Bunuel's Algebra Diagnostic Test Absolute Values/Modulus, Algebra, Arithmetic, Exponents/Powers, Fractions/Ratios/Decimals, Functions and Custom Characters, Inequalities, Min/Max Problems, Must or Could be True Questions, Percent and Interest Problems. https://gmatclub.com/forum/download/file.php?id=68660 Please note that the questions are ranked in ascending level of difficulty according to the timer stats. Grading & How to use this test: Please time yourself and allocate 2 mins per question 27 or more correct, your Algebra score is Q90 24 - 26 correct, your Algebra score is Q86 21 - 23 correct, your Algebra score is Q83 18 - 20 correct, your Algebra score is Q80 15 - 17 correct, your Algebra score is Q78 12 - 14 correct, your Algebra score is Q76 Note that levels 7 and 8 are very hard questions. These questions are well within the GMAT parameters but their difficulty is driven by the amount of time you will need to spend to solve each of the questions. Do not be discouraged if you are not able to solve any of the Level 7 or 8 questions. ------------------- EASY ------------------- LEVEL 1: 1. If the average of a, b, c, 14 and 15 is 12. What is the average value of a, b, c and 29 ? A. 12 B. 13 C. 14 D. 15 E. 16 D 2. If 22 - |y + 14| = 20, what is the sum of all possible values of y ? A. -28 B. -16 C. -12 D. -4 E. 4 A 3. Jeeves prepares a hangover cure using four identical cocktail shakers. The first shaker is 1/2 full, the second shaker is 1/3 full, the third shaker is 1/4 full and the last one is empty. If Jeeves redistributes all the content of the shakers equally into the four shakers, what fraction of each shaker will be filled? A. 13/48 B. 4/13 C. 13/36 D. 9/13 E. 35/48 A 4. If 7^8 = m and 8^7 = n , then what is the value of 56^{56} in terms of m and n ? A. mn B. m^7*n^8 C. m^8*n^7 D. (mn)^{56} E. 56^{mn} B 5. If ab^2 > 0 and ac 0 II. b^2c 0 A. I only B. II only C. III only D. II and III only E. I, II, and III D LEVEL 2: 6. What is the value of \sqrt{(49\%)} - \frac{7}{100} ? A. 0 B. 0.63% C. 0.063 D. 63% E. 6.3 D 7. What is the value of (12\%)^2 ? A. 0.144\% B. 1.44\% C. 0.144 D. 144\% E. 14.4 B 8. If \sqrt{2+\sqrt{2+x} } +\sqrt{2-\sqrt{2+x} }=\sqrt{2x} , what is the value of x ? A. -2 B. -\sqrt{2} C. 1 D. \sqrt{2} E. 2 E 9. If x > 0 and \sqrt{17^{\sqrt{x} } } = 17^{\frac{1}{\sqrt{x} } } , what is the value of x ? A. \frac{1}{2} B. \frac{1}{\sqrt{2 C. \sqrt{2} D. 2 E. 4 D 10. If m is a positive number and n is a negative number, and |m| > |n|, then which of the following has the greatest value ? A. |\frac{m - n}{n}| B. |\frac{m - n}{m}| C. |\frac{m + n}{m - n}| D. |\frac{m + n}{n}| E. |\frac{m + n}{m}| A ------------------- MEDIUM ------------------- LEVEL 3: 11. Jeeves prepares a hangover cure using four identical cocktail shakers. The first shaker is 1/2 full, the second shaker is 4/5 full, the third shaker is 1/k full and the last one is empty. After Jeeves redistributed all the content of the shakers equally into the four shakers, each shaker became 31/80 full. What is the value of k? A. 2 B. 3 C. 4 D. 5 E. 6 C 12. If the sum of all 21 terms of an arithmetic progression is zero, then which of the following MUST be true ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant) I. 11th smallest term is zero II. 11th largest term is zero III. The range of the numbers is 20 A. I B. II C. III D. I and II E. None of the above D 13. What is the value of \sqrt {2.7\%}-\sqrt {0.16\%} ? A. 0.001 B. 10\% C. 1 D. 1000\% E. 100 B 14. x_1+x_2+...+x_{100} = 1 ; x_1+x_2+...+x_{99} = 2 ; x_1+x_2+...+x_{98} = 3 ; ... x_1= 100 . What is the value of x_1*x_2*...*x_{100} ? A. -100 B. -1 C. 0 D. 1 E. 100 A 15. What is the value of (\sqrt {800\%})^2 A. 0.04\% B. 0.4 C. 100\% D. 2 E. 400\% E LEVEL 4: 16. A merchant offers a discount of 30% on her list price and ends up making a profit of 19% on her cost price. By what percentage was her list price more than the cost price? A. 76% B. 70% C. 57% D. 55% E. 49% B 17. If (x + y)^2 0 . What is the sum of the first 11 terms of the sequence? A. 43954414 B. 43954588 C. 43954675 D. 43954713 E. 43954780 D 19. What is the value of |\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}| ? A. 2\sqrt{5}-2\sqrt{7}-2\sqrt{3} B. 2\sqrt{5}-2\sqrt{7} C. 2\sqrt{7}-2\sqrt{5} D. 2\sqrt{5}-2\sqrt{7}+2\sqrt{3} E. 2\sqrt{7}+2\sqrt{5} B 20. If x > 0 and x^{(3*x^{12})}=4 , what is the value of x ? A. \sqrt {2} B. \sqrt {2} C. \sqrt {2} D. \sqrt{3} E. \sqrt{2} B ------------------- HARD ------------------- LEVEL 5: 21. If a > b > c > d > e and abcde > 0 , then which of the following must be true ? I. ab > 0 II. bc > 0 III. de > 0 A. I only B. II only C. III only D. II and III only E. I, II, and III D 22. If the sum of the first 31 terms of an arithmetic progression consisting 46 terms is zero, then which of the following MUST be true ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant) I. 31st smallest term is zero II. 16th largest term is zero III. The sum of the largest and smallest terms of the sequence is positive A. I only B. II only C. III only D. I and II only E. None of the above E 23. What is the value of 4^{3^{2^{1^{2^{3^4} } } } } ? A. 2,144 B. 262,142 C. 262,144 D. 262,146 E. 262,148 C 24. If x is an integer and \frac{(x^2 - 2)!