GMAT Review Questions
"Was the number of books sold at Bookstore X last week greater than the number of books sold at Bookstore Y last week? (1) Last week, more than 1,000 books were sold at Bookstore X on Saturday and fewer than 1,000 books were sold at Bookstore Y on Saturday (2) Last week, less than 20 percent of the books sold at Bookstore X were sold on Saturday and more than 20 percent of the books sold at Bookstore Y were sold on Saturday"
"y = t/q We need to determine whether x > y, or s/p > t/q, or sq > pt. Since s > t and q > p, and all values are positive, we can determine that sq is greater than pt, and thus that x > y. Answer: C "
"A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women? 1/14 1/7 2/7 3/7 1/2"
" 5! / 2!3! = 10 So, the number of ways that 2 women can be selected is 10. Now the men: Thus, the number of ways that 2 men can be selected is 3. Now we can multiply to get the total number of teams with exactly 2 women (and thus exactly 2 men): w = (10)(3) = 30. Since there are 30 possible teams with exactly 2 women, and 70 possible teams overall, w/t = 30/70 = 3/7. The correct answer is D."
"Eco Wildlife Preserve contains 5x zebras and 2x lions, where x is positive. If the lions kill z zebras, is the new ratio of zebras to lions less than 2 to 1? (1) z>x (2)z=4"
"(5x-z)/2x < 2/1 5x-z<4x is z>x?"
"Pascal has 96 miles remaining to complete his cycling trip. If he reduced his current speed by 4 miles per hour, the remainder of the trip would take him 16 hours longer than it would if he increased his speed by 50%. What is his current speed? A) 6 B) 8 C) 10 D) 12 E) 16"
"B The best way to solve this problem is to Work Backwards from the answer choices. When working backwards, start with answer (B) or (D), whichever looks easier. (Your default can be choice (B) unless that number looks annoying for some reason.)"
"A certain high school has 5,000 students. Of these students, x are taking music, y are taking art, and z are taking both music and art. How many students are taking neither music nor art? (A) 5,000 - z (B) 5,000 - x - y (C) 5,000 - x + z (D) 5,000 - x - y - z (E) 5,000 - x - y + z"
"Easier formula to remember is Group1+ Group2- Both +Neither = Total (Source: Kaplan) So, solution is x + y - z + Neither = 5000 Neither = 5000 - x - y + z ..Choice E!"
"Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails? (A) x/(x+y) (B) y/(x+y) (C) xy/(x+y) (D) xy/(x-y) (E) xy/(y-x)"
"Say x=1 hour and y=2 hours (notice that y must be greater than x, since the time for machine A to do the job, which is y hours, must be more than the time for machines A and B working together to do the same job, which is x hours). In this case, the time needed for machine B to do the job must also be 2 hours: 1/2+1/2=1. Now, plug x=1 and y=2 in the options to see which one yields 2. Only option E fits. Answer: E. "
Idiom NOT by x BUT by y
"Think of it this way for structure - the word ""NOT"" is where the two paths diverge, and each path needs to be directly connected to the ""shared"" entry. So... The goal was scored not by Drogba, but by Donovan. Is correct, because ""the goal was scored ... (stop at ""NOT"" and start after ""BUT"") by Donovan"" is a complete statement. "
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?
-(10)^20
A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
0.7a+0.5b=6.30.7a+0.5b=6.3 --> 7a+5b=637a+5b=63. After some trial and error you'll get that only two integer pairs of (a,b) satisfy this equation: (9,0) and (4,7) as we are told that "a customer purchased both apples and bananas" then the first pair is out and we'll have: a=4a=4 and b=7b=7 --> a+b=11a+b=11.
