Is a^2 > 3a – b^4?

(1) 3a – b^4 = -5
(2) a > 5 and b > 0

Kudos for a correct solution.
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Question:

Is \(a^2 > 3a - b^4?\)

Solution:

Is \(a^2 > 3a - b^4\)?

→ question becomes: Is \(a^2 + b^4 -3a > 0\)?


1) \(3a – b^4 = -5 → b^4 – 3a = 5.\) Substituting in inequality above, we obtain:

Is \(a^2 + 5 > 0?\) Since \(a^2\) is always positive and 5 is a positive number, their sum will always be greater than zero. Hence, statement I is sufficient.

2) a>5 and b>0.

Let a = 5. →\(a^2 = 25.\)
Let \(b=1 → b^4 = 1 → 3a = 15\)

→ Is 25 > 15-1 ? → Is 25 > 14? Answer is yes. This is an extreme case as a is always greater than 5. Integer a could be 5.001 for example.

With any value of a and b chosen that satisfy above condition in statement 2, inequality always hold. Hence, statement 2 is sufficient.

Answer: D.

Statement (1) makes right side of inequality -ve. Left side of the inequality will always be +ve or zero in case a=0.

hence sufficient.

Statement (2) a^2 will have values 36,49,64....etc.(for a=6,7,8........)
3a-b^4 will have values 15-somevalue,21-somevalue,24-somevalue (for a-6,7,8.....) , no impact of value of b as b^4 is always +ve
Hence Sufficient.

Ans: D
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VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No This question asks whether a^2 > 3a – b^4.

Statement 1: 3a – b^4 = -5. This statement may not appear sufficient at first. You cannot manipulate it algebraically to mirror the question stem as might be the first impulse. However, the important thing is that “3a – b^4 equals a negative number.” Remember that a^2 cannot be less than zero, as a squared number cannot be negative. The answer, then, is “yes,” as you can absolutely conclude that a^2 will be greater than -5. The correct answer is either A or D.

Statement 2: a > 5 and b > 0. Again, this statement may not appear to be sufficient. It does not give specific values for a or b. However, if you “Just Do It” and plug in the numbers then you will see that it is sufficient. A good strategy for these “greater than” statements is to use the actual numbers given. Although the statement says a > 5 and b > 0 you can just use 5 and 0 in the inequality. The inequality becomes “25 > 15 – 0?” The answer is clearly “yes”: 25 > 15. And as you increase both “a” and “b” the result becomes a stronger “yes.” For example, if “a = 6 and b = 2” the inequality is “36 > 18 – 16?” Conceptually it looks like this, so long as a is greater than 3 then a^2 will be greater than 3a. Whatever number “b” is can only take away from the 3a it cannot add to it. In fact, Statement 2 gives more information than strictly necessary; “a > 3” would be sufficient.

The correct answer is D.
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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics