Which of the following is/are correct regarding solution to the equation $(x+5)(x+6)(x+7)=504$?
(A) One real solution
(B) Two complex solutions
(C) Three real solutions
(D) No real solution
Continue reading $(x+5)(x+6)(x+7)=504$
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Sleepers Below the Rails
Which of the following reason(s) support(s) the usage of sleepers below the rails?
(A) To reduce the pressure
(B) To increase the normal reaction
(C) To increase the surface area
(D) To accommodate the thermal expansion/contraction of the rails Continue reading Sleepers Below the Rails
$\int\limits_{-1}^0 {\ln \left| x \right|dx} $
$\int {\frac{{{{\sec }^2}x}}{{{{(\sec x + \tan x)}^5}}}dx} $
$(a^2)^x+(b^2)^x+(c^2)^x=(ab)^x+(bc)^x+(ca)^x$
The number of real solutions for x for the equation $(a^2)^x+(b^2)^x+(c^2)^x=(ab)^x+(bc)^x+(ca)^x$ if a, b, c are unequal real numbers:
(1) 3 solutions
(2) 1 solution
(3) No solution
(4) Infinite solutions
Solution
We have, $a^{2x}+b^{2x}+c^{2x}=a^x b^x + b^x c^x + c^x a^x $
Or, $(a^x)^2+(b^x)^2+(c^x)^2=a^x b^x + b^x c^x + c^x a^x $
Let, $a^x = X, b^x = Y, c^x = Z$
So, $X^2+Y^2+Z^2=XY+YZ+ZX$
$\Rightarrow (X-Y)^2 + (Y-Z)^2 + (Z-X)^2 =0$
$\therefore X=Y=Z$ Or $a^x = b^x = c^x $
The above is only possible when x = 0 as $a \neq b \neq c $
Hence, option (2)