"In a certain class of students, 25% received A's on the final exam and 2/3 of these students received A's in the class as a whole. If 15 students in the class received A's on the final exam but did not receive an A in the class, how many students did not receive an A on the final exam? A) 45 B) 60 C) 105 D) 135 E) 180"
135
"100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors? 6 16 17 33 84"
16
if y is an integer, then the least possible value of |23-5y|
2
"List S includes elements {8, 2, 11, x, 3, y} and has a mean of 7 and a median of 5.5. If x < y, then which of the following is the maximum possible value of x? A) 0 B) 1 C) 2 D) 3 E) 4"
3
"Alexandra needs to mix cleaning solution in the following ratio: 1 part bleach for every 4 parts water. When mixing the solution, Alexandra makes a mistake and mixes in half as much bleach as she ought to have. The total solution consists of 27 mL. How much bleach did Alexandra put into the solution? "
3 ml
The ratio of roses to tulips in a bouquet is 3 to 5. If 2 roses are added to the bouquet and 6 tulips are removed, the new ratio of roses to tulips is 5 to 6. How many tulips were in the bouquet to begin with?
30
10 years ago, Sana was half as old as byron. if byron is now 15 years older than sana, how old will sana be in 7 years
32
If the sum of the last 3 integers in a set of 7 consecutive integers is 258, what is the sum of the first 4 integers?
330
On the planet Flarp, 3 floops equal 5 fleeps, 4 fleeps equal 7 flaaps, and 2 flaaps equal 3 fliips. How many floops are equal to 35 fliips?
35 fliips x 2 flaaps/3 fliips x 4 fleeps/7 flaaps x 3 floops/5 fleeps = 8 floops
Greg, Marcia, Peter, Jan, Bobby and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front row of the theater. If Marcia and Jan will not sit next to each other, in how many ways different arrangements can the 6 people sit?
480
"A list kept at Town Hall contains that town's average daily temperature in Fahrenheit, rounded to the nearest integer, for each day of a particular completed month. Does this month have 30 or 31 days? (1) The median temperature is 73.5. (2) The sum of the average daily temperatures is divisible by 3."
A
"A list of measurements in increasing order is 4,5,6,8,10,x. If the median of these measurements is 6/7 times their arithmetic mean, What is the value of x? A. 16 B. 15 C. 14 D. 13 E. 12 "
A
"Company Q plans to make a new product next year and sell each unit of this new product at a selling price of $2. The variable costs per unit in each production run are estimated to be 40% of the selling price, and the fixed costs for each production run are estimated to be $5,040. Based on these estimated costs, how many units of the new product will Company Q need to make and sell in order for their revenue to equal their total costs for each production run? A. 4,200 B. 3,150 C. 2,520 D. 2,100 E. 1,800"
A
"If m and n are positive integers, is (√m)^n an integer? (1) (√m) is an integer. (2) (√n) is an integer."
A
"If money is invested at r percent interest, compounded annually, the amount of investment will double in approximately 70/r years. If Pat's parents invested $ 5000 in a long term bond that pays 8 percent interest, compounded annually, what will be the approximate total amount of investment 18 years later, when Pat is ready for college? A. $20000 B. $15000 C. $12000 D. $10000 E. $9000"
A
"If x is a positive integer, is x^2 + 6x +10 odd? (1) x^2 + 4x + 5 is odd (2) x^2 + 3x + 4 is even"
A
"If x is negative, is x < -3 ? (1) x^2 > 9 (2) x^3 < -9"
A
"In one week a construction worker worked more than 40 hours, and was paid two times his regular hourly rate for every hour worked in excess of 40 hours. If he earned $780, how many hours did he work in that week? (1) If he had worked 6 more hours that week, his earnings would have been $960. (2) If he had worked 20 fewer hours that week, his earnings would have been $390 less that week."
A
"In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ? (1) The x-coordinate of point R is -1. (2) Point R lies on the line y = -3."
A
"Is (x + 1)/(x - 3) < 0 ? (1) -1 < x < 1 (2) x^2 - 4 < 0"
A
"Is q > t ? (1) qp^2 < tp^2 (2) qp^3 > tp^3"
A
"Is the integer z divisible by 6? (1) The greatest common factor of z and 12 is 3. (2) The greatest common factor of z and 15 is 3."
