PCAT Practice: Quantitative Reasoning

Suppose h(x)= 4 sin x - 2 cos x. Then h′(x) = A. 4 sin x - 2 sin x B. 4 cos x + 2 sin x C. 4 cos x + 2 cosx D. -4 cos x - 2 cos x

Answer: B

If x≠ 0, which of the following is equal to (x/3+3/x+x/3+3/x)/(x^2+9)/x ? A. 2/3 B. 3/2 C. 2x/3 D. 2/3x

Answer: A Explanation: Choice A is the correct answer. In this question, we are asked to determine a value for the expression x/3+3/x+x/3+3/x)/(x^2+9)/x. There are complex fractions in both the numerator and denominator, so we need to get rid of them. The common denominator for the fractions in the numerator is 3x, multiply these fractions with a denominator of 3 by x/x (This expression equals 1, so the value of the denominator will not change). We can also multiply the fractions with a denominator of x by 3/3, which also equals 1. Now all these fractions have the denominator 3x; therefore, the numerator of the original complex fraction is now: x23x+93x+x23x+93x, which can be simplified to 2(x2+9)3x. The denominator of the original complex fraction has the expression (x2+9)x, which cannot be simplified. The new expression reads 2(x2+9)3x/(x2+9)x/. Now you can multiply the numerator by the reciprocal of the denominator, giving you 2(x2+9)3x×xx2+9 . You can cancel x2+ 9 from each fraction, leaving you with 2x3x , or. 23

If log49(x) = 1/2, then x = A. 7 B. 24.5 C. 98 D. 2401

Answer: A Explanation: The key to answering questions like this one on logarithms is to translate it into an equivalent equation involving powers. The equation given, log49(x) = 1/2, can be rewritten as: 49^1/2 = x. Using the law of exponents, 49(1/2)=√49=7,which is the value of x.

Which is equivalent to 3i2 + 2i4? A. -1 B. 1 C. 5 D. 3 + 2i

Answer: A Explanation: This question asks you to manipulate complex numbers. The key to solving this problem is that i2 = -1. Substitute this into the expression: 3i2+2i4=3i2+2(i2)(i2)=3(−1)+2(−1)(−1)=−3+2=−1

Which of the following is equivalent to (-2 + i)(3 + 2i)? A. -8 -i B. 1 + 3i C. -6 + 2i D. 5 + 3i

Answer: A Explanation: To multiply two complex numbers, first distribute the parenthesis: (-2 + i)(3 + 2i) = -6 + 3i- 4i + 2i2 = -6 - i + 2i2. Then use the fact that i2 = -1: -6 - i + 2 i2 =-6 - i + 2(-1) = -8 - i.

A certain bread recipe calls for wheat flour, rye flour, and corn flour in a 9 : 5 : 2 ratio. One loaf uses a total of four cups of flour. How many cups of corn flour are needed to bake 6 loaves? A. 34 B. 3 C. 6 D. 12

Answer: B Explanation: Be aware that a ratio is just another way of writing a fraction, and that you can get all kinds of information out of a ratio. For example, a ratio with three parts can be broken up into three ratios with two parts, e.g., a ratio of A:B:C also gives you ratios of A:B, B:C, and A:C. Also, if you know the ratio of all the parts, you also know the ratios of each part to the whole, e.g., if two parts of a mixture have a ratio of 1:3, then to find the ratio of one part to the whole mixture, you just add the parts together to get a whole of 1 + 3 = 4, and get part-to-whole ratios of 1:4 and 3:4. Since the ratio of wheat to rye to corn flour is 9:5:2, for every 9 + 5 + 2 = 16 units of total flour used, 2 of those units will be corn flour. So corn flour represents 29+5+2=26=18 of the flour required for this recipe. If one loaf uses 4 cups of flour, 6 loaves will use 4 × 6 = 24 cups of flour. So the amount of corn flour needed to bake six loaves is 18×24=3 cups.

