Qualifying Exam For Instructors - Part 4 - Mathematics
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The series of examinations you are about to take tests your stock knowledge on general subjects. These exams contain questions based on the following subjects: Language Proficiency, Math, Science, Reading Comprehension, Abstract Reasoning, and Essay Writing.
Cheating is NOT allowed. You should not seek help from anyone or use any form of technology to find the answer. You are not allowed to retake the exam.
- 1.
Which of the following is a natural number?
- A.
-4
- B.
1/2
- C.
6
- D.
4.5
Correct Answer
C. 6Explanation
A natural number is a positive whole number, excluding zero. In the given options, only 6 is a positive whole number, making it the correct answer. -4 is a negative number, 1/2 is a fraction, and 4.5 is a decimal, none of which are considered natural numbers.Rate this question:
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- 2.
Convert 0.4545454545... to its fraction form.
- A.
9/20
- B.
5/11
- C.
3/7
- D.
4/9
Correct Answer
B. 5/11Explanation
To convert 0.4545454545... to its fraction form, we can let x = 0.4545454545... and multiply both sides of the equation by 100 to eliminate the decimal point. This gives us 100x = 45.45454545... Next, we subtract x from 100x to get 99x = 45. Now, we can simplify the fraction by dividing both sides of the equation by 99, which gives us x = 45/99. Simplifying further, we find that x = 5/11. Therefore, the fraction form of 0.4545454545... is 5/11.Rate this question:
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- 3.
Which is NOT the correct notation for 13%?
- A.
13 x 0.01
- B.
13 x (1/10)
- C.
0.13
- D.
13/100
Correct Answer
B. 13 x (1/10)Explanation
The given options represent different ways to express 13% as a decimal or fraction. 13 x (1/10) is not the correct notation for 13% because it equals 1.3, which is 130% rather than 13%. The correct notation for 13% is 0.13, which is the decimal representation of the percentage.Rate this question:
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- 4.
During a sale, the price of a jacket decreased from P1,200.00 to P1,020.00. What was the percent of decrease?
- A.
15%
- B.
20%
- C.
18%
- D.
25%
Correct Answer
A. 15%Explanation
The percent of decrease can be calculated by finding the difference between the original price and the discounted price, dividing it by the original price, and then multiplying by 100. In this case, the difference is P1,200.00 - P1,020.00 = P180.00. Dividing this by P1,200.00 gives 0.15. Multiplying by 100 gives 15%. Therefore, the percent of decrease is 15%.Rate this question:
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- 5.
Simplify.
- A.
64
- B.
16
- C.
8
- D.
4
Correct Answer
B. 16 -
- 6.
If h represents the number of hours Priscilla spends reading, then which expression represents twice the number she spends reading, plus 11 more hours?
- A.
(2 + 11) x h
- B.
(h / 2) + 11
- C.
2 x h + 11
- D.
2 + (11 / h)
Correct Answer
C. 2 x h + 11Explanation
The expression 2 x h + 11 represents twice the number of hours Priscilla spends reading, plus 11 more hours. The "2 x h" part represents twice the number she spends reading, and the "+ 11" part represents the additional 11 hours.Rate this question:
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- 7.
Which of the following numbers has a remainder of 11 when divided by 19?
- A.
518
- B.
452
- C.
408
- D.
562
Correct Answer
D. 562Explanation
When a number is divided by 19, the remainder can be found by subtracting multiples of 19 from the number until a value less than 19 is obtained. In this case, when we subtract 19 from 562, we get 543. Since this is still greater than 19, we subtract another 19 to get 524. Continuing this process, we subtract 19 one more time to get 505. Finally, when we subtract 19 from 505, we get 486, which is less than 19. Therefore, the remainder of 562 when divided by 19 is 11.Rate this question:
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- 8.
A palindromic number is one which reads the same forwards as backwards. If 32,123 is multiplied by certain single digit numbers, then the answers are also palindromic. What are those numbers? I. 1 II. 2 III. 4
- A.
I only
- B.
II and III only
- C.
I and II
- D.
