GRE Mathematics Preparation: Advanced Quantitative Reasoning
Welcome to this advanced-level GRE Mathematics preparation quiz, part of the September 2026 Series. This set of 50 questions is designed to challenge your quantitative reasoning skills, data analysis capabilities, and mathematical problem-solving agility—all essential for achieving a top score on the GRE Quantitative Reasoning section.
This quiz covers a broad spectrum of topics, including:
- Algebra and Functions: Tackle complex equations, inequalities, and function properties.
- Geometry and Coordinate Geometry: Solve problems involving shapes, lines, and spatial relationships.
- Data Analysis and Probability: Interpret data sets, calculate probabilities, and understand statistical measures.
- Number Properties and Arithmetic: Master the rules of integers, fractions, and exponents.
- Word Problems and Applied Mathematics: Translate real-world scenarios into mathematical models.
Each question is crafted to mirror the style and difficulty of the actual GRE, with a focus on multi-step reasoning and the application of fundamental concepts in novel ways. A detailed explanation follows every question to help you understand the underlying logic and refine your test-taking strategies.
Prepare to think critically, manage your time wisely, and elevate your mathematical proficiency. Good luck!
Q1. If ##f(x) = x^3 - 3x^2 + 2x## and ##g(x) = f(f(x))##, what is the sum of the coefficients of the polynomial ##g(x)##?
The sum of the coefficients of any polynomial is found by evaluating the polynomial at ##x = 1##. So, we need ##g(1) = f(f(1))##. First, ##f(1) = 1^3 - 3(1)^2 + 2(1) = 1 - 3 + 2 = 0##. Then, ##g(1) = f(0) = 0^3 - 3(0)^2 + 2(0) = 0##. Therefore, the sum of the coefficients is 0.
Q2. In a geometric sequence, the sum of the first 5 terms is 31 and the sum of the first 10 terms is 1023. What is the common ratio?
Let the first term be ##a## and the common ratio be ##r##. The sum of the first ##n## terms is ##S_n = a(r^n - 1)/(r - 1)##. We have ##S_5 = a(r^5 - 1)/(r - 1) = 31## and ##S_{10} = a(r^{10} - 1)/(r - 1) = 1023##. Dividing gives ##\dfrac{r^{10}-1}{r^5-1}=\dfrac{1023}{31}=33##. Since ##r^{10}-1=(r^5-1)(r^5+1)##, we obtain ##r^5+1=33##, so ##r^5=32## and therefore ##r=2##.
Q3. A circle with center ##(h, k)## is tangent to the x-axis and also tangent to the line ##y = x##. If the radius of the circle is 2, what is the sum of all possible values of ##h + k##?
Since the circle is tangent to the x-axis, ##|k|=2##, so ##k=\pm2##. Tangency to ##y=x## requires the distance from ##(h,k)## to ##x-y=0## to be 2: ##\dfrac{|h-k|}{\sqrt{2}}=2##, hence ##|h-k|=2\sqrt{2}##. For ##k=2##, the two values of ##h+k## are ##4\pm2\sqrt{2}##, whose sum is 8. For ##k=-2##, the two values are - 4##\pm2\sqrt{2}##, whose sum is -8. Therefore, the sum of all possible values is ##8+(-8)=0##.
Q4. If ##x## and ##y## are positive integers such that ##x^2 - y^2 = 45##, what is the maximum possible value of ##x + y##?
Factor the equation as ##(x-y)(x+y)=45##. Since ##x## and ##y## are positive integers, the factors ##x-y## and ##x+y## are positive and have the same parity. The factor pairs of 45 are ##(1,45)##, ##(3,15)##, and ##(5,9)##. The maximum possible value of ##x+y## is therefore 45, obtained from ##x-y=1## and ##x+y=45##, which give ##x=23## and ##y=22##.
Q5. The average (arithmetic mean) of a set of 10 numbers is 20. If the smallest number is removed, the average of the remaining numbers is 22. What is the smallest number?
The sum of the original 10 numbers is ##10\times20=200##. After the smallest number is removed, the remaining 9 numbers have total ##9\times22=198##. Therefore, the removed number is ##200-198=2##.
Q6. In a right triangle ##ABC##, the legs are ##AB = 6## and ##BC = 8##. A point ##D## lies on the hypotenuse ##AC## such that ##BD## is perpendicular to ##AC##. What is the length of ##BD##?
