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Question Number 27946 by das47955@mail.com last updated on 17/Jan/18 $$\boldsymbol{\mathrm{Q}}.\boldsymbol{\mathrm{N}}− \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{coefficint}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}^{\mathrm{4}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\left(\mathrm{1}+\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{10}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\left(\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{10}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\: \\ $$ Answered by…
Question Number 93474 by Rio Michael last updated on 13/May/20 $$\mathrm{Q}.\:\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\mathrm{4}{r}\:+\:\mathrm{5}\right)\:=\:\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{7}{n}\: \\ $$ Answered by eswar150933 last updated on 13/May/20…
Question Number 27930 by tawa tawa last updated on 17/Jan/18 Commented by çhëý böý last updated on 20/Jan/18 $$\left(\mathrm{1}\right)=\left(\mathrm{4}.\mathrm{0}×\mathrm{10}^{\mathrm{4}} \right)^{\mathrm{2}} \:=\mathrm{1}.\mathrm{6}×\mathrm{10}^{\mathrm{9}} \\ $$$$\left(\mathrm{2}\right)=\left(\mathrm{4}.\mathrm{0}×\mathrm{10}^{\mathrm{4}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} =\mathrm{14}.\mathrm{14}…
Question Number 27920 by math1967 last updated on 17/Jan/18 $$\int\frac{\left({x}−\mathrm{1}\right){dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}}} \\ $$ Answered by math1967 last updated on 18/Jan/18 $$\int\frac{\left({x}^{\mathrm{2}} −\mathrm{1}\right){dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +{x}^{\mathrm{2}}…
Question Number 158984 by LEKOUMA last updated on 11/Nov/21 $$\mathrm{1}.\:{Prove}\:{by}\:{recurrence}\:{that}\: \\ $$$${so}\:\:{n}\:\in\:{N}\:{and}\:\theta\:\in\: {R}\: \\ $$$$\left(\mathrm{cos}\:\left({n}\theta\right)+{i}\mathrm{sin}\:\left({n}\theta\right)=\mathrm{cos}\:\left({n}\theta\right)+{i}\mathrm{sin}\:\left({n}\theta\right)\right. \\ $$$$\mathrm{2}.\:{Prove}\:{that}\:{U}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{5}}\left({U}_{{n}} ^{\mathrm{2}} +\mathrm{6}\right)\:{and} \\ $$$${U}_{\mathrm{1}} =\frac{\mathrm{5}}{\mathrm{2}},\:{is}\:{decrease} \\ $$$$…
Question Number 27884 by das47955@mail.com last updated on 16/Jan/18 $$\left(\mathrm{3}\right)\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{independ}}− \\ $$$$\boldsymbol{\mathrm{ent}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\left(\:\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}−\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{12}} \\ $$ Answered by Rasheed.Sindhi last updated on 16/Jan/18 $$\:\:\:\left(\:\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{2}}…
Question Number 27885 by das47955@mail.com last updated on 16/Jan/18 $$\left(\mathrm{4}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{indepen}}− \\ $$$$\boldsymbol{\mathrm{dent}}\:\boldsymbol{\mathrm{of}}\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)^{\mathrm{6}} \\ $$ Commented by Rasheed.Sindhi last updated on 16/Jan/18…
Question Number 27882 by das47955@mail.com last updated on 16/Jan/18 $$\left(\mathrm{2}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{indepen}}− \\ $$$$\boldsymbol{\mathrm{dent}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{6}} \\ $$ Answered by Rasheed.Sindhi last updated on 16/Jan/18 $$\mathrm{T}_{\mathrm{r}+\mathrm{1}}…
Question Number 158945 by LEKOUMA last updated on 10/Nov/21 $$\left.\mathrm{1}\right)\:{Prove}\:{by}\:{absurd}\:{that}\: \\ $$$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:\mathrm{3}}\:\:{is}\:{irrational} \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{by}\:{absurd}\:{that} \\ $$$$\sqrt{\mathrm{2}}+\sqrt{\mathrm{6}\:}\leqslant\sqrt{\mathrm{15}} \\ $$ Answered by mr W last updated on…