1.
If a² = 12, then a^{4} = ?
Correct Answer
A. 144
Explanation
a4 (a to the fourth power) = a2 (a squared) x a2 (a squared) = 12 x 12 = 144
2.
A dress on sale in a shop is marked at $D. During the discount sale its
price is reduced by 15%. Staff are allowed a further 10% reduction on
the discounted price. If a staff member buys the dress what will she
have to pay in terms of D?
Correct Answer
C. 0.765D
Explanation
If the price is reduced by 15 %, then the new price will be 0.85D
If this new price is further reduced by 10%, the discounted price will be 0.9 x 0.85D = 0.765D
3.
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?
Correct Answer
E. 8
Explanation
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
4.
If n is even, which of the following cannot be odd?I n + 3II 3nIII n² - 1
Correct Answer
B. II only
Explanation
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.
5.
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 24II 20III 5
Correct Answer
D. II and III only
Explanation
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.
6.
N is an integer chosen at random from the set{5, 7, 9, 11 }p is chosen at random from the set{2, 6, 10, 14, 18}What is the probability that n + p = 23 ?
Correct Answer
A. 0.1
Explanation
Each of the integers in the first set could be combined with any from the second set, giving a total of 4 x 5 = 20 possible pairs.
Of these the combinations that could give a sum of 23 are (5 + 18), and (9 + 14)
This means that the probability of getting a sum of 23 is 2/20 = 1/10
7.
If the radius of the circle with centre O is 7 and the measure of angle
AOB is 100, what is the best approximation to the length of arc AB ?
Correct Answer
D. 12
Explanation
If the radius is 7, the circumference = 14π
The length of the arc is 100/360 of the circumference
Taking π as 22/7 we get
(100 x 14 x 22) / (360 x 7) which reduces to 440/ 36 = 12.22 (i.e. approx. 12)
8.
Sheila works 8 hours per day on Monday, Wednesday and Friday, and 6
hours per day on Tuesday and Thursday. She does not work on Saturday
and Sunday. She earns $324 per week. How much does she earn in dollars
per hour?
Correct Answer
C. 9
Explanation
Total hours worked = 8 x 3 + 6 x 2 = 36
Total earned = 324. Hourly wage = 324 / 36 = 9
9.
If the product of 6 integers is negative, at most how many of the integers can be negative?
Correct Answer
D. 5
Explanation
Multiplying two negatives (or any even multiple) results in a positive. But multiplying three negatives (or any odd multiple) gives a negative. If the result of multiplying 6 negatives is odd, the largest number of negative integers will be the largest odd number (i.e.5)
10.
If a positive integer n, divided by 5 has a remainder 2, which of the following must be true?
I n is odd
II n + 1 cannot be a prime number
III (n + 2) divided by 7 has remainder 2
Correct Answer
A. None
Explanation
You can find the integers which when divided by 5 have a remainder 2 by adding 2 to all multiples of 5. So we have n = 9 , 12, 17, 22 etc.
From this series we can see that n does not have to be odd.
Also n + 1 can be a prime because, for example, 12 + 1 = 13
And (n + 2) / 7 has a remainder 2 in some cases but not all.
Remember the question asks us for what MUST be true, and we see that none of the statements are true in all cases.
11.
A solid cube of side 6 is first painted pink and then cut into smaller
cubes of side 2. How many of the smaller cubes have paint on exactly 2
sides?
Correct Answer
C. 12
Explanation
When the larger cube is cut into smaller cubes, the corner cubes will have paint on three sides. The cubes in the middle of the faces will have paint on only one side, but the cubes cut from the edges will have paint on two sides. In this case, there will be only one cube one each edge (excluding the corners), and since there are 12 edges, there will be 12 cubes with paint on two sides. (Drawing a diagram will make this much clearer.)
12.
Line l contains the points (3,1) and (4,4). If line m is a different line, parallel to line l in the same coordinate plane, which of the following could be the equation of line m?
Correct Answer
D. Y = 3x + 1
Explanation
Slope of the line l = (4 – 1) / (4 – 3) = 3/1 = 3
Parallel lines have the same slope. In the equation of a straight line, y = mx + c, m represents the slope, so we are looking for an equation where m = 3
But both answer choices A and D have slope 3.
Now we calculate the intercept, c, for line l. Using the coordinate of one of the points on l, we get: 4 = (3 x 4) + c ; 4 – 12 = c; c = -8
Hence, answer A is the equation of line l, and answer D must be the equation of a parallel line.
13.
A triangle has a perimeter 13. The two shorter sides have integer
lengths equal to x and x + 1. Which of the following could be the
length of the other side?
Correct Answer
C. 6
Explanation
The measure of the third side of a triangle must lie between the sum and the difference of the other two sides. Using this fact along with the answer choices, we can eliminate the wrong answers.
(A) cannot be right because 2 would not be the longest side.
(B) cannot be right because again it would not be the longest side
(C) could be right because the other two sides would be 3 and 4.
(D) and (E) cannot be right because this third side would be greater than the sum of the other two sides.
14.
A machine puts c caps on bottles in m minutes. How many hours will it take to put caps on b bottles?
Correct Answer
B. Bm/60c
Explanation
Substitute sensible numbers and work out the problem. Then change the numbers back to letters. For example if the machine puts 6 caps on bottles in 2 minutes, it will put 6 /2 caps on per minute, or (6 /2) x 60 caps per hour. Putting letters back this is 60c/m.If you divide the required number of caps (b) by the caps per hour you will get time taken in hours. This gives bm/60c (Remember dividing by a fraction is the same as multiplying by its inverse.)
15.
Paint needs to be thinned to a ratio of 2 parts paint to 1.5 parts
water. The painter has by mistake added water so that he has 6 liters
of paint which is half water and half paint. What must he add to make
the proportions of the mixture correct?
Correct Answer
A. 1 liter paint
Explanation
At the moment the paint is 3 liters of paint and 3 liters of water. We need to add to this to make the new ratio 2 liters paint to 1.5 liters water. As this is equivalent to 4 : 3 we can see that we have the right amount of water, and just need 1 liter of paint to make it correct.
16.
6 pints of a 20 percent solution of alcohol in water are mixed with 4
pints of a 10 percent alcohol in water solution. The percentage alcohol
in the new solution is ___.
Correct Answer
A. 16
Explanation
Total new volume is 10 pints. The first solution contributes 0.2 x 6 pints alcohol, and the second contributes 0.1 x 4. Total alcohol = 1.6. Percentage alcohol = (1.6 / 10) x 100 = 16