A
"QUESTION 4 OF 31 A scientist is studying bacteria whose cell population doubles at constant intervals, at which times each cell in the population divides simultaneously. Four hours from now, immediately after the population doubles, the scientist will destroy the entire sample. How many cells will the population contain when the bacteria is destroyed? (1) The population just divided and, since the population divided two hours ago, the population has quadrupled, increasing by 3,750 cells. (2) The population will double to 40,000 cells with one hour remaining until the scientist destroys the sample."
A
"Raffle tickets numbered consecutively from 101 through 350 are placed in a box. What is the probability that a ticket selected at random will have a number with a hundreds digit of 2 ? (A) 2/5 (B) 2/7 (C) 33/83 (D) 99/250 (E) 100/249"
A
"What is the value of 2a4b? (1) a = -2b (2) b = 4`"
A
"If q, s, and t are all different numbers, is q < s < t ? (1) t - q = |t - s| + |s - q| (2) t > q"
A (Revised from first iteration)
"If c and d are integers, is c-3d even? (1) c and d are odd (2) c-2d is odd"
A (Shortened from first iteration)
"According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum? a) 5:30 b) 7:00 c) 7:30 d) 8:00 e) 9:00"
Answer = B
"Is xy > y^2? (1) yx<1 (2) |x| > y"
Answer = E
How many of the integers that satisfy the inequality (x+2)(x+3) / (x-2) >= 0 are less than 5?
Answer is (D) - 4
"For each student in a certain class, a teacher adjusted the student's test score using the formula y = 0.8x + 20, where x is the student's original test score and y is the student's adjusted test score. If the standard deviation of the original test scores of the students in the class was 20, what was the standard deviation of the adjusted test scores of the students in the class? A. 12 B. 16 C. 28 D. 36 E. 40"
Answer: B " (Shortened from first iteration)
If xy not equal 0 and x^2*y^2 -xy = 6, which of the following could be y in terms of x? I. 1/2x II. -2/x III. 3/x A. I only B. II only C. I and II D. I and III E. II and III
Answer= E
" order to complete a reading assignment on time, Terry planned to read 90 pages per day. However, she read only 75 pages per day at first, leaving 690 pages to be read during the last 6 days before the assignment was to be completed. How many days in all did Terry have to complete the assignment on time? (A) 15 (B) 16 (C) 25 (D) 40 (E) 46"
B
"Company P had 15 percent more employees in December than it had in January. If Company P had 460 employees in December, how many employees did it have in January? (A) 391 (B) 400 (C) 410 (D) 423 (E) 445"
B
"For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301? A. 10,100 B. 20,200 C. 22,650 D. 40,200 E. 45,150"
B
"If 5^21 X 4^11 = 2x10^n. What is the value of n? A. 11 B. 21 C. 22 D. 23 E. 32"
B
"If x = , then the value of is A. between −5×106 and −4×106 B. between −2.5×10−7 and −2×10−7 C. equal to −100 D. between 2×10−7 and 2.5×10−7 E. between 4×106 and 5×106"
B
"In Country C, the unemployment rate among construction workers dropped from 16 percent on September 1, 1992, to 9 percent on September 1, 1996. If the number of construction workers was 20 percent greater on September 1, 1996, than on September 1, 1992, what was the approximate percent change in the number of unemployed construction workers over this period? (A) 50% decrease (B) 30% decrease (C) 15% decrease (D) 30% increase (E) 55% increase"
B
"Is x < 5? (1) x^2 > 5 (2) x^2 + x < 5 "
B
"Is |x - y | > |x | - |y | ? (1) y < x (2) xy < 0"
B
"Pat bought 5 pounds of apples. How many pounds of pears could Pat have bought for the same amount of money? (1) One pound of pears costs $0.50 more than one pound of apples. (2) One pound of pears costs 1.5 times as much as one pound of apples."
