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1) x\sqrt{y}+y\sqrt{x}=20
x\sqrt{x}+y\sqrt{y}=65
2) 3x+4x+5x =6x
3) Find all the roots of the equation : 4x4 – 24x3 + 57x2 + 18x – 45 = 0 if one root is 3 + i√ 6 .
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14 Answers
62First Part...
ab(a+b)=20
(a+b)(a^2+b^2-ab)=65
Divide you will get
a/b+b/a-1=13/4
So a/b+b/a=17/4
Now a quadratic which solves to a/b=4 or 1/4
Now the remaining part is quite simple :)
622nd question
I can give a hint.. Try to prove that there is a unique solution and then also observe one solution is 3....
1708$\textbf{(2):: Here $\mathbf{3^x+4^x+5^x=6^x}$}\\\\ \textbf{Divide both side by $\mathbf{(5.5)^x}$. Here $\mathbf{(5.5)^x>0\forall x\in \mathbb{R}}$ }\\\\ $\mathbf{\left(\frac{3}{5.5}\right)^x+\left(\frac{4}{5.5}\right)^x+\left(\frac{5}{5.5}\right)^x=\left(\frac{6}{5.5}\right)^x}$\\\\ \textbf{So Here L.H.S is a Combination of decreasing function and R.H.S\\\\ is an Increasing function. Using the formula $\mathbf{a^x}$ is an Increasing\\\\ function for $\mathbf{a>1}$ and Decreasing function for $\mathbf{0 < a<1}$}\\\\ $\textbf{So L.H.S and R.H.S Intersect at one point.}$\\\\ \textbf{So Using Inspection We Get $\boxed{\boxed{\mathbf{x=3}}}$}
1number of positive integral solutins of x2-y2=9876543210 :
a) 0
b) 45
c) 91
d) 90
1Number of positive integral solutins of x2-y2=9876543210 :
a) 0
b) 45
c) 91
d) 90
62no solution..
Hint: Prove the difference of 2 perfect squares is a multiple of 4
62When will the product of (x-y)(x+y) be even? if x and y are both odd or both even..
Now try to prove that the product is a multiple of 4
1how can we prove (x+y)(x-y) is multiple of 4?
please tell me , its urgent for me.....
please.............