The hypotenuse is ##AC=\sqrt{6^2+8^2}=10##. The area of the triangle is ##\frac12(6)(8)=24##. Using ##AC## as the base, the same area is ##\frac12(10)(BD)##. Thus ##24=5BD##, giving ##BD=\dfrac{24}{5}=4.8##.
Q7. If ##a## and ##b## are integers such that ##a^2 + b^2 = 100##, how many ordered pairs ##(a, b)## satisfy this equation?
The nonnegative solutions arise from ##6^2+8^2=100## and ##0^2+10^2=100##. From ##(6,8)## there are 8 ordered signed pairs: ##(\pm6,\pm8)## and ##(\pm8,\pm6)##. From ##(0,10)## there are 4 pairs: ##(0,\pm10)## and ##(\pm10,0)##. Hence the total number is ##8+4=12##.
Q8. A bag contains 4 red marbles, 5 blue marbles, and 6 green marbles. If three marbles are drawn at random without replacement, what is the probability that all three marbles are of different colors?
There are ##\binom{15}{3}=455## ways to choose 3 marbles from 15. To obtain one marble of each color, there are ##4\times5\times6=120## choices. Therefore, the probability is ##\dfrac{120}{455}=\dfrac{24}{91}##.
Q9. The function ##h(x) = 2x^2 - 8x + 5## has its vertex at ##(p, q)##. What is the value of ##p + q##?
For ##h(x)=2x^2-8x+5##, the x-coordinate of the vertex is ##p=-\dfrac{b}{2a}=\dfrac{8}{4}=2##. Then ##q=h(2)=2(2)^2-8(2)+5=8-16+5=-3##. Therefore, ##p+q=2-3=-1##.
Q10. If ##\log_{10}(x) + \log_{10}(x - 3) = 1##, what is the value of ##x##?
Using the product rule, ##\log_{10}(x(x-3))=1##. Hence ##x(x-3)=10##, giving ##x^2-3x-10=0##. Factoring, ##(x-5)(x+2)=0##, so the algebraic candidates are ##x=5## and ##x=-2##. The logarithms require ##x>3##, so the only valid solution is ##x=5##.
Q11. A cylindrical tank with a radius of 3 feet and a height of 10 feet is filled with water. If water leaks out at a rate proportional to the volume of water remaining, and after 1 hour, half the water has leaked, how much water (in cubic feet) remains after 2 hours?
The initial volume is ##V_0=\pi r^2h=\pi(3)^2(10)=90\pi##. Because the leakage rate is proportional to the volume remaining, ##V(t)=V_0e^{-kt}##. After 1 hour half remains, so ##e^{-k}=\frac12##. Hence after 2 hours, ##V(2)=90\pi\left(\frac12\right)^2=\dfrac{90\pi}{4}=\dfrac{45\pi}{2}##.
Q12. In a class of 30 students, 18 play soccer, 15 play basketball, and 8 play both. How many students play neither sport?
By inclusion-exclusion, the number who play at least one sport is ##18+15-8=25##. Therefore, the number who play neither is ##30-25=5##.
Q13. What is the sum of all integers ##n## such that ##n^2 - 10n + 21 < 0##?
Factor the inequality: ##n^2-10n+21=(n-3)(n-7)##. The product is negative when ##3<n<7##. Thus the integer values are ##4,5,6##, and their sum is ##4+5+6=15##.
Q14. If the median of a set of 9 distinct integers is 15, what is the maximum possible value of the largest integer?
With 9 distinct integers in increasing order, the median is the fifth value. Thus the fifth value is 15, while the sixth through ninth values only need to be distinct integers greater than 15. The largest value can therefore be made arbitrarily large, so it is unbounded.
Q15. A line in the xy-plane has a slope of 2 and passes through the point ##(3, 4)##. What is the x-intercept of this line?
Using point-slope form, ##y-4=2(x-3)##, so ##y=2x-2##. At the x-intercept, ##y=0##. Thus ##0=2x-2##, which gives ##x=1##.
Q16. If ##x## is a positive integer and ##x^3## is divisible by 24, what is the smallest possible value of ##x##?