If f(x)=x+25 , then what is f-1(x)? A. x−2/5 B. 5x- 2 C. 5/x+2 D. 2−x/5

Answer: B Explanation: If the function is y=f(x)=x+2/5. To find the inverse function, switch x and y and then solve for y: x = y+2/5 5x = y+2 y = 5x-2

A group of seven friends decide to play a board game together. In the game, one of them will act as the game master and the rest will work in pairs. How many different pairs of people can be selected from a group of 7 people? A. 14 B. 21 C. 28 D. 35

Answer: B Explanation: There are 7 different people that can be chosen first. For each of these 7 people chosen first, there are 6 people that can be chosen second. So there is a total of 7 × 6, or 42 ordered pairs of people that can be chosen. However, the order in which two people are chosen does not matter. Thus, choosing A first and then B is the same as choosing B first and then A. So to find the number of different pairs of people that can be chosen, the number of ordered pairs, 42, must be divided by 2. So the number of different pairs of people that can be chosen is 42/2, or 21, and choice B is correct.

log8(64) - log64(8) = A. −1/2 B. 1/2 C. 3/2 D. 5/2

Answer: B Explanation: Using the definition of logarithms, we know that log8(64) = 2, since 82 = 64. Similarly, log64(8) = 1/2, since 64^1/2 = 8. Therefore, the expression given in the question, log8(64) - log64(8) = 2 - 1/2 = 3/2.

A grocery store has exactly 24 bagels. If one customer buys 1/3 of the bagels, and a second customer buys 1/4 of the remaining bagels, how many bagels are left in the store? A. 6 B. 10 C. 12 D. 14

Answer: C Explanation: The first customer buys 1/3(24)=8, bagels, so 24 - 8 = 16 remain in the store. The second customer buys 1/4(16)=4 bagels, so 16 - 4 = 12 bagels are left

A group of twelve students participate in a fundraiser where they sell frozen pizzas. The twelve students sell 3, 3, 4, 7, 7, 7, 10, 10, 14, 18, 21, and 24 pizzas, respectively. What is the mode number of pizzas sold? A. 3 B. 4 C. 7 D. 9

Answer: C Explanation: The number 7 occurs 3 times, and no other number occurs more than twice.

What is the range of the function below? f(x) = 3x2- 5 A. (-∞, ∞) B. (0, ∞) C. (-5, ∞) D. (5, ∞)

Answer: C Explanation: The range of a function is the list of possible outputs of the function. It is impossible for this function to generate any number less than -5, and there is no upper limit to the output of this function. The range is (-5, ∞).

A new law will impose a 18% sales tax on certain products sold in a state, where all other products in the state are subject to a 6% sales tax when purchased. 18 percent of 18 is equal to 6 percent of A. 6 B. 9 C. 54 D. 108

Answer: C Explanation: There are 2 separate percentage problems in this question; however, any unnecessary calculations can be prevented by using logic. For example, we are asked for 18% of a number and 6% of another number. We know that 18% is the same as 3 × 6%; therefore, our answer is going to have to be 3 × 18%, or 54%. Choice C is correct. If you did not understand this logic, here is a different solution. In the form of an algebraic equation: (18%)(18) = 6%(x). When both sides of the equation are divided by 6%, we will have the expression: (18%)(18)/6%= x. Solving for x : x= 54.

What is ∫(4x + e^x)dx? A. 4 + ex + C B. 2x^2 + e^1 + C C. 2x^2 + e^x + C D. x^2 + e^x + C

Answer: C Explanation: To find the integral, calculate the integral of each term separately. Remember, the integral of ex is ex. ∫(4x+ex)dx=∫4xdx+∫exdx+C=4x22+ex+C=2x2+ex+C

A certain stereo at Store X costs 10 percent less than the same stereo at Store Y. If the stereo costs $360 at Store X, how much does it cost, in dollars, at Store Y? A. 330 B. 396 C. 400 D. 440

Answer: C Explanation:Choice C is the correct answer. In a question like this, it is important to identify the "whole" so you can calculate the correct percentage. We are told that the price of a stereo at store xis 90% of the price of the same stereo at store y. Based on this information, it can be concluded that the stereo costs less at store x. Therefore, the whole will be the price of the stereo at store y and the part will be the price at store x. Since part equals percent times whole, we can set up an equation: $360 = 90%y. Solving for the stereo's price at store y: y = $360/0.9., which is $400.

Which of the following is equal to (3x - 7y)^2? A. 9x2 + 49y2 B. 9x2 + 21xy + 49y2 C. 9x2- 21xy + 49y2 D. 9x2- 42xy + 49y2

Answer: D

The temperatures in city X on Monday through Sunday of last week were 26, 14, 18, 31, 33, 29, and 31, respectively. What was the median temperature in city X for these 7 days? A. 17 B. 19 C. 26 D. 29

Answer: D Explanation: Arrange the temperatures in increasing order. The temperatures are 14, 18, 26, 29, 31, 31, and 33. There is an odd number of terms, so the median is the middle value, 29.

