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Re: HOT Competition 26 Aug/8AM: If p < 0 and the value of 1 - p/q is betwe
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26 Aug 2020, 22:55
Constraints:
1. P < 0
2. 0 < (1-p/q) < 1
Let’s further simplify the 2nd constraint.
0 < 1-p/q < 1
-1 < -p/q < 0 (subtracting 1)
0 < p/q < 1 (multiplying by -1, signs flipped) -------- (1)
Now, from equation (1) we can infer that
A. Since p/q > 0 and p<0, therefore q = negative i.e. q < 0
B. Since p/q < 1, therefore |q| > |p|
Note – We are told nothing about the nature of p and q (integers, fractions etc.), so we will treat p and q as real numbers
To check “must be true” condition for option, the best strategy is to think of a scenario in which you can falsify the condition and hence reject that option. With this strategy, let’s jump into analysing each statement:
1. p^2 + q^2 > 1
If p = -1/3 and q = -1/2, then
(-1/2) ^ + (-1/3) ^2 = 13/36 < 1 ---- NOT TRUE (REJECT)
3. q – p < 0
Since p and q < 0 and |q| > |p|, q-p will always be negative. Let’s take two cases to prove it.
Case1: Let p = -2, q = -3, then
(-3) – (-2) = -3+2 = -1 < 0 ---- TRUE
Case2: Let p = -1/3, q = -1/2, then
(-1/2) – (-1/3) = -1/2+1/3 = -1/6 < 0 ---- TRUE
Therefore, statements 2 & 3 = MUST BE TRUE
CORRECT option is E
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