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Properties of Determinants

a^1/a=b^1/b=c...

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a^{1/a}=b^{1/b}=c^{1/c} and a^{bc}+b^{ca}+c^{ab}=729 then find value of b^{1/b

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A^{\frac{1}{A}} = B^{\frac{1}{B}} = C^{\frac{1}{C}}

A^{B C} + B^{C A} + C^{A B} = 729

, then the value of

B^{\frac{1}{B}}

A^{1 / A} = B^{1 / B} = C^{1 / C}, A^{B C} + B^{A C} + C^{A B} = 729

Which of the following equals

A^{1 / A}

Q. Assertion :If

\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}

A . P

a_{1}, b_{1}, c_{1}

G . P

a x^{2} + b x + c

a_{1} x^{2} + b_{1} x^{2} + c_{1} = 0

have a common root &

\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}

A . P

a_{1}, b_{1}, c_{1}

G . P

Q. In an acute-angled triangle ABC, the altitudes from A,B and C when extended intersect the circumcircle again at points

A_{1}, B_{1}, C_{1}

, respectively. If

∠ A B C = 45^{o}

∠ A_{1}, B_{1}, C_{1}

0 < a, b, c < 1

a + b + c = 2

, then the greatest value of

\frac{a}{1 - a} \times \frac{b}{1 - b} \times \frac{c}{1 - c}

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