B
"Six shipments of machine parts were shipped from a factory on two trucks, with each shipment entirely on one of the trucks. Each shipment was labeled either S1 (1/4), S2 (1/5), S3 (1/6), S4(3/20), S5(2/15), or S6 (1/10). The table shows the value of each shipment as a fraction of the total value of the six shipments. If the shipments on the first truck had a value greater than 1/2 of the total value of the six shipments, was S3 shipped on the first truck? (1) S2 and S4 were shipped on the first truck. (2) S1 and S6 were shipped on the second truck."
B
"The interior of a rectangular carton is designed by a certain manufacturer to have a volume of x cubic feet and a ratio of length to width to height of 3:2:2. In terms of x, which of the following equals the height of the carton, in feet? A. cube root of x B. cube root of (2x/3) C. cube root of (3x/2) D. 2/3 * cube root of x"
B
"Which of the following is greatest? A. 10√3 B. 9√4 C. 8√5 D. 7√6 E. 6√7"
B
"Working simultaneously and independently at an identical constant rate, 4 machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days? (A) 24 (B) 18 (C) 16 (D) 12 (E) 8"
B
"s= {1,2,5,7,X} If x is a positive integer, is the mean of set S greater than 4? (1) the median of set s is greater than 2 (2) the median of set s is equal to the mean of set S"
B
"A pharmaceutical company received $3 million in royalties on the first $20 million in sales of the generic equivalent of one of its products and then $9 million in royalties on the next $108 million in sales. By approximately what percent did the ratio of royalties to sales decrease from the first $20 million in sales to the next $108 million in sales? (A) 8% (B) 15% (C) 45% (D) 52% (E) 56%"
C
"A store reported total sales of $385 million for February of this year. If the total sales for the same month last year was $320 million, approximately what was the percent increase in sales? (A) 2% (B) 17% (C) 20% (D) 65% (E) 83%"
C
"A store sells a certain product at a fixed price per unit. At the product's current price, q units cost a total of exactly $300. If the price were lowered by $5 from its current value, then q + 2n units would cost exactly $300; if the price were raised by $5, then q - n units would cost exactly $300. What is the value of q? A) 10 B) 15 C) 20 D) 25 E) 30
C
"Aaron will jog from home at x miles per hour and then walk back home by the same route at y miles per hour. How many miles from home can Aaron jog so that he spends a total of t hours jogging and walking? (A) xt/y (B) (x+t)/xy (C) xyt/(x+y) (D) (x+y+t)/xy (E) (y+t)/x-t/y"
C
"Beginning in January of last year, Carl made deposits of $120 into his account on the 15th of each month for several consecutive months and then made withdrawals of $50 from the account on the 15th of each of the remaining months of last year. There were no other transactions in the account last year. If the closing balance of Carl's account for May of last year was $2,600, what was the range of the monthly closing balances of Carl's account last year? (1) Last year the closing balance of Carl's account for April was less than $2,625. (2) Last year the closing balance of Carl's account for June was less than $2,675."
C
"If (1/3+1/4+1/5+1/6)=r(1/9+1/2+1/15+1/18), then r = A. 1/3 B. 4/3 C. 3 D. 4 E. 12"
C
"If [x] denotes the least integer greater than or equal to x, is [x] = 0? (1) -1< x< 1 (2) x < 0"
C
"If x/y = c/d and d/c = b/a, which of the following must be true? I. y/x = b/a II. x/a = y/b III. y/a = x/b (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III"
C
"In a certain deck of cards, each card has a positive integer written on it, in a multiplication game a child draws a card and multiplies the integer on the card with the next large integer. If the each possible product is between 15 and 200, then the least and greatest integer on the card would be A. 3 and 15 B. 3 and 20 C. 4 and 13 D. 4 and 14 E. 5 and 14"
C
"On a camping trip, x campers each paid y dollars for food. What percent of the total food expenses did each camper pay? (1) If there were one fewer camper, each camper would owe 1 dollar more. (2) If there were half as many campers, each camper would owe 7 dollars more."