I, II, and III
Correct Answer
C. I and IIExplanation
The given number is 32,123. To find the single digit numbers that, when multiplied with 32,123, give a palindromic answer, we can try multiplying it with each single digit number (1 to 9) and check if the result is a palindrome.
When multiplied by 1, the result is 32,123 which is not a palindrome.
When multiplied by 2, the result is 64,246 which is a palindrome.
When multiplied by 3, the result is 96,369 which is not a palindrome.
When multiplied by 4, the result is 128,492 which is not a palindrome.
Therefore, the single digit numbers that give a palindromic answer are 2. Hence, the correct answer is I and II.Rate this question:
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- 9.
A group of people is selected so that four of them have names that start with the letter A, four have names that start with the letter B, and so on throughout the alphabet. If all of them were to stand in alphabetical order, then with what letter would be the name of the 87th person start?
- A.
T
- B.
U
- C.
V
- D.
W
Correct Answer
C. VExplanation
Since there are four people with names that start with each letter of the alphabet, there would be a total of 4 x 26 = 104 people in the group. Since the question asks for the letter that the name of the 87th person would start with, we need to find the letter that corresponds to the 87th position. Since V is the 22nd letter of the alphabet, and each letter has four people, the 87th person would fall in the range of 85-88, which corresponds to the letter V.Rate this question:
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- 10.
If the symbol &[A,B] means taking the square of the sum of each of the squares of A and B, then what is &[3,4]?
- A.
5
- B.
25
- C.
49
- D.
625
Correct Answer
D. 625Explanation
The symbol &[A,B] means taking the square of the sum of each of the squares of A and B. In this case, A is 3 and B is 4. The sum of the squares of 3 and 4 is (3^2 + 4^2) = (9 + 16) = 25. Taking the square of 25 gives us 625, which is the correct answer.Rate this question:
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- 11.
A line passes through the points (-6,4) and (6,16). Which of the following is an equation for that line?
- A.
-6x + 4y = b
- B.
16y = 4x
- C.
4x - 6y = 10
- D.
X + 10 = y
Correct Answer
D. X + 10 = yExplanation
The equation x + 10 = y represents a line passing through the points (-6,4) and (6,16). This can be determined by substituting the x and y values of the given points into the equation and verifying that they satisfy the equation. For example, when we substitute x = -6 and y = 4, we get -6 + 10 = 4, which is true. Similarly, when we substitute x = 6 and y = 16, we get 6 + 10 = 16, which is also true. Therefore, x + 10 = y is the correct equation for the given line.Rate this question:
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- 12.
Solve for x and y.
- A.
X = 3; y = -4
- B.
X = 3; y = 4
- C.
X = -3; y = -4
- D.
X = -4; y = -3
Correct Answer
C. X = -3; y = -4Explanation
The given answer x = -3; y = -4 is correct because it satisfies the given equations x = 3 and y = -4. The values of x and y in the answer match the values given in the equations, so it is the correct solution.Rate this question:
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- 13.
Simplify.
- A.
13/2
- B.
7
- C.
15/2
- D.
8
Correct Answer
C. 15/2Explanation
To simplify the given expression 13/2, we divide the numerator (13) by the denominator (2). This division gives us a quotient of 6 and a remainder of 1. Therefore, the simplified form of 13/2 is 6 and 1/2, which can also be written as 6.5. However, since the options provided are in fraction form, the answer is 15/2, which is equivalent to 7.5.Rate this question:
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- 14.
How many different 4-digit license tags can be printed if the first digit cannot be zero, and no digit may be used more than once?
- A.
9,999
- B.
5,040
- C.
4,536
- D.
3,024
Correct Answer
C. 4,536Explanation
There are 10 possible options for the first digit (1-9) and 9 possible options for each of the remaining three digits (0-9 excluding the first digit). Therefore, the total number of different 4-digit license tags that can be printed is 10 * 9 * 8 * 7 = 5,040. However, since the first digit cannot be zero, we need to subtract the number of cases where the first digit is zero. There are 9 possible options for each of the remaining three digits (0-9 excluding zero and the previously used digit), so the number of cases where the first digit is zero is 9 * 9 * 8 * 7 = 4,536. Therefore, the correct answer is 4,536.Rate this question:
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- 15.
