Qualifying Exam For Instructors - Part 4 - Mathematics

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.

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.

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.

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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.

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%.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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