Date:  Math 105/110  Name:
Start:  Gateway Test  ID #:
End:  Section:

This test is printed on both sides. Circle the letter of the correct answer to each question. Calculators are not allowed. Passing is 20. You should have been given two blank sheets of scratch paper. Use none of your own paper. Allowed time: 60 minutes.

7. Which of the following is equivalent to \frac{3 − 5a}{\frac{1}{2} + \frac{2}{a}}?

(a) \frac{6a^{2} + 10a}{a − 2}  (b) \frac{3a − 5a^{2}}{a + 2}  (c) 6  (d) 3a − 5a^{2}  (e) \frac{6a − 10a^{2}}{a + 4}

20. Which of the following is equivalent to \frac{5z^{8}}{25z^{n+7}} ?

(a) \frac{1}{5}z^{−n− \frac{1}{15}}  (b) \frac{1}{5}z^{n+15}  (c) \frac{1}{5}z^{1−n}  (d) \frac{1}{20}z^{\frac{8}{n+7}}  (e) \frac{1}{5}z^{\frac{8}{n+7}}

21. Given that \frac{1}{a} + \frac{1}{b} = 4 and \frac{1}{a} − b = 2, then a =

(a) \frac{1}{3}  (b) \frac{3}{4}  (c) \frac{33}{20}  (d) 1  (e) \frac{2}{3}

34. \sqrt{60}\sqrt{\frac{y}{4}} =

(a) \frac{15}{4} \sqrt{y}  (b) \sqrt{15y}  (c) 15\sqrt{y}  (d) \sqrt{15} y  (e) \sqrt{30y}

47. If 4^{y} = \frac{2^{1/2}}{\sqrt{32}}, then y =

(a) −\frac{1}{2}  (b) 0  (c) \frac{1}{2}  (d) 1  (e) −1

56. Find all values of y that satisfy |−y + 2| ≤ 4.

(a) −2 ≤ y ≤ 6  (b) y ≤ 6  (c) 2 ≤ y ≤ 6  (d) −4 ≤ y ≤ 4  (e) −2 ≤ y ≤ 2

65. What is the distance between the two points (3, 2) and (7, −1)?

(a) \sqrt{17}  (b) 7  (c) \sqrt{7}  (d) 10  (e) 5

71. Suppose that x + y = 1 and x + 2y = 3. Then x − y =

(a) −3  (b) −2  (c) −1  (d) 0  (e) 1

81. If g(x) = (x − 2)^{2} and h(s) = 2s − 2, then g(h(s)) =

(a) 4s^{2} − 4s + 4  (b) 4s^{2} + 8s + 16  (c) 4s^{2}  (d) 4s^{2} − 16s + 16  (e) s^{2} − 2s + 4

92. A water tank is initially \frac{1}{8} full. After adding 13 gallons of water, it is \frac{2}{3} full. What is the capacity of the tank in gallons?

(a) 20  (b) 19\frac{1}{2}  (c) 24  (d) 104  (e) 26

THIS TEST IS CONTINUED ON THE BACK

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107. The slope of the line having equation x = −3y − 2 is

(a) −2  (b) \frac{1}{2}  (c) −\frac{1}{3}  (d) −3  (e) 3

114. If g(y) = log_{3} y, then g(\frac{1}{3}) =

(a) −1  (b) −3  (c) 3^{1/3}  (d) 1  (e) \frac{1}{3}

124. Suppose that the line with equation y = mx + b passes through the points (−2, 3) and (6, −9). Then m + b =

(a) \frac{3}{2}  (b) 2  (c) 3  (d) 0  (e) −\frac{3}{2}

138. The domain of the function defined by the formula f(x) = \frac{(x + 1)(x + 2)}{(x + 2)(x + 3)} is

(a) All real numbers but −3 (b) All real numbers but −2 (c) All real numbers but −3, −2, and −1 (d) All real numbers but −1 and −2 (e) All real numbers but −2 and −3

150. \frac{|(−5 − (−8))^{3}|}{−|−3|} =

(a) −3  (b) 3  (c) −9  (d) 9  (e) −\frac{2197}{3}

157. Which of the following equals (x^{2} − 2x − 1)^{2}?

(a) x^{4} − 4x^{3} + 6x^{2} − 4x + 1 (b) x^{4} + 4x^{2} + 1 (c) x^{4} + 4x^{3} − 4x^{2} + 4x + 1 (d) x^{4} − 4x^{2} − 1 (e) x^{4} − 4x^{3} + 2x^{2} + 4x + 1

168. What is the range of the function defined by the formula f(x) = 4x^{2} + 1?

(a) All real numbers ≥ 1 (b) All real numbers ≥ 5 (c) All real numbers (d) All real numbers ≥ 3 (e) All real numbers ≥ −\frac{1}{4}

177. Which of the following equals \frac{e}{(e^{−1} − e^{−2})^{−2}} ?

(a) e^{3} − e^{5}  (b) e^{−1} − e^{−3}  (c) e^{3} − 2e^{4} + e^{5}  (d) e^{−2} − 2 + e^{−3}  (e) e^{−1} − 2e^{−2} + e^{−3}

182. The perimeter of a rectangle is eight times its width. If the length of the rectangle is 90 inches,what is its width in inches?

(a) 11\frac{1}{4}  (b) 30  (c) 720  (d) 98  (e) 22\frac{1}{2}

199. Which of the following equals \frac{1}{(1 − \sqrt{a})(1 + \sqrt{a})}?

(a) 1  (b) \frac{1 + \sqrt{a}}{1 − \sqrt{a}}  (c) \frac{1}{1 − a}  (d) \frac{1 − \sqrt{a}}{1 + \sqrt{a}}  (e) \frac{1}{1 + a}

204. If \frac{1}{2x} − \frac{1}{3x} = 3 − \frac{1}{6x}, then x =

(a) \frac{1}{9}  (b) −\frac{1}{3}  (c) \frac{1}{2}  (d) \frac{1}{6}  (e) \frac{1}{3}

211. Let x be Imogene’s present age in years. In eight years Imogene will be three times as old as she was two years ago. Find x^{2} − x.

(a) 20  (b) 42  (c) 342  (d) 90  (e) 6

221. Let T be the right triangle with vertices at (0, 0), (3, 0), and (3, 6). How long is the hypotenuse of T?

(a) 9  (b) 3\sqrt{5}  (c) 3  (d) 2\sqrt{2}  (e) 2\sqrt{5}

234. Which of the following is equivalent to \frac{x^{2} + 2x − 15}{x^{2} + 7x + 10}?

(a) \frac{x + 3}{x + 5}  (b) \frac{x − 5}{x + 2}  (c) \frac{x + 2}{x + 7}  (d) \frac{x − 3}{x + 2}  (e) \frac{x + 3}{x + 2}

248. A line with equation y = mx+b passes through the point (1, −1) and is parallel to the line with equation y = −x − 5. Then m + b =

(a) 0  (b) 1  (c) −1  (d) 5  (e) −5

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