Since ##24=2^3\times3##, the prime factorization of ##x## must contain at least one factor of 2 and one factor of 3 so that ##x^3## contains ##2^3## and at least one factor of 3. The smallest such positive integer is ##x=6##. Indeed, ##6^3=216=24\times9##.
Q17. The sum of the infinite geometric series ##1 + 1/3 + 1/9 + 1/27 + \ldots## is ##S##. What is the sum of the series ##1/2 + 1/6 + 1/18 + 1/54 + \ldots## in terms of ##S##?
The second series is obtained by multiplying every term of the first series by ##\frac12##. Therefore, its sum is ##\dfrac{S}{2}##.
Q18. A rectangular box has positive integer dimensions ##x##, ##y##, and ##z##. If the surface area is 94 and the sum of the lengths of all edges is 48, what is the volume of the box?
The surface-area condition gives ##2(xy+yz+xz)=94##, so ##xy+yz+xz=47##. The edge-length condition gives ##4(x+y+z)=48##, so ##x+y+z=12##. For positive integer dimensions, ##x=3##, ##y=4##, and ##z=5## satisfy both conditions: ##3+4+5=12## and ##3\cdot4+4\cdot5+3\cdot5=47##. Therefore, the volume is ##xyz=3\times4\times5=60##.
Q19. If ##f(x) = \sqrt{x - 1}## and ##g(x) = x^2 + 1##, what is the domain of ##f(g(x))##?
We have ##f(g(x))=\sqrt{(x^2+1)-1}=\sqrt{x^2}=|x|##. Since ##x^2\ge0## for every real ##x##, the composition is defined for all real numbers.
Q20. A circle is inscribed in a square with a side length of 10. What is the area of the shaded region outside the circle but inside the square?
The square has area ##10^2=100##. The inscribed circle has radius ##5##, so its area is ##\pi(5)^2=25\pi##. Therefore, the required area is ##100-25\pi##.
Q21. If ##2^x = 3## and ##3^y = 4##, what is the value of ##xy##?
Taking natural logarithms gives ##x\ln2=\ln3## and ##y\ln3=\ln4##. Thus ##x=\dfrac{\ln3}{\ln2}## and ##y=\dfrac{\ln4}{\ln3}##. Therefore, ##xy=\dfrac{\ln4}{\ln2}=2## because ##\ln4=2\ln2##.
Q22. In a survey, 60% of people like coffee, 50% like tea, and 30% like both. What percentage of people like neither coffee nor tea?
By inclusion-exclusion, the percentage who like at least one is ##60\%+50\%-30\%=80\%##. Hence the percentage who like neither is ##100\%-80\%=20\%##.
Q23. What is the value of ##\sin(75^\circ)##?
Using the angle-addition identity, ##\sin75^\circ=\sin(45^\circ+30^\circ)##. Therefore, ##\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ##. Substituting the standard values gives ##\dfrac{\sqrt2}{2}\dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\frac12=\dfrac{\sqrt6+\sqrt2}{4}##.
Q24. If the roots of the equation ##x^2 + px + q = 0## are ##r## and ##s##, and the roots of ##x^2 + rx + s = 0## are ##p## and ##q##, and at least one of ##q## or ##s## is zero, what is the value of ##p + q + r + s##?
By Vieta's formulas, ##r+s=-p##, ##rs=q##, ##p+q=-r##, and ##pq=s##. From the two sum relations, ##s=q##. Since at least one of ##q## or ##s## is zero, we have ##q=s=0##. Then ##r=-p##. Therefore, ##p+q+r+s=p+0-p+0=0##.
Q25. If ##x## and ##y## are integers such that ##|x - 3| + |y + 2| = 1##, how many ordered pairs ##(x, y)## satisfy this equation?
Since the absolute values are nonnegative integers and their sum is 1, the only possibilities are ##(|x-3|,|y+2|)=(1,0)## or ##(0,1)##. The first gives ##(x,y)=(2,-2)## and ##(4,-2)##. The second gives ##(3,-1)## and ##(3,-3)##. Thus there are 4 ordered pairs.
Q26. A data set consists of 5 numbers: 2, 4, 6, 8, and ##x##. If the mean of the data set is equal to its median, what is the sum of all possible values of ##x##?
