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Questions (211)

 10 Prove that $\begin{vmatrix} xa&yb&zc\\ yc&za&xb\\ zb&xc&ya\\ \end{vmatrix}=xyz\begin{vmatrix} a&b&c\\ c&a&b\\ b&c&a\\ \end{vmatrix}$ if $x+y+z=0$ 7 Integrate $\sin^{-1}\frac{2x}{1+x^2}$ 6 Evaluate $\int_0^1x(\tan^{-1}x)^2~\textrm{d}x$ 6 If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}$ 5 Real roots of the equation $\log_{(5x+4)}(2x+3)^3-\log_{(2x+3)}(10x^2+23x+12)=1$

Reputation (2,154)

 +10 The Median Minimizes the Sum of Absolute Deviations (The ${L}_{1}$ Norm) +5 $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\frac{x+y+z-xyz}{1-xy-yz-zx}$ true for all $x$? +5 Find the ratio $p:q:r$, if $p,q,r$ are in H.P, and their squares are in A.P. +5 Proof of the Associative property of the Least common multiple

 4 The Median Minimizes the Sum of Absolute Deviations (The ${L}_{1}$ Norm) 3 Computing the integral of $\log(\sin x)$ 3 Proofs of AM-GM inequality 2 Proof for the formula of sum of arcsine functions $\arcsin x + \arcsin y$ 2 Derive the conditions $xy<1$ for $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ and $xy>-1$ for $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy}$

Tags (99)

 10 trigonometry × 70 4 optimization × 3 6 calculus × 44 4 convex-optimization 6 integration × 23 4 median 4 functions × 9 3 inequality × 17 4 absolute-value × 8 3 logarithms × 10

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