GMATPrep Tough Questions

2) If x ≠ 0, then √(x^2)/x =
a. -1
b. 0
c. 1
d. x
e. |x|/x

first off, this problem doesn't even come close to 700 level (contra your stated purpose in the original post).

you should know the following fact from memory:
FACT:
√(x^2) = |x|

if you know this fact, then you're done with the problem, as the given expression reduces immediately to the formulation in answer (e).

if you don't know this fact, simply plug in a number for x.
try x = 3. if x = 3, then the expression yields 3/3, or 1, eliminating everything except (c) and (e).
now, try a negative number (since the problem involves absolute values, you should strongly suspect that the difference between positives and negatives is crucial). if x = -3, then the expression yields 3/-3, or -1. of the two remaining answers, only (e) works.

varunkh70 wrote:3) If n is a positive integer less than 200 and 14n⁄60 is also an integer, then n has how many different positive prime factors
a. 2
b. 3
c. 5
d. 6
e. 8

there's an easier way and a harder way to do this one.

EASIER WAY:
since the answer choices don't depend on n, you can deduce that you'll get the same answer for ANY valid value of n. (if this were not the case - i.e., if the value depended on the particular value of n chosen - then n would have to appear in the answer.)

therefore, if you can find ANY n satisfying the problem statement, you'll have the answer.

since the denominator of the given expression is 60, it's clear that n = 60 is a valid option.
n = 60 has three prime factors - 2, 3, and 5 - so the answer is (b).

HARDER WAY:
14n/60 reduces to 7n/30. therefore, 30 is a factor of 7n; i.e., the prime factors of 30 (2, 3, and 5) are contained within the prime factorization of 7n.
since none of those prime factors is a '7', all three of them must be contained within the prime factorization of n itself (i.e., the '7' is irrelevant).
therefore, n is a multiple of 30.
within the given range, n could be 30, 60, 90, 120, 150, or 180.
all of these have the same three prime factors: namely, 2, 3, and 5. answer (b).

varunkh70 wrote:1) (-1^(k+1)).(½^k). T is the sum of the first 10 k, is t
a. > 2
b. between 1 and 2
c. between ½ and 1
d. between ¼ and ½
e. < ¼

this is an alternating series - i.e., a series of numbers in which the signs are + - + - + - ..., alternating every time. that is important in the discussion that follows.

the first 2 terms are 1/2 - 1/4. i will now prove that the solution is pinned down between those 2 values:

start with 1/2.
the next 2 terms are as follows: 1/2, minus something smaller, plus something even smaller than that. this must lead to something smaller than 1/2.
the same thing happens every following 2 terms, because the terms keep getting smaller and smaller. therefore, everything in the entire series will remain smaller than 1/2.

now start with 1/2 - 1/4 = 1/4 (the sum of the first 2 terms).
the next 2 terms are as follows: plus something smaller, minus something even smaller than that. this must lead to something bigger than 1/4.
the same thing happens every following 2 terms, because the terms keep getting smaller and smaller. therefore, everything in the entire series will remain bigger than 1/4.

taking these 2 observations together, we have that the final sum must be between 1/4 and 1/2.

ans (d)

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of course, it's easier just to do pattern recognition: just find the sums for the first 4-5 examples, notice that ALL of them are between 0.25 and 0.5, and generalize accordingly. the gmat may be tricky about some things, but they are surprisingly foursquare about creating patterns that continue throughout the duration of a problem (i.e., they are unlikely to give you a problem in which the first twenty series are between 0.25 and 0.5 and then, surprise!).