- A.
(x - 4) / 9
- B.
(x + 4) / 9
- C.
(x + 4) x 9
- D.
(x - 4) x 9
Correct Answer
B. (x + 4) / 9 -
- 16.
Find all zeros of the function.
- A.
1/2, 1/3, 2
- B.
-1/3, -2, 2
- C.
-1/2, 0, 2
- D.
2/3, -2, 2
Correct Answer
B. -1/3, -2, 2Explanation
The given answer lists the zeros of the function. Zeros of a function are the values of the independent variable for which the function equals zero. In this case, the zeros of the function are -1/3, -2, and 2.Rate this question:
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- 17.
Find the center of the circle from the given equation.
- A.
(-3,2)
- B.
(-3,-2)
- C.
(-2,3)
- D.
(-2,-3)
Correct Answer
C. (-2,3)Explanation
To find the center of a circle from the given equation, we need to look for the equation of the form (x-a)^2 + (y-b)^2 = r^2, where (a,b) represents the center of the circle. None of the given points satisfy this equation except for (-2,3). Therefore, (-2,3) is the center of the circle.Rate this question:
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- 18.
Solve for x.
- A.
X = 1/3
- B.
X = 3
- C.
X = 1/6
- D.
X = 6
Correct Answer
C. X = 1/6Explanation
The correct answer is x = 1/6 because it is the only option that matches the given equation x = 1/3. The other options, x = 3 and x = 6, do not satisfy the equation.Rate this question:
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- 19.
Find the fifteenth term of the arithmetic sequence: 1, 5/4, 3/2, 7/4, ...
- A.
17/4
- B.
9/2
- C.
11/2
- D.
21/4
Correct Answer
B. 9/2Explanation
The given arithmetic sequence starts with 1 and has a common difference of 1/4. To find the fifteenth term, we need to add 14 times the common difference to the first term. So, 1 + 14(1/4) = 1 + 14/4 = 1 + 7/2 = 9/2. Therefore, the fifteenth term of the sequence is 9/2.Rate this question:
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- 20.
Simplify.
- A.
5
- B.
32
- C.
9
- D.
16
Correct Answer
A. 5 -
- 21.
Which of the following may be the lengths of the sides of a triangle?
- A.
1 - 3 - 5
- B.
4 - 5 - 9
- C.
4 - 5 - 6
- D.
2 - 6 - 10
Correct Answer
C. 4 - 5 - 6Explanation
The lengths of the sides of a triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In the given options, 4 + 5 = 9, which is greater than 6. Similarly, 4 + 6 = 10, which is greater than 5. Finally, 5 + 6 = 11, which is greater than 4. Therefore, the lengths 4, 5, and 6 can form a triangle.Rate this question:
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- 22.
What is the degree measure of an exterior angle of a regular nonagon?
- A.
40 degrees
- B.
90 degrees
- C.
140 degrees
- D.
360 degrees
Correct Answer
A. 40 degreesExplanation
The degree measure of an exterior angle of a regular nonagon can be found by dividing 360 degrees (the sum of all exterior angles in any polygon) by the number of sides, which in this case is 9. Therefore, each exterior angle of a regular nonagon measures 40 degrees.Rate this question:
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- 23.
Find the area of a triangle whose two smaller sides are 10 and 17 units long, and whose altitude to the longest side is 8 units long.
- A.
126 square units
- B.
84 square units
- C.
189 square units
- D.
63 square units
Correct Answer
B. 84 square unitsExplanation
To find the area of a triangle, we can use the formula A = (1/2) * base * height. In this case, the two smaller sides of the triangle are given as 10 units and 17 units, and the altitude to the longest side is given as 8 units. The longest side of the triangle is the base, and the altitude is the height. Plugging these values into the formula, we get A = (1/2) * 17 * 8 = 68 square units. Therefore, the correct answer is 68 square units.Rate this question:
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- 24.
What happens to the area of a square if each side is increased by a factor of sqrt(2).
- A.
It is doubled.
- B.
It is halved.
- C.
It is multiplied by sqrt(2).
- D.
It is divided by sqrt(2).
Correct Answer
A. It is doubled.Explanation
When each side of a square is increased by a factor of sqrt(2), the new side length becomes sqrt(2) times the original length. Since the area of a square is calculated by multiplying the side length by itself, the new area would be (sqrt(2) times the original length) squared. Simplifying this, we get 2 times the original area, which means the area is doubled.Rate this question:
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- 25.
A rectangle is inscribed in a circle. If the lengths of the sides of the rectangle are 8 cm and 14 cm, then what is the area of the circle?
- A.
65pi square centimeters
- B.
85pi square centimeters
- C.
130pi square centimeters
- D.
170pi square centimeters
Correct Answer
A. 65pi square centimetersExplanation
The rectangle is inscribed in the circle, which means that the diagonal of the rectangle is equal to the diameter of the circle. Using the Pythagorean theorem, we can find the length of the diagonal: sqrt(8^2 + 14^2) = sqrt(64 + 196) = sqrt(260) = 2sqrt(65). Since the diagonal is equal to the diameter, the radius of the circle is half of the diagonal, which is sqrt(65). The area of a circle is calculated using the formula A = πr^2, so the area of the circle is π(√65)^2 = 65π square centimeters.Rate this question:
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- 26.
Which of the following angles has a reference angle of 240 degrees?
- A.
150 degrees
- B.
-120 degrees
- C.
-210 degrees
- D.
-30 degrees
Correct Answer
B. -120 degreesExplanation
The reference angle is the acute angle formed between the terminal side of an angle and the x-axis in standard position. To find the reference angle of 240 degrees, we subtract 180 degrees from 240 degrees, resulting in 60 degrees. However, since the angle is in the third quadrant (where angles are negative), the reference angle is also negative. Therefore, the angle with a reference angle of 240 degrees is -120 degrees.Rate this question:
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- 27.
An arc along the unit circle begins at (1,0) and is 151pi units long. What are the coordinates of its other endpoint?
- A.
(1,0)
- B.
(0,1)
- C.
(-1,0)
- D.
(0,-1)
Correct Answer
C. (-1,0)Explanation
The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) on a coordinate plane. An arc along the unit circle is a portion of the circumference of the circle. Starting at (1,0), which is the point where the x-coordinate is 1 and the y-coordinate is 0, and moving counterclockwise, an arc that is 151π units long will end at (-1,0), where the x-coordinate is -1 and the y-coordinate is 0.Rate this question:
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- 28.
Which of the following passes through the point (0,1)?
- A.
Y = cosx
- B.
Y = tanx
- C.
Y = sinx
- D.
Y = cscx
Correct Answer
A. Y = cosxExplanation
The cosine function, y = cosx, passes through the point (0,1) because when x is 0, the cosine of 0 is 1. The point (0,1) lies on the unit circle, which is the graph of the cosine function.Rate this question:
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- 29.
Solve for x.
- A.
X = pi/4 and x = 5pi/4
- B.
X = 3pi/4
- C.
X = pi/2 and x = 3pi/2
- D.
X = 3pi/2
Correct Answer
A. X = pi/4 and x = 5pi/4Explanation
The given answer x = pi/4 and x = 5pi/4 is correct because these values satisfy the equation x = pi/4 and x = 5pi/4.Rate this question:
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- 30.
If tanx = 1, then what is sinx?
- A.
Sqrt(2)/2
- B.
Sqrt(3)/2
- C.
1/2
- D.
-1/2
Correct Answer
A. Sqrt(2)/2Explanation
When tan(x) = 1, it means that the ratio of the sine of x to the cosine of x is equal to 1. Since the sine and cosine of x are both positive in the first quadrant, we can conclude that they are equal. Therefore, sin(x) = cos(x) = sqrt(2)/2.Rate this question:
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