N2000 P1 Q11
N2000 P1 Q11
6 marks
Past Year National Exams
Free
It is given that \(x\) and \(y\) satisfy the equation \({{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}\left( xy \right)=\frac{7}{12}\pi \).
- Find the value of \(y\) when \(x=1\).[2]
- Express \(\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\tan }^{-1}}\left( xy \right) \right)\) in terms of \(x\), \(y\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\).[2]
- Show that, when \(x=1\), \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{3}-\frac{1}{2\sqrt{3}}\).[2]
Video Solution:
Default solution
Answer:(i) \(\frac{1}{\sqrt{3}}\) (ii) \(\frac{1}{1+{{\left( xy \right)}^{2}}}\left( x\frac{\mathrm{d}y}{\mathrm{d}x}+y \right)\) \text{ or } \(-\frac{1}{1+{{x}^{2}}}-\frac{1}{1+{{y}^{2}}}\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)\)
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