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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.
Please note that the questions are ranked in ascending level of difficulty according to the timer stats.
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.
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 \(ac < 0\), 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
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 \(\sqrt[3]{2.7\%}-\sqrt[4]{0.16\%}\) ?
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 \((\sqrt[3]{800\%})^2\)
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 \((x + y)^2 < x^2\), 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
E
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
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. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
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 [x] is the greatest integer less than or equal to x, and [√x] = 5 and [√y] = 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. \(x^2 < 1\)
III. \(x^3 > 0\)
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 < 0
B. x ≠ 2
C. y > -1
D. y ≠ 1
E. x + y > 3
C
38. If \(\frac{m^4}{|m|}<\sqrt{m^2}\), then which of the following must be true?
I. \(m < \pi\)
II. \(m^2<1\)
III. \(m^3>-8\)
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 \(a_{n-1} - 7 < a_n < a_{n-1} - 2\) for \(n > 1\). 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 \(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\), 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. \(10^{n}\)
D
Check Word Problems Diagnostic set HERE.
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11 Mar 2023, 12:45
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Question 30 - Official Solution:
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. \(2\)
We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.
Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:
Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::
Answer: B
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. \(2\)
We want to find the value of the expression \((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})\). To simplify this expression, we can start by squaring it.
\(=((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10}))^2=\)
\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)
\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)
\(=(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} - \sqrt{10})^2=\)
\(=(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)
Applying the difference of squares identity \((a - b)(a+b)=a^2-b^2\) to \((4-\sqrt{15})(4 + \sqrt{15})\) we can simplify to:
\(=(16 -15)(4 + \sqrt{15})(\sqrt{6} - \sqrt{10})^2=\)
Now using the identity \((a - b)^2=a^2-2ab+b^2\) to simplify the squared term::
\(=(4 + \sqrt{15})(6 - 2\sqrt{60}+10)=\)
\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)
\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)
\(=4(16-15)=4\).
\(=(4 + \sqrt{15})(16 - 4\sqrt{15})=\)
\(=(4 + \sqrt{15})4(4 - \sqrt{15})=\)
\(=4(16-15)=4\).
Answer: B
11 Mar 2023, 08:45
Kudos
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Question 19 - Official Solution:
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. \(2\sqrt{7}+2\sqrt{5}\)
To answer this question we should recall the property of the absolute value:
\(|x| = x\), when \(x \geq 0\);
\(|x| = -x\), when \(x < 0\).
So, we should evaluate the expressions in the modulus to see whether they are positive or negative.
STEP 1:
Since \(\sqrt{5}-\sqrt{7} < 0\), then \(|\sqrt{5}-\sqrt{7}|=-(\sqrt{5}-\sqrt{7})=\sqrt{7}-\sqrt{5}\);
Since \(\sqrt{3} - \sqrt{5} < 0\), then \(|\sqrt{3} - \sqrt{5}|=-(\sqrt{3} - \sqrt{5})= \sqrt{5} -\sqrt{3}\).
Thus, \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) will become:
\(|\sqrt{3} -(\sqrt{7}-\sqrt{5})| - |(\sqrt{5} -\sqrt{3})-\sqrt{7}|=|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\).
STEP 2:
\(\sqrt{3} +\sqrt{5}-\sqrt{7}\) must be positive because \(\sqrt{3} +\sqrt{5}=1.something + 2.something=3.something\), while \(\sqrt{7} < 3.something\). So, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| =\sqrt{3} +\sqrt{5}-\sqrt{7}\);
\(\sqrt{5} -\sqrt{3}-\sqrt{7}\) is obviously negative. So, \(\sqrt{5} -\sqrt{3}-\sqrt{7}=-(\sqrt{5} -\sqrt{3}-\sqrt{7})=\sqrt{7}+\sqrt{3}-\sqrt{5}\).
Thus, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\) will become:
\((\sqrt{3} +\sqrt{5}-\sqrt{7})-(\sqrt{7}+\sqrt{3}-\sqrt{5})=2\sqrt{5}-2\sqrt{7}\).
Answer: B
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. \(2\sqrt{7}+2\sqrt{5}\)
To answer this question we should recall the property of the absolute value:
\(|x| = x\), when \(x \geq 0\);
\(|x| = -x\), when \(x < 0\).
So, we should evaluate the expressions in the modulus to see whether they are positive or negative.
STEP 1:
Since \(\sqrt{5}-\sqrt{7} < 0\), then \(|\sqrt{5}-\sqrt{7}|=-(\sqrt{5}-\sqrt{7})=\sqrt{7}-\sqrt{5}\);
Since \(\sqrt{3} - \sqrt{5} < 0\), then \(|\sqrt{3} - \sqrt{5}|=-(\sqrt{3} - \sqrt{5})= \sqrt{5} -\sqrt{3}\).
Thus, \(|\sqrt{3} - |\sqrt{5}-\sqrt{7}|| - ||\sqrt{3} - \sqrt{5}|-\sqrt{7}|\) will become:
\(|\sqrt{3} -(\sqrt{7}-\sqrt{5})| - |(\sqrt{5} -\sqrt{3})-\sqrt{7}|=|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\).
STEP 2:
\(\sqrt{3} +\sqrt{5}-\sqrt{7}\) must be positive because \(\sqrt{3} +\sqrt{5}=1.something + 2.something=3.something\), while \(\sqrt{7} < 3.something\). So, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| =\sqrt{3} +\sqrt{5}-\sqrt{7}\);
\(\sqrt{5} -\sqrt{3}-\sqrt{7}\) is obviously negative. So, \(\sqrt{5} -\sqrt{3}-\sqrt{7}=-(\sqrt{5} -\sqrt{3}-\sqrt{7})=\sqrt{7}+\sqrt{3}-\sqrt{5}\).
Thus, \(|\sqrt{3} +\sqrt{5}-\sqrt{7}| - |\sqrt{5} -\sqrt{3}-\sqrt{7}|\) will become:
\((\sqrt{3} +\sqrt{5}-\sqrt{7})-(\sqrt{7}+\sqrt{3}-\sqrt{5})=2\sqrt{5}-2\sqrt{7}\).
Answer: B
