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Chapter 10: Problem 832
If two variables \(x\) and \(y\) and \(x>0 . x y=1\) then minimum value of \(\mathrm{x}+\mathrm{y}\) is (a) 1 (b) 2 (c) \(2(1 / 2)\) (d) \(3(1 / 3)\)
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If \(\mathrm{y}=\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+\mathrm{x}+1\right)\right] \tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+3 \mathrm{x}+3\right)\right]\) \(+\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+5 \mathrm{x}+7\right)\right] \ldots \ldots\) to \(\mathrm{n}\) terms then \((\mathrm{dy} / \mathrm{d} \mathrm{x})\) \(=\) (a) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (b) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]-\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (c) \(\left[2 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (d) \(\sum \mathrm{n}\)
If \(\mathrm{x}=\mathrm{a}\left(1-\cos ^{3} \theta\right), \mathrm{y}=\mathrm{a} \sin ^{3} \theta\) and \(\left|\left(\mathrm{d}^{2} \mathrm{y}\right) /\left(\mathrm{dx}^{2}\right)\right|(\pi / 6)\) \(=(\mathrm{A} / \mathrm{a})\) then \(\mathrm{A}=\) (a) \((27 / 32)\) (b) \((32 / 27)\) (c) - (32 / 27) (d) \((27 / 32)\)
If \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})=\mathrm{f}^{\prime}(\mathrm{x})\) and \(\mathrm{F}(\mathrm{x})=|[\mathrm{f}(\mathrm{x} / 2)]|^{2}+|[\mathrm{g}(\mathrm{x} / 2)]|^{2}\) and \(F(5)=5\) then \(F(10)=\) (a) 5 (b) 10 (c) (d) 15
Equation of the tangent for the curve \(\mathrm{y}=\mathrm{a} \log \sec (\mathrm{x} / \mathrm{a})\) at \(\mathrm{x}=\mathrm{a}\) is (a) \((\mathrm{y}-\mathrm{a} \log \sec 1) \tan 1=\mathrm{x}-\mathrm{a}\) (b) \((\mathrm{x}-\mathrm{a}) \tan 1=(\mathrm{y}-\mathrm{a} \log \sec 1)\) (c) \((\mathrm{x}-\mathrm{a}) \cos 1=((\mathrm{y}-\mathrm{a}) \log \sec 1) \tan 1\) (d) None of these
If the curves \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}=6\) and \(\mathrm{ax}^{2}+4 \mathrm{y}^{2}=4\) intersect orthogonally than a = (a) 2 (b) 1 (c) 3 (d) \(-3\)
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