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

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 Solution: ab^2 > 0 => a > 0 & b could either be positive or negative ac We already deduced a > 0 from the previous inequality so here c 0, c 0 a > 0 but b could either be + or - , so this CANNOT BE MUST BE TRUE II. b^2c 0 We already know that a > 0, c 0 , since both parts of the product are > 0, expression will always be > 0. MUST BE TRUE II and III MUST BE TRUE Answer: D

13. What is the value of \sqrt {2.7\%}-\sqrt {0.16\%} ? A. 0.001 B. 10\% C. 1 D. 1000\% E. 100 \sqrt {2.7\%}-\sqrt {0.16\%} = \sqrt {0.027}-\sqrt {0.0016} = 0.3 - 0.2 = 0.1 = 10\% Answer: B

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Bunuel's Algebra PS Diagnostic Test

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06 Oct 2022, 18:56

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5. If \(ab^2 > 0\) and \(ac < 0\), then which of the following must be true?

I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

Solution:

\(ab^2 > 0\) => \(a > 0\) & \(b\) could either be positive or negative

\(ac < 0\) => We already deduced \(a > 0\) from the previous inequality so here \(c < 0\)

So, we have \(a > 0, c < 0\) and \(b\) is inconclusive

I. \(ab >0\)

\(a > 0\) but \(b\) could either be + or - , so this CANNOT BE MUST BE TRUE

II. \(b^2c < 0\)

We already know that \(c < 0\) and \(b^2\) will always be positive so expression will always be < 0. MUST BE TRUE

III. \(a*c^2 > 0\)

We already know that \(a > 0, c < 0\) so \(c^2 > 0\), since both parts of the product are > 0, expression will always be > 0. MUST BE TRUE

II and III MUST BE TRUE

Answer: D

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6. What is the value of \(\sqrt{(49\%)} - \frac{7}{100}\) ?

A. 0
B. 0.63%
C. 0.063
D. 63%
E. 6.3

Solution:

\(\sqrt{(49\%)} - \frac{7}{100} = \frac{7}{10} - \frac{7}{100} = \frac{70-7}{100} = \frac{63}{100} = 63%\)

Answer: D

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7. What is the value of \((12\%)^2\) ?

A. \(0.144\%\)
B. \(1.44\%\)
C. \(0.144\)
D. \(144\%\)
E. \(14.4\)

Solution:

\((12\%)^2 = \frac{12^2}{100^2} = \frac{144}{10000} =\frac{1.44}{100} = 1.44\%\)

Answer: B

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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. \(2\)

Solution:

Best approach is to plug in values and those first which give integer results as we need to equate LHS and RHS

We see a Sqrt(2+x) so safe to first plus in E (2) into the entire equation and see if things match

LHS = \(\sqrt{2+\sqrt{2+2}} +\sqrt{2-\sqrt{2+2}} = \sqrt{4} +\sqrt{0} = 2\)
RHS = \(\sqrt{2x} = \sqrt{4} = 2\)

LHS = RHS

Answer: E

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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. \(4\)

Solution:

\(\sqrt{17^{\sqrt{x}}} = 17^{\frac{1}{\sqrt{x}}}\)

Squaring both sides

\(17^{\sqrt{x}} = 17^{\frac{2}{\sqrt{x}}}\)

Since both sides are equal and both bases are also equal => Both exponents must also be equal

\(\sqrt{x} = \frac{2}{\sqrt{x}}\)
\(\sqrt{x} * \sqrt{x} = 2\)
\(x = 2\)

Answer: D

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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. \(|\frac{m + n}{m}|\)

Solution:

POE the best way to go about this question.

m = Positive
n = Negative

For example: m = 5, n = -4 => m + n = 1, m - n = 9

So, any answer choice with (m - n) in the numerator will be higher as compared to any other value &
any answer choice with (m - n) in the denominator will be lower as compared to other

ELIMINATE C, D, E

Between A and B, the only difference is the denominator |m| and |n|, Since |m| > |n|, we need the bigger number which will be the one with |n| in the denominator

Answer: A

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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?4

A. 2
B. 3
C. 4
D. 5
E. 6

Solution:

Total capacity filled = \(\frac{1}{2} + \frac{4}{5} + \frac{1}{k}\)
Total capacity = 4

Total capacity filled per shaker = Total filled/Total capacity = \(\frac{1}{4}*[\frac{1}{2} + \frac{4}{5} + \frac{1}{k}]\)

Total capacity filled per shaker given = \(\frac{31}{80}\)

