1. If a² = 12, then a4 =

A. 144

B. 72

C. 36

D. 24

E. 16

Correct Answer: A

解析:

a4 = a2 x a2 = 12 x 12 = 144

2. If n is even, which of the following cannot be odd?

I n + 3
II 3n
III n² - 1

A. I only

B. II only

C. III only

D. I and II only

E. I, II and III

Correct Answer: B

解析:

In case I , even plus odd will give odd. In case II, odd times even will give even. In case III even squared is even, and even minus odd is odd. (You can check this by using an easy even number like 2 in place of n). Only case II cannot be odd.

3. One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?

I 24
II 20
III 5

A. I only

B. II only

C. III only

D. II and III only

E. I, II and III

Correct Answer: D

解析:

The maximum area of the triangle will come when the given sides are placed at right angles. If we take 8 as the base and 5 as the height the area = ½ x 8 x 5 = 20. We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20.

4. A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?

A. 12

B. 11

C. 10

D. 9

E. 8

Correct Answer: E

解析:

Food consumed per day = 39/6. In the remaining days it will consume 91 - 39 pounds = 52 pounds. Now divide the food by the daily consumption to find the number of days. 52 / (39/6) = 52 x (6 / 39) = 8

5. A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?

A. 8p

B. pq

C. pq + 27

D. -p

E. (p - q)6

Correct Answer: C

解析:

A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5. Consider the answer choices in turn. 8 is the cube of 2, and p is a cube, and so the product will also be a cube. pq will also be a cube as shown above.pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube. -p will be a cube. Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).