as [latex]x\bigotimes y=xy-y[latex].  If [latex]x[latex] and [latex]y[latex] are positive integers, which of the following CANNOT be zero? 

(A) [latex]x\bigotimes y[latex]
(B) [latex]y\bigotimes x[latex]
(C) [latex](x-1)\bigotimes y[latex] 
(D) [latex](x+1)\bigotimes y[latex] 
(E) [latex]x\bigotimes (y-1)[latex]


15. [latex]f(x) = 2x-3[latex]
[latex]f(m) = -11[latex]

Quantity A Quantity B
The value of [latex]m[latex] Half the value of[latex]f(m)[latex]




1 B
2 C
3 E
4 B
5 C
6 D
7 B
8 C
9 E
10 E
11 A
12 E
13 (a) Domain: the set of all real numbers. The graph is a horizontal line with [latex]y[latex] intercept [latex]-4[latex]  and no [latex]x[latex]-intercept.
(b) Domain: the set of all real numbers. The graph is a line with slope [latex]-900[latex], [latex]y[latex]-intercept [latex]100[latex], and [latex]x[latex]-intercept [latex]1/9[latex]
(c) Domain: the set of all real numbers. The graph is a parabola opening downward with vertex at [latex](-20,5)[latex] line of symmetry [latex]x=-20, y[latex]-intercepts [latex]-395,x[latex]-intercepts [latex](-20, \sqrt{ 5 })[latex]. 
(d) Domain: the set of numbers greater than or equal to [latex]-2[latex]. The graph is half a parabola opening to the right with vertex at [latex](-2,0), x[latex]-intercept [latex]-2[latex], and [latex]y[latex]-intercept [latex]\sqrt{ 2 }[latex]. 
(e) Domain: the set of all real numbers. The graph is two half-lines joined at the origin: one half-line is the negative [latex]x[latex]-axis and the other is a line starting at the origin with slope [latex]2[latex]. Every nonpositive number is an [latex]x[latex]-intercept, and the y-intercept is [latex]0[latex]. The function is equal to the following piecewise-defined function


14 D
15 A