=> \(\frac{31}{80} = \frac{1}{4}*[\frac{1}{2} + \frac{4}{5} + \frac{1}{k}]\)

=> \(\frac{31}{80} = \frac{1}{8} + \frac{1}{5} + \frac{1}{4k}\)

=> \(\frac{31}{80} - \frac{1}{8} - \frac{1}{5} = \frac{1}{4k}\)

=> \(\frac{5}{80} = \frac{1}{4k}\)

=> \(k = 4\)

Answer: C

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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

Solution:

This is a MUST BE TRUE question so any statement if true, MUST BE TRUE under any and all circumstances

Given: An Arithmetic Progression of 21 terms & Sum of AP = 0

Analysis: For sum of all terms of AP to be 0, there can be two cases

a) Half the terms are Negative, Half are Positive and the middle term = 0
b) All terms = 0 (AP with Common Difference = 0)

I. 11th smallest term is zero

In either of the 2 above cases, 11th smallest term will always be 0. MUST BE TRUE

II. 11th largest term is zero

In either of the 2 above cases, 11th largest term will always be 0 (21 terms AP, 11th smallest term = 11th largest term). MUST BE TRUE

III. The range of the numbers is 20

No, the range depends on the first and last terms of the AP

Range of consecutive integers from -10 to + 10 = 20
Range of consecutive even integers from -20 to +20 = 40

NOT MUST BE TRUE

So, I & II are ALWAYS TRUE

Answer: D

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13. What is the value of \(\sqrt[3]{2.7\%}-\sqrt[4]{0.16\%}\) ?

A. \(0.001\)
B. \(10\%\)
C. \(1\)
D. \(1000\%\)
E. \(100\)

\(\sqrt[3]{2.7\%}-\sqrt[4]{0.16\%} = \sqrt[3]{0.027}-\sqrt[4]{0.0016} = 0.3 - 0.2 = 0.1 = 10\%\)

Answer: B

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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

Solution:

\(x_1 = 100\)
All other terms from \(x_2\) to \(x_{100}\) = -1

\(x_1*x_2*...*x_{100} = 100 * (-1)^{99} = 100*(-1) = -100\)

Answer: A

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15. What is the value of \((\sqrt[3]{800\%})^2\)

A. \(0.04\%\)
B. \(0.4\)
C. \(100\%\)
D. \(2\)
E. \(400\%\)

Solution:

\((\sqrt[3]{800\%})^2 = (\sqrt[3]{8})^2 = 2^2 = 4 = 400\%\)

Answer: E

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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%

Let the list price = 100
Price after discount of 30% = 70
Let her cost price = x

According to the question: 19% profit (wrt cost price) on price after discount

=> \(x + \frac{19x}{100} = 70\)
=> \(\frac{119x}{100} = 70\)
=> \(x = \frac{7000}{119} = \frac{1000}{17}\)

% change ={ (100 - 1000/17) / (1000/17) } * 100 = {(1700 - 1000)/17}/{1000/17} * 100 = 70\%[/m]

Answer: B

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06 Oct 2022, 23:42

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17. If \((x + y)^2 < x^2\), which of the following must be true?

I. \(y(y + 2x) < 0\)
II. \(y < x\)
III. \(xy < 0\)

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

\((x + y)^2 < x^2 => x^2 + y^2 + 2xy < x^2\)
\(y^2 + 2xy < 0\)
\(y(y+2x) < 0\)

MUST BE TRUE: True in every circumstance

I. \(y(y + 2x) < 0\)

Proved above. MUST BE TRUE

II. \(y < x\)

For \((x + y)^2\) to be LESS THAN \(x^2\), one of x or y must be negative so that LHS < RHS

Case 1: \(x = 4, y = -3 => 1^2 < 4^2\)
Case 2: \(x = -3, y = 4 => (-1)^2 < 4^2\)

Not Must be True

III. \(xy < 0\)

For \((x + y)^2\) to be LESS THAN \(x^2\), one of x or y must be negative so that LHS < RHS

For this statement, it does not matter which remains positive and which remains negative as long as its profitable

MUST BE TRUE

Answer: I and III are true
Answer: E

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07 Oct 2022, 06:49

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Bunuel, thank you so much for this, this is super helpful for one to gauge their ability level and identify where they need to improve. I'm all in for similar tests on other concepts !!

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07 Oct 2022, 11:14

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18. The infinite sequence \(a_1, \ a_2, \ …, \ a_n, \ …\) is such that \(a_n = n!\) for \(n > 0\). What is the sum of the first 11 terms of the sequence?

