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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote Is xy+zt+yz+tx positive? tagged by: fskilnik@GMATH This topic has 2 expert replies and 0 member replies GMAT/MBA Expert Timer 00:00 Your Answer A B C D E Global Stats Difficult GMATH practice exercise (Quant Class 14) Is xy+zt+yz+tx positive? (1) |x| = |y| = |z| = t (2) x+y+z+t = 0 Answer: ____(B)__ _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 fskilnik@GMATH wrote: GMATH practice exercise (Quant Class 14) Is xy+zt+yz+tx positive? (1) |x| = |y| = |z| = t (2) x+y+z+t = 0 $$\underline {xy} + \underline{\underline {zt}} + \underline {yz} + \underline{\underline {tx}} \,\,\mathop > \limits^? \,\,0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,y\left( {x + z} \right) + t\left( {z + x} \right) = \left( {x + z} \right)\left( {y + t} \right)\,\,\,\mathop > \limits^? \,\,0\,\,\,\,\,\left( * \right)\,$$ $$\left( 1 \right)\,\,\,\left| x \right| = \left| y \right| = \left| z \right| = t\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {x,y,z,t} \right) = \left( {0,0,0,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {x,y,z,t} \right) = \left( {1,1,1,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$ $$\left( 2 \right)\,\,\,x + y + z + t = 0\,\,\,\, \Rightarrow \,\,\,\,x + z = - \left( {y + t} \right)\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,?\,\,\,:\,\,\,\left( {x + z} \right)\left( {y + t} \right) = - {\left( {y + t} \right)^2} \le 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle$$ The correct answer is (B). We follow the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12985 messages Followed by: 1249 members Upvotes: 5254 GMAT Score: 770 fskilnik@GMATH wrote: [size=17]GMATH practice exercise (Quant Class 14) Is xy+zt+yz+tx positive? (1) |x| = |y| = |z| = t (2) x+y+z+t = 0 Target question: Is xy + zt + yz + tx positive? Statement 1: |x| = |y| = |z| = t Let's TEST some values. There are several values of x, y, z and t that satisfy statement 1. Here are two: Case a: x = y = z = t = 1. In this case, xy + zt + yz + tx = (1)(1) + (1)(1) + (1)(1) + (1)(1) = 4. So, the answer to the target question is YES, xy + zt + yz + tx IS positive Case b: x = y = z = t = 0. In this case, xy + zt + yz + tx = (0)(0) + (0)(0) + (0)(0) + (0)(0) = 0. So, the answer to the target question is NO, xy + zt + yz + tx is NOT positive Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT Statement 2: x + y + z + t = 0 Take: x + y + z + t = 0 Subtract y from both sides to get: x + z + t = -y Subtract t from both sides to get: x + z = -y - t Rewrite as follows: x + z = -(y + t) Now take the expression in the target question: xy + zt + yz + tx Rearrange the terms as follows: xy + yz + tx + zt Factor the terms in PAIRS: y(x + z) + t(x + z) Simplify to get: (y + t)(x + z) Since we already know that x + z = -(y + t), we can replace (x + z) with -(y + t) to get: (y + t)[-(y + t)] Notice that, if (y + t) is POSITIVE, then -(y + t) is NEGATIVE, which means (y + t)[-(y + t)] is NEGATIVE Similarly, if (y + t) is NEGATIVE, then -(y + t) is POSITIVE, which means (y + t)[-(y + t)] is NEGATIVE Finally, if (y + t) = 0, then -(y + t) = 0, which means (y + t)[-(y + t)] is ZERO IMPORTANT: Notice that, in all 3 possible cases above, the answer to the target question is NO, xy + zt + yz + tx is NOT positive Since we can answer the target question with certainty, statement 2 is SUFFICIENT Answer: B Cheers, Brent _________________ Brent Hanneson – Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months! • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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