C
"Positive integer y is 50 percent of 50 percent of positive integer x, and y percent of x equals 100. what is the value of x ? A) 50 B) 100 C) 200 D) 1000 E) 2000"
C
"S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices? (1) The number of distinct points in S is 5. (2) No three of the points in S are collinear."
C
"Stores L and M each sell a certain product at a different regular price. If both stores discount their regular price of the product, is the discount price at Store M less than the discount price at Store L ? (1) At Store L the discount price is 10 percent less than the regular price; at Store M the discount price is 15 percent less than the regular price. (2) At Store L the discount price is $5 less than the regular store price; at Store M the discount price is $6 less than the regular price."
C
"What is the total number of coins that Bert and Claire have? (1) Bert has 50 percent more coins than Claire. (2) The total number of coins that Bert and Claire have is between 21 and 28."
C
"For which of the following functions f is f(x) = f(1-x) for all x? A. f(x)=1−x B. f(x)=1−x^2 C. f(x)=x^2- (1-x)^2 D. f(x)=x^2∗(1−x)^2 E. f(x)=x / (1−x)"
D
"If a and b are positive integers, is a/b < 9/11 ? (1) a/b < 0.818 (2) b/a > 1.223 "
D
"If x2/9 - 4/y2 = 12, what is the value of x? (1) x/3 + 2/y = 6 (2) x/3 - 2/y = 2 "
D
"If xy = 1, what is the value of 2^(x + y)^2 / 2^(xy)^2 A. 2 B. 4 C. 8 D. 16 E. 32"
D
"Is the average of n consecutive integers equal to 1 ? (1) n is even (2) if S is the sum of the n consecutive integers, then 0 < S < n "
D
"Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, 2/3 dislike lima beans; and of those who dislike lima beans, 3/5 also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans? (1) 120 students eat in the cafeteria (2) 40 of the students like lima beans"
D
"Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to lineup male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups? (A) 9 (B) 12 (C) 15 (D) 36 (E) 720"
D
"The sum of the positive integers from 1 to 27 is equivalent to the sum of the integers from -27 to 54 0 to 28 15 to 45 38 to 46 48 to 54"
D
"What is the average (arithmetic mean) of eleven consecutive integers? (1) The average of the first nine integers is 7. (2) The average of the last nine integers is 9."
D
"What is the cube root of w ? (1) The 5th root of w is 64. (2) The 15th root of w is 4."
D
"Yesterday's closing prices of 2,420 different stocks listed on a certain stock exchange were all different from today's closing prices. The number of stocks that closed at a higher price today than yesterday was 20 percent greater than the number that closed at a lower price. How many of the stocks closed at a higher price today than yesterday? (A) 484 (B) 726 (C) 1,100 (D) 1,320 (E) 1,694"
D
"Boomtown urban planners expect the city's population to increase by 10% per year over the next two years. If that projection were to come true, the population two years from now would be exactly double the population of one year ago. Which of the following is closest to the percent population increase in Boomtown over the last year? A) 20% B) 40% C) 50% D) 65% E) 75%"
D-65%
"(R is negative on number line) The number line shown contains three points R, S, and T, whose coordinates have absolute values r, s, and t, respectively. Which of the following equals the average (arithmetic mean) of the coordinates of the points R, S, and T ? A. s B. s + t - r C. (r - s - t)/3 D. (r + s + t)/3 E. (s + t - r)/3"
E
"A candy wholesaler needs to quickly sell some candy bars that are nearing their expiration date, so he reduced the price of the candy bars. By what percent did he reduce the price of the candy bars? (1) The price of a candy bar was reduced by 36 cents. (2) If a candy retailer purchases a case of 144 candy bars from the wholesaler, she will save $51.84 as a result of the price reduction."
E
"At a certain food stand, the price of each apple is ¢ 40 and the price of each orange is ¢ 60. Mary selects a total of 10 apples and oranges from the food stand, and the average (arithmetic mean) price of the 10 pieces of fruit is ¢ 56. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is ¢ 52? A. 1 B. 2 C. 3 D. 4 E. 5"
E
"At a certain school, the ratio of the number of second graders to the number of fourth graders is 8 to 5, and the ratio of the number of first graders to the number of second graders is 3 to 4. If the ratio of the number of third graders to the number of fourth graders is 3 to 2, what is the ratio of the number of first graders to the number of third graders? A. 16 to 15 B. 9 to 5 C. 5 to 16 D. 5 to 4 E. 4 to 5"
E
"At a two-day seminar, 90 percent of those registered attended the seminar on the first day. What percent of those registered did not attend the seminar on either day? (1) A total of 1,000 people registered for the two-day seminar. (2) Of those registered, 80 percent attended the seminar on the second day."
E
"For a party, three solid cheese balls with diameters of 2 inches, 4 inches, and 6 inches, respectively, were combined to form a single cheese ball. What was the approximate diameter, in inches, of the new cheese ball? (The volume of a sphere is 43πr343πr3, where r is the radius.) (A) 12 (B) 16 (C) cube root of 16 (D) 2* cube root of 8 (E) 2* cube root of 36"
E
"If x and y are positive numbers such that x + y = 1, which of the following could be the value of 100x + 200y? I. 80 II. 140 III. 199 (A) II only (B) III only (C) I and II (D) I and III (E) II and III"
E
"If x, y, and z are positive numbers, what is the value of the average (arithmetic mean) of x and z ? (1) x - y = y - z (2) x^2 - y^2 = z"
E
"If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be (A) 2 (B) 5 (C) 6 (D) 7 (E) 14"
E
"In order to play a certain game, 24 players must be split into n teams, with each team having an equal number of players. If there are more than two teams, and if each team has more than two players, how many teams are there? (1) If thirteen new players join the game, one must sit out so that the rest can be split up evenly among the teams. (2) If seven new players join the game, one must sit out so that the rest can be split up evenly among the teams."
E
"Jackie has two solutions that are 2 percent sulfuric acid and 12 percent sulfuric acid by volume, respectively. If these solutions are mixed in appropriate quantities to produce 60 liters of a solution that is 5 percent sulfuric acid, approximately how many liters of the 2 percent solution will be required? A) 18 B) 20 C) 24 D) 36 E) 42"
E
"Kevin invested $8,000 for one year at a simple annual interest rate of 6 percent and invested $10,000 for one year at an annual interest rate of 8 percent compounded semiannually. What is the total amount of interest that Kevin earned on the two investments? A. $880 B. $1,088 C. $1,253 D. $1,280 E. $1,296"
E
"On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line? (1) xyz < 0 (2) xy < 0"
E
"What is the number of cans that can be packed in a certain carton? (1) The interior volume of this carton is 2,304 cubic inches. (2) The exterior of each can is 6 inches high and has a diameter of 4 inches."
E
"A furniture dealer purchased a desk for $150 and then set the selling price to the purchase price plus markup that was 40 percent of the selling price. If the dealer sold the desk at the selling price, what was the amount of the dealer's gross profit from the purchase and the sale of the desk? 1) $40 2) $60 3) $80 4) $90 5) $100"
E-100
"If R is the set of all numbers that, when squared, have a units digit of 1 or 9, then which of the following must be true about set R? I. All members of R are prime. II. If N is a member of R, -N is a member of R. III. If N is a member of R, is a member of R. Which statements are true?"
II and III only
x^y=y^x, where x is prime and y is even
If Y is even, Y^x must be even as well. It must be true then that x^y is also even. If x^y is even, X itself must be even. Since x is both even and prime, it must be true that x=2 since 2 is the only even prime
range of f(x)= (x+sq(3))^4
Range of function can never be negative. Raising anything to an even power will result in a non-negative value