At a fast food restaurant there are a group of a friends. When seven of them leave, only one is left behind. At the only other table that had people, the b individuals at that table each order three cheeseburgers, for a total of fifteen cheeseburgers at that table. What is a + b? A. 9 B. 11 C. 12 D. 13

Answer: D Explanation: Choice D is the correct answer. In this question, we are given two equations, one in terms of a and the other in terms of b. We are asked to determine a value for a + b. The simplest approach would most likely be to solve each equation for a and b, then add the two quantities. In the first equation, a- 7 = 1; therefore, a = 8. In the second equation, 3b = 15; hence, b = 5. Therefore, a + b = 8 + 5 = 13.

A candy store has 75 pounds of candy in stock at the beginning of the day. If x is the percentage of the stock sold when 24 one-quarter pound blocks of fudge, 51 one-third pound bags of gummies, and 84 one-twelfth pound bags of marshmallows are sold, what is x? A. 24 B. 30 C. 35 D. 40

Answer: D Explanation: First let's find the value of 24/4+ 51/3+ 84/12. We have that 24/4= 6, 51/3= 17, and 84/12= 7. So 24/4+ 51/3+ 84/12is equal to 6 + 17 + 7, which is 30. So the question is now what is the value of x if 30 is x% of 75? The fractional equivalent of x% is x100. So 30 = x/100(75). Canceling a factor of 25 from the 75 and the 100 on the right side of this equation, we have that 30 = x/4(3). Then 30 = 3x/4, 120 = 3x, and x = 120/3= 40.

g(x) = |x| I. Which of the following are true statements? I. g(x) is continuous at x = 0. II. g(x) is differentiable at x = 0. III. limx→0g(x)=0 A. I only B. III only C. I and II D. I and III

Answer: D Explanation: The graph of g is unbroken but has a sharp corner at x = 0 (graph it to see). Thus, I is true, which means, limx→0g(x)=g(0)=|0|=0 so III is also true. However, g is not differentiable at x = 0 so II is false.

For f(x) = 2e^-3x, what is f″(x)? A. 18e^x B. -6e^-3x C. 9e^1 D. 18e^-3x

Answer: D Explanation: The question asks you to find the second derivative of the function f(x) - 2e^-3x. Find the first derivative: f'(x) = (-3)2e^-3x = -6e^-3x Remember that when you take the derivative of an exponential function, the exponent does not change. You only bring down the derivative of the exponent. Now find the second derivative: f"(x) = (-3) - 6e^-3x = 18e^-3x

Which of the following sets of numbers has the smallest standard deviation? A. {12, 14, 18} B. {31, 7, 43} C. {150, 175, 200} D. {325, 326, 327}

Answer: D Explanation: The set with the numbers closest together will have the smallest standard deviation. Answer choice (D) consists of consecutive integers. They are only one number apart.

What is the sum of the two vectors (3 + 2i) and (-2 + i)? A. 2 + 2i B. -8 C. -6 + 2i D. 1 + 3i

Answer: D Explanation: To add vectors, group the real parts together and the imaginary parts together, and then add: (3 + 2i) + (-2 + i) = (3 + -2) + (2i + i) = 1 + 3i

Three partners divided a profit of $14,000 so that one partner received $1,400 and one partner received $2,800. What is the range, in dollars, of the amounts the three partners received? A. 1,400 B. 4,200 C. 5,600 D. 8,400

Answer: D Explanation: To find the range, we must first find the amount that the third partner received. The amount that the third partner received was 14,000 - (1,400 + 2,800) which equals 14,000 - 4,200 or 9,800. The range is 9,800 - 1,400 or 8,400. Choice (D) is correct.

Elaine had an average (arithmetic mean) score of 76 on the first three exams she took and a score of 84 on her fourth exam. What was her average (arithmetic mean) score on all four exams? A. 77 B. 78 C. 79 D. 80

Answer: D Explanation: You need to know the average formula: Average = Sum of terms/Number of terms. In this question, it is also important to be familiar with the average formula in the rearranged form Sum of terms = Average Number of terms. Elaine's average score on all 4 exams was sum of the four scores/4. So the sum of all 4 scores must first be found. We can obtain the sum of all 4 scores. Elaine's average on the first 3 exams was 76, so the sum of her scores on the first 3 exams was 76 × 3, or 228. The sum of her scores on all 4 exams is equal to the sum of her scores on the first 3 exams plus her score on the fourth exam. So the sum of her scores on all 4 exams was 228 + 84, which is 312. Her average on all 4 exams was 312/4, which is 78. Choice B is correct.