Question Number 133764 by liberty last updated on 24/Feb/21 Answered by EDWIN88 last updated on 24/Feb/21 $$\:\mathrm{let}\:\mathrm{x}\:\mathrm{be}\:\mathrm{a}\:\mathrm{page}\:\mathrm{number}\:\mathrm{was}\:\mathrm{counted}\:\mathrm{twice} \\ $$$$\mathrm{Assuming}\:\mathrm{the}\:\mathrm{pages}\:\mathrm{start}\:\mathrm{counting}\:\mathrm{at}\:\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{count}\:\mathrm{continously}\:\mathrm{up}\:\mathrm{we}\:\mathrm{use}\:\mathrm{formula} \\ $$$$\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{n}\right)+\mathrm{x}\:=\:\mathrm{1999}\: \\ $$$$\Rightarrow\mathrm{x}\:+\frac{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)}{\mathrm{2}}\:=\:\mathrm{1999}\:;\:\mathrm{since}\:\mathrm{x}\:\mathrm{is}\:\mathrm{positive}\:…
Question Number 68212 by peter frank last updated on 07/Sep/19 Answered by $@ty@m123 last updated on 07/Sep/19 $${Let}\:{required}\:{equation}\:{of}\:{line}: \\ $$$${y}={m}_{\mathrm{1}} {x}+{c}\:\:\:…..\left(\mathrm{1}\right) \\ $$$${Given}\:{line}:\:\mathrm{4}{x}+\mathrm{3}{y}=\mathrm{21} \\ $$$${Its}\:{slope}:\:{m}_{\mathrm{2}}…
Question Number 68210 by peter frank last updated on 07/Sep/19 Answered by $@ty@m123 last updated on 08/Sep/19 $${Let}\:\frac{\mathrm{sin}\:{A}}{{a}}=\frac{\mathrm{sin}\:{B}}{{b}}=\frac{\mathrm{sin}\:{C}}{{c}}={R} \\ $$$$\Rightarrow\mathrm{sin}\:{A}={aR},\:\mathrm{sin}\:{B}={bR},\:\mathrm{sin}\:{C}={cR}\:…\left(\mathrm{1}\right) \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{sin}\:\left({A}−{B}\right)\mathrm{sin}\:{C}}{\mathrm{1}+\mathrm{cos}\:\left({A}−{B}\right)\mathrm{cos}\:{C}} \\ $$$$=\:\frac{\mathrm{sin}\:\left({A}−{B}\right)\mathrm{sin}\:\left({A}+{B}\right)}{\mathrm{1}−\mathrm{cos}\:\left({A}−{B}\right)\mathrm{cos}\:\left({A}+{B}\right)} \\…
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Question Number 68203 by peter frank last updated on 07/Sep/19 Commented by Cmr 237 last updated on 07/Sep/19 $$\eth\mathrm{f}=\frac{\eth\mathrm{f}}{\eth\mathrm{x}}+\frac{\eth\mathrm{f}}{\eth\mathrm{y}}+\frac{\eth\mathrm{f}}{\eth\mathrm{z}} \\ $$$$\:\:\:\:=\mathrm{2xy}+\mathrm{cosh}\left(\mathrm{yz}\right)+\mathrm{x}^{\mathrm{2}} +\mathrm{xzsinh}\left(\mathrm{yz}\right)+\mathrm{xysinh}\left(\mathrm{yz}\right) \\ $$ Commented…
Question Number 68191 by peter frank last updated on 06/Sep/19 Answered by mind is power last updated on 06/Sep/19 $${sin}\left({a}−{b}\right)={sin}\left({a}\right){cos}\left({b}\right)−{cos}\left({a}\right){sin}\left({b}\right) \\ $$$$\frac{{sin}\left({a}−{b}\right)}{{cos}\left({a}\right){cos}\left({b}\right)}={tg}\left({a}\right)−{tg}\left({b}\right) \\ $$$$\Rightarrow\frac{{sin}\left({a}−{b}\right)}{{cos}\left({a}\right){cos}\left({b}\right)}={tg}\left({b}\right)\left({k}−\mathrm{1}\right) \\…
Question Number 68189 by peter frank last updated on 06/Sep/19 Commented by peter frank last updated on 06/Sep/19 $${Qn}\:\mathrm{9},\:\:\mathrm{10} \\ $$ Commented by mind is…
Question Number 2589 by Filup last updated on 23/Nov/15 $${f}_{{n}} =\frac{\mathrm{1}}{{n}}\left({n}+{f}_{{n}−\mathrm{1}} \right) \\ $$$${f}_{\mathrm{1}} =\mathrm{1} \\ $$$$ \\ $$$$\mathrm{Evaluate}: \\ $$$${S}=\underset{{i}=\mathrm{1}} {\overset{{m}} {\sum}}{f}_{{i}} \\ $$…
Question Number 2400 by Filup last updated on 19/Nov/15 $$\mathrm{According}\:\mathrm{to}\:\mathrm{Wikipedia}\:\mathrm{and}\:\mathrm{WolframAlpha}, \\ $$$$\mathrm{the}\:\mathrm{sign}\:\mathrm{function},\:\mathrm{sgn}\left({x}\right),\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}: \\ $$$$ \\ $$$$\mathrm{sgn}\left({x}\right)\equiv\frac{{x}}{\mid{x}\mid}=\frac{\mid{x}\mid}{{x}}\:\:\:\mathrm{for}\:{x}\neq\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{satisfies}: \\ $$$$\mathrm{sgn}\left({x}\right)=\sqrt{{x}}\sqrt{\frac{\mathrm{1}}{{x}}} \\ $$$$\boldsymbol{{but}} \\ $$$$\mathrm{sgn}\left(\mathrm{0}\right)=\mathrm{0} \\…
Question Number 67871 by rrebo5637@gmail.com last updated on 01/Sep/19 $$\mathrm{8}=\mathrm{4x} \\ $$$$\mathrm{x}=? \\ $$ Commented by Prithwish sen last updated on 01/Sep/19 $$\mathrm{2} \\ $$…