Praxis- Mathematics
17. If the result of 10^33 - 33 is written as an integer, what is the sum of the digits of the integer?
10^2 - 33 = 67 10^3 - 33 = 967 10^4 - 33 = 9967 . . . 10^n -33 = 9999...967 (n-2 copies) Based on the preceding pattern, the integer representing 10^33-33 starts with 31 occurrences of the digit 9 and ends with the digits 6 and 7. The sum of the digits is 31 x 9+6+7 which is equivalent to 292.
18. The line shown in the preceding xy-plane is the graph of y=f(x). What is the value of f(12)+f(-12)?
6. The line passes through the points with coordinates (0,3) and (3,0), so the equation of the line is x+y=3. Since function f is given by f(x)= 3 -x for all values of x, then f(x)+f(-x)=(3-x)+(3+x)=6 for all values of x, so in particular f(12) + f(-12) = 6.
22. "I don't know why x^0 = 1. To me, it seems like it should be 0 because anything times 0 is 0". Ms. Swain wants to given an explanation that addresses the underlying mathematics and is accessible to the students in her class. Which of the THREE of the following explanations meets Ms. Swain's goals?
Correct answers B, C, E.
Mr. Platt is evaluating potential final exam review problems for his Algebra 1 class He considers the following problem: Jeremy finds a beanstalk that is 10 cm high when planted. He observes that the beanstalk grows straight up at a constant rate of 4 millimeters per day. a. Write an equation that expresses the height of the beanstalk as a function of time since planting. b. How long does it take for the beanstalk to reach a height of 34 cm? c. How high will the beanstalk be 10 full days after it is planted? d. Jeremy learns of a different beanstalk that is only 3 cm high when planted. This beanstalk grows at a rate of 5 mm per day. If both beanstalks were planted at the same time, after how much time will they be the same height? Which of the following topics are addressed in the review problem? Creating a linear model, solving a linear equation and solving an exponential equation.
Creating a linear model and solving a linear equation. Creating a linear model is addressed in parts a and d. Solving a linear equation is addressed in parts b and d. Solving an exponential equation is not addressed in the problem.
19. For which of the following radian angle measures is the terminal side of the corresponding angle in the standard position located in quadrant II?
Since (37π)/8 = 4π +(5π)/8, the terminal side of the angle (37π)/8 is the same as the terminal side of the angle (5π)/8. Since (5π)/8 is between π/2 and π, the terminal side of the angle (5π)/8 is in quadrant II.
In the proceeding figure, P and T are points on like k, and points Q, R and S are points on line l. Lines k and l are parallel. What is the measure of the angle TRS?
Since angles PTR and TRS are alternate interior angles, they are congruent and they have the same angle measure. The equation: 5x + 10 + 110 = 8x+102 x=6. The measure of angle TRS is 8 x 6 + 102 = 150 degrees.
27. How many of the three preceding diagrams represent y as a function of x?
TWO. If y is a function of x, then each value of x in the domain is mapped to at most one value of y. This is true for diagram A and diagram C but not for diagram B. In diagram B, there is at least one x-value that is mapped to two different y-values (that is, the graph in the xy‐plane fails the vertical line test).
26. Laura is creating a triangular garden in her backyard by laying wooden edging and filling the space with soil. She has already laid one side of the wooden edging, measuring 21 feet in length. If Laura wants the garden to be an obtuse triangle, which of the following is a pair of possible lengths for the other two sides of the garden?
10 feet and 24 feet. If a, b and c are positive numbers such that a≤b≤c, then there exists a triangle with sides of lengths a, b and c, if and only if a+b>c. Also recall that if the lengths of a triangle are a, b, and c, where a≤b≤c, the triangle is a right triangle if and only if a^2 + b^2 = c^2; the triangle is acute if and only if a^2 + b^2 > c^2; the triangle is obtuse if and only if a^2 + b^2 < c^2.
24. The following xy-plane shows the graph of polynomial h with degree 3 and with real coefficients. At which of the following values of x is h'(x)>0 and h"(x)<0?
A. -3. The condition h'(x)>0 means that the graph of h is increasing at that value of x, and the condition h"(x)<0 means that the graph of h is concave down at that value of x. The graph of h is both increasing and concave down at x=-3.
13. The students in Mr. Gleason's class are working on tasks that will help them understand the meaning of rational exponents. While working in pairs without using calculators, two of Mr. Gleason's students make different guesses about the value of 3^(1.5). Alison guesses that 3^(1.5)=3^(1.5)=6. Brian thinks the value is less than 6 and makes the following calculations on paper. After completing the calculations, Brian says "Since 3 is less than 4, 3^(1.5) is less than 6." Which of the following statements best characterizes the validity of Brian's approach to the problem?
Brians approach is mathematically valid.
21. The graph of the function f is shown in the preceding x y planexy‐plane. Five values, a, b, c, d, and e, are indicated on the x-axis. At how many of the five values does the function f appear to be continuous but not differentiable?
