Bunuel's Algebra DS Diagnostic Test

1. Is \(x = 0\)?

(1) \(xy = x\)
(2) \(x + y = y\)



2. Is \(Q > 0\) ?

(1) \(P*Q^3*R^4 < 0\)
(2) \(P*Q^2*S^6 > 0\)



3. Is \(|x + y| < |x| + |y|\) ?

(1) \(xy < 0\)
(2) \(x < y\)



4. If x and y are integers, is \(x = 0\)?

(1) \((-2)^x*y^3 > 0\)
(2) \((-3)^x*y^2 < 0\)



5. If m and n are positive integers, is \(4^m*(\frac{1}{3})^n < 1\)?

(1) n = 2m
(2) n = 4




6. If \(x + y ≠ 0\), is \(\frac{1}{(x + y)} < 2\)?

(1) \(x^3 = y^3\)
(2) \(\frac{1}{x} < 2\)



7. Is \(x = 0\) ?

(1) \(x^2*y^3 > 0\)
(2) \(x^3*y^2 < 0\)



8. Is \(x > y\)?

(1) \(x + y > 0\)
(2) \(y^2 > x^2\)



9. Is \(|x + 1| < 2\) ?

(1) \((x - 1)^2 < 1\)
(2) \(x^2 - 2 < 0\)



10. Is \(x > y\) ?

(1) \(x^2 < y^2\)
(2) \(y < 0\)





11. Is \(|x - y| = ||x| - |y||\) ?

(1) \(xy > 0\)
(2) \(x < y < 0\)



12. What is the value of positive integer n ?

(1) \(n^{5!} = 1\)
(2) \(n^5 = n!\)



13. If m and n are positive integers, and \(x = 2^m3^n\), is m < n ?

(1) x is divisible by 144
(2) x is not divisible by 648



14. Is \(xy < 0\) ?

(1) \(x^2 = y^2\)
(2) \(\frac{1}{(x + y)} < 1\)



15. Is \(|x| + x > |y| + y\) ?

(1) \(xy > 0\)
(2) \(x + y < 0\)




16. Is \(x^2 < x^3\)?

(1) \(x < x^2\)
(2) \(x < 1\)



17. Is \(ac + \sqrt{(a^2 - 1)(c^2 - 1)} \leq 1\) ?

(1) \(a^2 + b^2 = 1\)
(2) \(c^2 + d^2 = 1\)



18. If n is an integer such that \(\frac{1}{6} < \frac{1}{(n-1)} < \frac{1}{3}\), what is the value of n?

(1) \((n - 6)(n - 7) = 0\)
(2) \((n - 5)(n - 3) ≠ 0\)



19. If m and n are positive two-digit integers, what is the value of the tens digit of m minus the tens digit of n ?

(1) m - n = 42.
(2) The units digit of m minus the units digit of n is not a multiple of 3.



20. If \(x + y ≠ 0\), is \(\frac{1}{(x + y)} < 1\)?

(1) \(x^2 = y^2\)
(2) \(\frac{1}{x} < 2\)





21. Is p an odd integer?

(1) \((p^5)^5\) is an odd integer.
(2) \(\sqrt[5]{\sqrt[5]{p\) is an odd integer.



22. Does \(x = \)y ?

(1) \(\frac{xy}{(x + y)} = 0\)
(2) \(xy = 0\)



23. If \(x\) and \(y\) are integers, \(m = 3^{x-y}*5^{2y-1}*7^{4-x}\) and \(n = 105^y\), what is the value of \(y\)?

(1) \(n\) is NOT a multiple of \(m\).
(2) \(m\) is a multiple of \(n\).



24. Is \(p + q > pq\) ?

(1) \(p > 0 > q\)
(2) \(|q| = p\)



25. If m and n are positive two-digit integers, what is the value of the tens digit of m minus the tens digit of n ?

(1) m - n = 43.
(2) The units digit of m minus the units digit of n is not a multiple of 3.




26. If m and n are integers, is [b]mn > 0 ?[/b]

(1) \(\frac{m}{n}\) is an integer.
(2) \(|m| < |n|\)



27. If \(x^2 +y^2 \leq 25\), is \(x^2 < x\) ?

(1) \(y^2 > 9\).
(2) \(x = y + 3\).



28. If \(s - \frac{1}{s} < \frac{1}{t} - t\), then is \(s > t\) ?

(1) \(s > 1\)
(2) \(t > 0\)



29. The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_{n} < a_{n-1}\), for all n > 1. Does the sequence have more than 12 positive numbers ?

(1) \(\frac{a_{14{a_{13=\frac{\sqrt{10{\pi}\).
(2) The product of any two of the first 14 terms is positive.



30. If a, b, and c are three-digit positive integers and if a = b + c, is the hundreds digit of a greater than the sum of the hundreds digits of b and c ?

(1) The tens digit of a is equal to the sum of the tens digits of b and c.
(2) The tens digit of a is equal to the product of the tens digits of b and c.





31. Positive integers a, b and c are all less than 10. If the sum of all the distinguishable three digit numbers that can be formed by juxtaposing these integers is 3108, is a=b?

(1) c = 9
(2) a = 2



32. If \(xy ≠ 0\), is \(x < y\)?

(1) \(x^8 < y^8\)
(2) \(x^{(-9)} < y^{(-9)}\)



33. If \(x ≠ 0\), is \(x < |x|\) ?

(1) \(|x|< 1\)
(2) \(\frac{x}{|x|} < x\)



34. If x is an integer and \(|x - 3| + |x + 4| \leq |x + 8|\), what is the value of x ?

(1) \(x < 1\)
(2) \(|x| > x\)



35. If a and b are single-digit positive numbers and a/b is NOT a recurring decimal, what is the value of a?

(1) \(-\frac{1}{3} > -\frac{a}{b} > -\frac{4}{5}\)
(2) b is equal to the sum of its positive divisors excluding b itself




36. If \(a + b + c + d = 12\), what is the value of \(\sqrt{a^2+b^2+c^2+d^2}\) ?

(1) \(ab = cd = ad\)
(2) \(|a| = |b| = |c| = |d|\)



37. If \(xy \neq 0\), is \(x + y < 0\)?

(1) \(\frac{x}{\sqrt{x^2-\sqrt{-y*|y|}=y-1\)
(2) \((x+3)^2+(y+4)^2<15\)



38. Positive integers a, b and c are all less than 10. If the sum of all the distinguishable three digit numbers that can be formed by juxtaposing these integers is 1332, is a = b?

(1) c = 3
(2) b = 2



39. If \(|a| > |b| > |c|\), is \(a*b^3*c^3 > a*b^4*c^2\) ?

(1) \(a > b > c\)
(2) \(a + b > 0\)



40. If n is an integer such that \(\frac{1}{9} < \frac{1}{(n^2-1)} < \frac{1}{2}\), what is the value of n?

(1) \(\frac{1}{3} > \frac{1}{(1 - n)} > 1/7\)

(2) n is not an even integer.