Number of solutions of Determinant

Algebra Level 4

∣ arcsin ⁡ x arcsin ⁡ 2 x arcsin ⁡ 3 x arcsin ⁡ 3 x arcsin ⁡ x arcsin ⁡ 2 x arcsin ⁡ 2 x arcsin ⁡ 3 x arcsin ⁡ x ∣ = 0 \left| \begin{matrix} \arcsin { x } & \arcsin { 2x } & \arcsin { 3x } \\ \arcsin { 3x } & \arcsin { x } & \arcsin { 2x } \\ \arcsin { 2x } & \arcsin { 3x } & \arcsin { x } \end{matrix} \right| =0 If the number of solutions of the above determinant is A A , find 2016 A \dfrac{2016}A .


The answer is 2016.

Rishabh Deep Singh by Rishabh Deep Singh , 23, India

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1 solution

Joe Mansley
Mar 15, 2021

After some algebra we get ( arcsin ⁡ x + arcsin ⁡ 2 x + arcsin ⁡ 3 x ) ( ( arcsin ⁡ x − arcsin ⁡ 2 x ) ) 2 + ( arcsin ⁡ 2 x − arcsin ⁡ 3 x ) 2 + ( arcsin ⁡ 3 x − arcsin ⁡ x ) 2 ) = 0 (\arcsin x + \arcsin 2x + \arcsin 3x ) ((\arcsin x - \arcsin 2x))^{2}+ (\arcsin 2x - \arcsin 3x )^{2} + (\arcsin 3x - \arcsin x)^{2}) = 0

The only solution to this is x = 0 x=0

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