ONE. Of the five values, f is continuous but not differentiable only at x=a. Function f is continuous at x=a (the graph near x=a can be drawn without lifting the pen from the paper) but is not differentiable at x=a (the graph has a sharp change of direction). Function f is not continuous at x=b because the graph has a jump discontinuity at x=b. Function f is not continuous at x=c because f is not defined at x=c and has a vertical asymptote at x=c. Function f appears to be both continuous and differentiable at x=d. Function f is not continuous at x=e because f is not defined at x=e.
15. Which of the following is equivalent to the preceding expression for all positive values of x and h?
Option D is correct.... put work together = -1/((x+1)(x+1+h))
Mr. Christensen asked Carla to use division to simplify the expression (10x +4)/2x. Mr. Christensen then asked the students in the class to explain what was wrong with Carla's solution. Which of the following student answers is the best explanation for what was most likely wrong with Carla's solution?
She cannot cancel like this because you can only cancel a factor of both the numerator and denominator. Carla incorrectly divided the 4 in the numerator by the 2 in the denomination to obtain the quotient 2, ignoring the factor of 2 in 10x. Then Carla incorrectly divided the 10x in the numerator by the variable x in the denominator to obtain the quotient 5, ignoring the variable x is not a factor of the other summand in the numerator. Since (10x+4)/2x = (2(5x+2))/2x, only the 2 in the denominator is a factor of the numerator and since the variable x is not a factor of both terms in the numerator, the x's cannot be canceled.
25. The following table is the complete table of values for functions f and g. Based on the table, what is the domain of the composition function. (f o g)(x)?
The domain of the composition function (f∘g)(x)consists of the values of x in the domain of g for which g(x) is in the domain of f. The domain of function g is {−2,−1,0,1,2,3}. The domain of function f is also {−2,−1,0,1,2,3}. There are three values of x in {−2,−1,0,1,2,3} for which g(x) is also in {−2,−1,0,1,2,3}; specifically, x equals negative x=−1, x=0, x=1.
The preceding box plot represents the distribution of the distances for 20 bicycling trips. If another trip of 5 miles is added to the data set, which of the following must change? The mean, median, mode or range?
The mean of the original data set is greater than 5 miles. When another trip of 5 miles is added to the data set, the mean of the new data set decreases. The median and the mode may or may not change when the 5-mile trip is added to the data set. The range does not change when the 5-mile trip is added to the data set.
The following table shows the distribution of 50 cookies sold in a store yesterday by flour type and topping. What is the probability that a randomly selected cookie is not made with cake flour and does not have a sprinkles topping?
The number of cookies that are not made with cake flour and do not have a sprinkles topping is 13 + 7, or 20. The probability that a randomly selected cookie is not made with cake flour and does not have a sprinkles topping is 20/50 or 0.40.
Marcie paid $8.10 for a shirt, including sales tax. The sales tax was 8%. What was the original price of the shirt before tax was added? Ms. Stull graded a quiz and found that many students had difficulty solving the preceding problem. She wants to discuss two problems with a similar structure with her students during the next class and has already selected a problem involving percents. Which of the following problems involving fractions has a structure that is most similar to the structure of the problem on the quiz?
The problem on the quiz involves increasing an unknown amount by a multiple of that unknown amount to arrive at a given total and can be solved by the equation x+(8%)x=8.10, where x represents the original price, in dollars, of the shirt before tax was added; that is, divide the given total by the sum of 1 and the given percent or fraction. The problem in option (B) can be solved by the equation x+2/5x=48, where x represents the number of ounces of food that George feeds his large dog. Option B has a structure that is most similar to the structure of the problem on the quiz.
Ms. Moore is analyzing the proof that a student submitted on the following problem: In the preceding figure, the measure of the arc BC in the circle with center O is 60 degrees. Prove that triangle OBC is an equilateral triangle by showing that the measure of each angle in triangle OBC is 60 degrees. The student's proof follows: Since the measure of arc BC is 60 degrees, the measure of angle BOC is also 6p degrees because angle BOC is the central angle subtended by arc BC. Segments OB and OC are congruent, since each of them is a radius of the circle and all radii of a circle are congruent. Therefore, triangle OBC is an equilateral triangle.
The student should use the isosceles triangle theorem to prove that angles OBC and OCB are congruent and then use the triangle sum theorem to find the measure of these angles.
12. The function f is defined by f(x)= (3x^2 + 4x + 1)/(x^3 - 1) for all numbers x greater than 1. Which of the following is true?
as x approached infinity, lim f(x)=0, and the graph of f in the xy-plane approaches the x-axis from above as x approaches infinity. For large values of x, the cubic polynomial in the denominator of the rational function f grows much faster than the quadratic polynomial in the numerator of f as x increases without bound. Therefore, the limit of f(x) as x approaches infinity is 0. Since both the numerator and the denominator of f are positive for all numbers x greater than 1, it follows that f(x)>0 for all x>1. Therefore, the graph of f in the xy-plane is everywhere above the x-axis. In particular, the graph of f in the xy-plane approaches the x-axis from above as x approaches infinity.