}{x^2-2}=33! , what is the range of all possible values of x ? A. 0 B. 2 C. 6 D. 11 E. 12 E 25. If 9^x + 9^{-x} = 62 , then what is the value of 3^x + 3^{-x} ? A. 1/64 B. 1/8 C. 8 D. 62 E. 64 C LEVEL 6: 26. If x^x=\sqrt{\frac{\sqrt{2} }{2} } , which of the following could be a value of x ? A. \frac{1}{16} B. \frac{1}{4} C. \frac{1}{\sqrt{2 D. \frac{1}{2} E. \sqrt{2} A 27. A cloth merchant professes to sell at a loss of 5%. However, the merchant falsifies the meter scale and gains 25%. What is the measure of the scale that he uses when measuring 1 meter of cloth? A. 70 B. 72 C. 75 D. 76 E. 77 D 28. If 45x = 121y, which of the following must be true? I. x > y II. x^2 > y^2 III. x/11 is an integer (A) I only (B) II only (C) III only (D) I and III only (E) None of the above E 29. If x and y are integers 35x = 69y, which of the following must be true? I. x > y II. y/7 is an integer III. x/23 is an integer (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III D 30. What is the value of (\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}) ? A. -4 B. -2 C. -1 D. 1 E. 2 B ------------------- VERY HARD ------------------- LEVEL 7: 31. If is the greatest integer less than or equal to x, and = 5 and = 6, where x and y are positive integers, what is the greatest possible value of x + y ? A. 61 B. 81 C. 82 D. 83 E. 85 D 32. If \frac{|x| - 2}{|x - 2|}= 1 , then which of the following must be true? I. |x| > 2 II. x^2 0 A. I only B. II only C. III only D. I and III only E. None D 33. {x, y, z} is an increasing sequence of numbers such that the ratio between the consecutive terms is constant. {x + 2y, 2x + y + z, x + 3y + z} is an increasing sequence of numbers such that the difference between the consecutive terms is constant. What is the value of x/z ? A. 1/4 B. 1/2 C. 1 D. 2 E. 4 A 34. Mbappe withdraws \frac{100}{x}% of the amount in his account each time he visits the bank, where x is an integer greater than 1. If, after 5 visits, he has less than \frac{1}{x}^{th} of the initial amount left in the bank, what is the range of possible values for x ? A. 1 B. 2 C. 3 D. 4 E. 5 B 35. If the sum of the first 11 terms of an arithmetic progression consisting 16 terms is zero, then which of the following COULD be true ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant) I. 11th smallest term is zero II. 6th largest term is zero III. All terms are non negative A. I only B. II only C. III only D. I and II only E. I, II, and III E LEVEL 8: 36. If \frac{x+500}{510}+\frac{x+510}{515} + \frac{x+520}{520} + \frac{x+530}{525}+ \frac{x+540}{530}=10 , what is the value of x ? A. 515 B. 520 C. 525 D. 530 E. 535 B 37. If x and y are consecutive integers, x > y, and x^2 - 1 > y^2 - 4y + x - 1, then which of the following must be true? A. x -1 D. y ≠ 1 E. x + y > 3 C 38. If \frac{m^4}{|m|} -8 A. I only B. II only C. III only D. I and II only E. I, II, and III E 39. The infinite sequence of integers a_1 , a_2 , ..., a_n , ... is such that a_1 = -17 and a_{n-1} - 7 1 . If a_x=-53 , then how many different values can x take? A. 1 B. 4 C. 5 D. 6 E. 7 E 40. If n is an integer greater than 1, what is the value of 10*\sqrt {10*\sqrt {10*\sqrt {10*\sqrt {...}}}} , where the given expression extends to an infinite number of roots? A. 10 B. 10^{\frac{1}{n}} C. 10^{\frac{n-1}{n}} D. 10^{\frac{n}{n-1}} E. 10^{n} D Check Word Problems Diagnostic set HERE . algebra.jpg

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Question 40 - Official Solution:

If \(n\) is an integer greater than 1, what is the value of \(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\), where the given expression extends to an infinite number of roots?

A. \(10\)
B. \(10^{\frac{1}{n}}\)
C. \(10^{\frac{n-1}{n}}\)
D. \(10^{\frac{n}{n-1}}\)
E. \(10^{n}\)

Let \(x=10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\)

Now, re-write above as \(x=10*\sqrt[n]{(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...})}}}\).

Since the expression extends to an infinite number of roots, then the expression in brackets would also equal to \(x\). Thus we can replace the expression in brackets with \(x\) and rewrite the expression as: \(x=10*\sqrt[n]{x}\)

Take above to the \(n^{th}\) power:

\(x^n=10^n*x\)

\(x^{n-1}=10^n\)

Take \(n-1^{th}\) root:

\(x=10^{\frac{n}{n-1}}\)

Answer: D

if i take \(\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\)=k

then I can make the equation as 10k=10*{(10k)^(1/n)}..now canceling 10 on both the sides we get k=(10k)^(1/n)
Solvin further we get k=10^(1/(n-1))..
I dont understand where am i going wrong. Please help.

Bunuel

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