The mean is ##\dfrac{20+x}{5}##. Consider the position of ##x## in the ordered data set. If ##x\le2##, the median is 4, so ##\dfrac{20+x}{5}=4## gives ##x=0##. If ##4\le x\le6##, the median is ##x##, so ##\dfrac{20+x}{5}=x## gives ##x=5##. If ##x\ge8##, the median is 6, so ##\dfrac{20+x}{5}=6## gives ##x=10##. The intervening ranges yield no additional solutions. Thus the possible values are ##0,5,10##, whose sum is ##15##.
Q27. If ##\sin\theta + \cos\theta = 1/2##, what is the value of ##\sin\theta\cos\theta##?
Squaring the given equation, ##(\sin\theta+\cos\theta)^2=\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta##. Since ##\sin^2\theta+\cos^2\theta=1##, we get ##\frac14=1+2\sin\theta\cos\theta##. Thus ##2\sin\theta\cos\theta=-\frac34##, so ##\sin\theta\cos\theta=-\frac38##.
Q28. A rectangular field has a perimeter of 100 meters. If the length is 10 meters more than the width, what is the area of the field in square meters?
Let the width be ##w## and the length be ##w+10##. Then ##2(w+w+10)=100##, so ##2w+10=50## and ##w=20##. The length is 30, giving area ##20\times30=600## square meters.
Q29. If ##f(x) = \dfrac{x}{x - 1}## for ##x \neq 1##, what is ##f(f(f(2)))##?
First, ##f(2)=\dfrac{2}{2-1}=2##. Thus 2 is a fixed point of ##f##, so ##f(f(2))=2## and ##f(f(f(2)))=2##.
Q30. How many different ways can the letters of the word "MISSISSIPPI" be arranged?
The word MISSISSIPPI has 11 letters, with I repeated 4 times, S repeated 4 times, and P repeated 2 times. Hence the number of distinct arrangements is ##\dfrac{11!}{4!\,4!\,2!}=34650##.
Q31. If the sum of the first ##n## terms of an arithmetic sequence is given by ##S_n = 3n^2 + 2n##, what is the 10th term?
The nth term is ##a_n=S_n-S_{n-1}##. Hence ##a_{10}=S_{10}-S_9##. We have ##S_{10}=3(10)^2+2(10)=320## and ##S_9=3(9)^2+2(9)=261##. Therefore, ##a_{10}=320-261=59##.
Q32. A circle has a diameter of length 10. A chord of length 6 is drawn in the circle. What is the maximum distance from the center of the circle to the chord?
The radius is 5. A perpendicular from the center to the chord bisects the chord, giving a half-chord of 3. If ##d## is the distance from the center to the chord, then ##d^2+3^2=5^2##. Thus ##d=\sqrt{25-9}=4##.
Q33. If ##x## and ##y## are positive integers such that ##x + y = 10##, what is the maximum value of ##xy##?
For a fixed sum, the product is maximized when the two numbers are as equal as possible. Here ##x=y=5##, so the maximum product is ##xy=5\times5=25##.
Q34. The probability that it rains on a given day is 0.3, independently from day to day. What is the probability that it rains on exactly 2 out of 5 days?
Using the binomial model, the probability of exactly 2 rainy days out of 5 is ##\binom{5}{2}(0.3)^2(0.7)^3=10\times0.09\times0.343=0.3087##.
Q35. If ##\log_2 (x - 1) + \log_2 (x + 1) = 3##, what is the value of ##x##?
Using the product rule, ##\log_2((x-1)(x+1))=3##. Thus ##x^2-1=2^3=8##, so ##x^2=9## and ##x=\pm3##. The logarithms require ##x-1>0##, so ##x>1##. Therefore, only ##x=3## is valid.
Q36. In a triangle, the lengths of the sides are consecutive integers. If the perimeter is 30, what is the area of the triangle?
Let the consecutive side lengths be ##n-1##, ##n##, and ##n+1##. Their perimeter is ##3n=30##, so ##n=10## and the sides are ##9,10,11##. The semiperimeter is ##s=15##. By Heron's formula, ##A=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{15\cdot6\cdot5\cdot4}=\sqrt{1800}=30\sqrt{2}##.
Q37. If ##f(x) = x^2 + 2x## and ##g(x) = 2x - 1##, what is the value of ##f(g(2))##?
