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Re: If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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30 Mar 2015, 08:54
It seems that the only exact method to find u and v is solving for quadratic (4th power) equation, but it's also too time consuming. If someone knows quick method of finding u and v without assuming that tose numbers are integers please share it.[/quote]
If \(x = u^2  v^2\), \(y = 2uv\) and \(z = u^2 + v^2\), and if \(x = 11\), what is the value of z?
Given: \(u^2  v^2=11\) and \(y = 2uv\). Question: \(u^2 + v^2=?\)
(1) y = 60 > \(2uv=60\) > \(4u^2v^2=3,600\).
Square \(u^2  v^2=11\): \(u^42u^2v^2+v^2=121\);
Add \(4u^2v^2\) to both sides: \(u^4+2u^2v^2+v^4=121+3,600\);
Apply \(a^2+2ab+b^2=(a+b)^2\): \((u^2+v^2)^2=121+3,600\).
\(u^2+v^2=\sqrt{121+3,600}=61\).
Sufficient.
(2) u = 6 > \(36  v^2=11\) > \(v^2=25\) > \(u^2+v^2=36+25=61\). Sufficient.
Answer: D.
Hope it helps.[/quote]
Hi Bunuel
Stmtn 1 Can we use Differences of Square approach
Property  The difference between squares grows as the squares themselves get larger
Hence given u^2  v^2 = x = 11
Thus the only numbers that fit this equation are when u = 6 and v = 5. Hence we can find Z  Sufficient
Similalry Stmtn 2 = Suff
Hence D
Pls mention if this approach is correct
Thanks



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Re: If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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10 Feb 2016, 23:49
vogelleblanc wrote: If \(x = u^2  v^2\), \(y = 2uv\) and \(z = u^2 + v^2\), and if \(x = 11\), what is the value of z? (1) y = 60 (2) u = 6 my approach towards statement 2: x=11 and u=6, thus 11=6^2v^2 25=v^2 v=5 or v=5 Since z=u^2+v^2, it does not matter if v is 5 or 5, so z=55 Please correct me if something is wrong!/spoiler] I do not understand statement 1, please help me Responding to a pm: Quote: I have a quick question about how ( u + v)^2 * ( u  v)^2 = 11^2 in the problem below. If (u  v)^2 = (u + v) * (u  v) = 11 , then does ( u + v)^2 * ( u  v)^2 = 11^2 ? Is this correct? If not, can you please help me understand this?
Note that \(x = u^2  v^2\) and x is given as 11. So, u^2  v^2 = 11 \((u + v)*(u  v) = 11\) So \([(u + v)*(u  v)]^2 = 11^2\) \((u+v)^2 * (u  v)^2 = 11^2\) Does this help?
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Re: If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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14 Mar 2016, 12:44
kinjiGC wrote: mau5 wrote: vogelleblanc wrote: If \(x = u^2  v^2\), \(y = 2uv\) and \(z = u^2 + v^2\), and if \(x = 11\), what is the value of z?
(1) y = 60 (2) u = 6
Another approach, for F.S 1, in which you don't have to calculate for anything We know that \(x = u^2  v^2\) = 11,Squaring on both sides, we have \(u^4 + v^4  2*v^2*u^2\) = 121 Thus, adding\(4*v^2*u^2\) on both sides, we have \((u^2 + v^2)^2 = 121+4*v^2*u^2 = z^2\) Hence, this statement is sufficient to calculate the value of z. As Z can be negative as well as +ve, Z value cannot be uniquely determined by using the above method. It is better to solve equations and determine. Z can't be negative as its a sum of squares....



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If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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17 Feb 2017, 18:57
x=u^2−v^2..........(1).......x=(u+v)(uv).........(u+v)=x/(uv) y=2uv.........(2) z=u^2+v^2.......(3) Adding eqn (2)+(3) results....... z+y=u^2+v^2+2uv=>>>>>(u+v)^2=>>>>((x/(uv))^2=>>>>>x^2/u^22uz+v^2 z+y=x^2/zy (z+y)(zy)=x^2 z^2y^2=x^2
(1) y = 60 z^2y^2=x^2 z^260^2=11^2 z^2=60^2+11^2 we get the value of z
(2) u = 6 From eqn (1) we get the value of v and substituting v in eqn (2) we get y using z^2y^2=x^2 we get the value of z.
Ans.D



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Re: If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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16 Aug 2017, 18:52
If u^2  v^2 = 11 > difference of squares > 36  25 = 11 > z = u^2 + v^2 = 36 + 25
Thus z is known before either statements 1 or 2. Correct?



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Re: If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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02 Mar 2018, 08:13
Analyze: z = 2u^2 + 11 (2) u=6 => v^2 = 11 6^2 ==> z solved (1) y=30 => u = 30/v put it in v^2 + 11 = u^2 => (v^2)^2 + 11v^2  900 = 0. Analyze: P = product of 2 roots =  900, S = sum of roots = 11 => Equation has 2 roots : one v^2 is positive and one v^2 is negative. But v^2 is always positive => v^2 is only one value ==> z solved ==> D



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If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11, [#permalink]
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02 Mar 2018, 20:22
VeritasPrepKarishma wrote: jlgdr wrote: Could someone please elaborate a little bit more on this problem? How do you get to the values of (u,v) ? Is it just by ballparking? Or is there an elegant algebraic approach for this one? Thanks Cheers! J Given in the stem: \(11 = u^2  v^2 = (u + v)(u  v)\) \(y = 2uv\) \(z = u^2 + v^2\) \(z + y = u^2 + v^2 + 2uv = (u + v)^2\) \(z  y = u^2 + v^2  2uv = (u  v)^2\) \((z + y)(z  y) = (u + v)^2*(u  v)^2 = 11^2 = 121\) 121 can be written as product of two numbers in 2 ways: 1, 121 and 11, 11 So (z + y)(z  y) = (61 + 60)*(61  60) or = (11 + 0)(11  0) So z can be 61 or 11 depending on whether y is 60 or 0. Statement 1 gives y = 60. So z must be 61. Sufficient Statement 2 gives u is 6 which means 2uv (= y) is not 0. So z must be 61. Sufficient Answer (D) I solved it as follows. Please let me know if my method is incorrect. Statement 1. y=60 uv = 30 v=30/u substituting v=30/u in z = u2+v2 ..we get the value of u and subsequently use that value in uv = 30 to get the value of v.Hence sufficient Statement 2 u given we know uv=30 hence sufficient. Hence D.




If x = u^2  v^2, y = 2uv and z = u^2 + v^2, and if x = 11,
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02 Mar 2018, 20:22



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