Bunuel's Algebra PS Diagnostic Test

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




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




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




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. \(56^{mn}\)




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






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

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




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

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




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. \(2\)




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. \(4\)




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








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




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




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

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




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




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

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






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%




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




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




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




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. \(\sqrt{2}\)









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




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




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




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




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






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. \(\sqrt{2}\)




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




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




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




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








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




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




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




34. Mbappe withdraws \(\frac{100}{x}\%\) of his money each time he visits the bank, where \(x > 1\). If after 5 visits, he has less than \(\frac{1}{x}^{th}\) of the initial amount in the bank, what is the range of all possible value of \(x\) ? (Assume x is an integer)

A. 1
B. 2
C. 3
D. 4
E. 5




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






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




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




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




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




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. \(10^{n}\)