First, ##g(2)=2(2)-1=3##. Then ##f(3)=3^2+2(3)=9+6=15##.
Q38. A bag contains 3 red balls and 5 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?
The probability is ##\frac38\times\frac27=\dfrac{6}{56}=\dfrac{3}{28}##. Equivalently, ##\dfrac{\binom32}{\binom82}=\dfrac{3}{28}##.
Q39. What is the value of ##\int_{0}^{1} (3x^2 + 2x)\, dx##?
An antiderivative is ##x^3+x^2##. Therefore, ##\int_0^1(3x^2+2x)\,dx=[x^3+x^2]_0^1=1+1=2##.
Q40. If ##x^2 + y^2 = 25## and ##xy = 12##, what is the value of ##(x + y)^2##?
Using ##(x+y)^2=x^2+y^2+2xy##, we obtain ##(x+y)^2=25+2(12)=49##.
Q41. A train travels at a speed of 60 miles per hour. How many minutes will it take to travel 45 miles?
The travel time is ##\dfrac{45}{60}## hour, which equals ##\frac34## hour. Since 1 hour is 60 minutes, ##\frac34\times60=45## minutes.
Q42. If the average of 5 numbers is 12, and the average of the first 3 numbers is 10, what is the average of the last 2 numbers?
The sum of all 5 numbers is ##5\times12=60##. The sum of the first 3 is ##3\times10=30##. Thus the last 2 numbers have sum ##60-30=30##, and their average is ##\dfrac{30}{2}=15##.
Q43. If ##\tan \theta = 3/4## and ##\theta## is in the third quadrant, what is the value of ##\sin \theta##?
Since ##\tan\theta=\frac34##, a reference triangle may be taken with opposite side 3, adjacent side 4, and hypotenuse 5. In the third quadrant, both sine and cosine are negative. Therefore, ##\sin\theta=-\frac35##.
Q44. A sequence is defined by ##a_1 = 2## and ##a_{n+1} = 3a_n - 1## for ##n \ge 1##. What is the value of ##a_4##?
Using the recurrence, ##a_2=3(2)-1=5##, ##a_3=3(5)-1=14##, and ##a_4=3(14)-1=41##.
Q45. If the volume of a cube is 64 cubic centimeters, what is the total surface area of the cube in square centimeters?
If the side length is ##s##, then ##s^3=64##, so ##s=4##. The total surface area is ##6s^2=6(4^2)=6(16)=96## square centimeters.
Q46. If ##f(x) = \dfrac{1}{x}## and ##g(x) = \dfrac{x}{x+1}##, what is the value of ##f(g(2))##?
First, ##g(2)=\dfrac{2}{2+1}=\frac23##. Therefore, ##f(g(2))=f\left(\frac23\right)=\dfrac{1}{2/3}=\frac32##.
Q47. A circle has a circumference of ##12\pi##. What is the area of a sector with a central angle of 60 degrees?
From ##2\pi r=12\pi##, the radius is ##r=6##. The circle's area is ##\pi r^2=36\pi##. A ##60^\circ## sector is ##\dfrac{60}{360}=\frac16## of the circle, so its area is ##\frac16(36\pi)=6\pi##.
Q48. If ##x## and ##y## are integers such that ##x^2 + y^2 = 50##, what is the maximum possible value of ##x + y##?
Since ##x^2+y^2=50##, we have ##(x+y)^2=50+2xy##. The integer solution ##x=y=5## satisfies ##25+25=50## and gives ##x+y=10##. By ##(x-y)^2\ge0##, we have ##x^2+y^2\ge2xy##, so ##50\ge2xy## and hence ##(x+y)^2\le100##. Thus the maximum possible value is ##10##.
Q49. If the probability of an event occurring is 0.4, what are the odds in favor of the event?
The probability of success is ##0.4## and the probability of failure is ##1-0.4=0.6##. Thus the odds in favor are ##0.4:0.6=4:6=2:3##.
Q50. A line passes through the points ##(1, 2)## and ##(3, 8)##. What is the equation of the line in slope-intercept form?
The slope is ##m=\dfrac{8-2}{3-1}=\frac62=3##. Using ##(1,2)## in ##y=mx+b## gives ##2=3(1)+b##, so ##b=-1##. Therefore, the equation is ##y=3x-1##.