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17 Mar 2020, 08:22
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Difficulty:
55%
(hard)
Question Stats:
65% (02:30) correct
35%
(02:30)
wrong
based on 296
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If x > 0 and y > 0, what is the value of x − 2y?
(1) \(16y^2 = \frac{x^4 - 240}{y^2}\)
(2) \(x^2 + 4y^2 = \frac{120}{x + 2y}\)
(1) \(16y^2 = \frac{x^4 - 240}{y^2}\)
(2) \(x^2 + 4y^2 = \frac{120}{x + 2y}\)
Bunuel
Asked: If x > 0 and y > 0, what is the value of x − 2y?
(1) \(16y^2 = \frac{x^4 - 240}{y^2}\)
16y^4 = x^4 - 240
x^4 - 16y^4 = 240 = 256 - 16 = 4^4 - -4*1^4
x = +- 4; y = +-1
x - 2y can not be determined
NOT SUFFICIENT
(2) \(x^2 + 4y^2 = \frac{120}{x + 2y}\)
(x^2 + 4y^2)(x+2y) = 120
(x^2 + 4y^2)(x+2y)(x-2y) = 120(x-2y)
x^4 - 16y^4 = 120(x-2y)
NOT SUFFICIENT
(1) + (2)
(1) \(16y^2 = \frac{x^4 - 240}{y^2}\)
16y^4 = x^4 - 240
x^4 - 16y^4 = 240 = 256 - 16 = 4^4 - -4*1^4
x = +- 4; y = +-1
(2) \(x^2 + 4y^2 = \frac{120}{x + 2y}\)
(x^2 + 4y^2)(x+2y) = 120
(x^2 + 4y^2)(x+2y)(x-2y) = 120(x-2y)
x^4 - 16y^4 = 120(x-2y) = 240
x - 2y = 2
SUFFICIENT
IMO C
17 Mar 2020, 23:03
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Bunuel
Solution
Step 1: Analyse Question Stem
- • \( x > 0 \)
• \( y > 0\)
• We need to find the value of \(x- 2y.\)
- o So, we can find \(x – 2y \) if
- We find the exact values of x and y or
We directly find the value of \(x -2y\) using relation between x and y.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: \(16y^2=\frac{x^4−240}{y^2}\)
- • According to this statement: \(16y^4 = x^4 – 240\)
- o Or, \(x^4 – (2y)^4 = 240\)
\(⟹(x^2 + (2y)^2)(x^2 – (2y)^2) = 240\)
\(⟹ (x^2 + 4y^2)(x+2y)(x – 2y) = 240\)
Note: we know is that x > 0 and y >0, which means they are positive, however, we don't know if they are integers or not, so we shouldn't assume only integral values here to solve the statements.
Statement 2: \(x^2+4y^2=\frac{120}{x+2y}\)
- • According to this statement:\( (x^2 + 4y^2)(x+2y) = 120\)
• From this statement we cannot find the values of x and y or the value of x- 2y.
Step 3: Analyse Statements by combining.
- • From statement 1: \((x^2 + 4y^2) (x+2y) (x – 2y) = 240\)
• From statement 2: \((x^2 + 4y^2) (x+2y) = 120\)
• Combining both the statements, we get, \(120*(x – 2y) = 240 ⟹ x – 2y = 2\)