A. 43954414
B. 43954588
C. 43954675
D. 43954713
E. 43954780

\(a_1 = 1!, a_2 = 2!, a_3 = 3!, a_4 = 4!, a_5 = 5!, a_6 = 6!, a_7 = 7!, a_8 = 8!, a_9 = 9!, a_10 = 10!, a_11 = 11!\)

Sum\( = 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! + 11!\)

Best way to approach this would be to analyze the answer choices and use POE:

If you notice, every answer choice has a different units digit, so let us work around that

Now, we know that every term of the above sum from 5! onwards to 11! ends with 0 in the units digit place

So, we basically need to calculate the Units digit of \(1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33\), so Units digit = 3

Only one answer choice has a Units digit = 3

Answer: D

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07 Oct 2022, 11:32

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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. \(2\sqrt{7}+2\sqrt{5}\)

Solution:

This question involves a modulus within a modulus so we need to to open one modulus at a time

\(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\)

\(|\sqrt{3} + \sqrt{5} - \sqrt{7}| - |- \sqrt{3} + \sqrt{5} - \sqrt{7}|\)

\( \sqrt{3} + \sqrt{5} - \sqrt{7} - \sqrt{3} + \sqrt{5} - \sqrt{7}\)

\( \sqrt{5} - \sqrt{7} + \sqrt{5} - \sqrt{7}\)

\( 2\sqrt{5} - 2\sqrt{7}\)

Answer: B

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07 Oct 2022, 11:51

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20. If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?

A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)

Solution:

Best approach in questions like these for me is plugging in answer choices. As we can see that options are increasing in value, I like to start with option C and then move accordingly based on how far above or below we are

We plug in and solve LHS and see how far (if we are) from the actual RHS

C) \(\sqrt[3]{2}\)

\(x^{(3*x^{12})} = 2^{2^4} \neq 4\)

We are way above the RHS, so we should go to a smaller number so Option B

B) \(\sqrt[6]{2}\)

\(x^{(3*x^{12})}= \sqrt{2}^{2^2} = 4\) = RHS

Answer: B

p.s. Apologies for not able to show more steps of the calculation, because the formula writing was becoming very complex and coming out wrong each time, but if you solve it by hand, you will see the values and will be much easier as well

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07 Oct 2022, 12:06

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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

Solution:

If \(a > b > c > d > e\) & \(abcde > 0\): We can infer that for a product of 5 numbers to be positive, either 1 or 3 or 5 of them have to be positive so based on that information and the descending order of these numbers, we can say the following

1) a > 0 and Rest < 0
2) a, b, c > 0 and Rest < 0
3) All > 0

Let us now examine the statements (MUST BE TRUE: Always true in every circumstance):

I. \(ab > 0\)

a > 0 in each of the above cases
But b > 0 (ab > 0) or b < 0 (ab < 0)
This is NOT Must be True

II. \(bc > 0\)

Either b and c will be both > 0 (Case 2 and 3) or both < 0 (Case 1)
In either way, bc > 0 ALWAYS
MUST BE TRUE

III. \(de > 0\)

Either d and e will be both > 0 (Case 3) or both < 0 (Case 1 and 2)
In either way, de > 0 ALWAYS
MUST BE TRUE

So, II and III are MUST BE TRUE

Answer: D

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07 Oct 2022, 12:13

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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

Solution:

Since we are dealing with a MUST BE TRUE question, it is easy to find the true part, but if we are able to find a possible solution where it is false, that can be ruled out because MUST BE TRUE should always be true no matter any condition

I. 31st smallest term is zero

Consider an AP of consecutive integers from -15 to 30: 31st smallest term = 15 (not equal to 0) (NOT Must be True)

II. 16th largest term is zero

Consider an AP of consecutive integers form -15 to 30: 16th largest term = 15 (not equal to 0) (NOT Must be True)

III. The sum of the largest and smallest terms of the sequence is positive

Consider an AP where all terms = 0 (Common difference = 0): Sum of largest and smallest = 0 (Not positive) (Not Must be True)

No statement is MUST BE TRUE

Answer: E (None of the above)

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07 Oct 2022, 12:18

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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

Solution:

\(4^{3^{2^{1^{2^{3^4}}}}}\)

What can we deduce about this? Ok the cyclicity of 4 is 2 => Powers of 4 either end in 4 or in 6
Based on this information, we can ELIMINATE options B and E

Also, any number which is 4 to the power of something MUST BE DIVISIBLE BY 4
Based on this information, we can ELIMINATE D (Because last two digits are 46, which is not divisible by 4)

We are left with A and C: Now A is a very small number and cannot be the answer to the expression

Answer: C