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Author: Tinku Tara

x-1-x-2-x-12-

Question Number 151999 by otchereabdullai@gmail.com last updated on 24/Aug/21 $$\int\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{12}} \\ $$ Answered by Olaf_Thorendsen last updated on 25/Aug/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{{x}−\mathrm{1}}{{x}^{\mathrm{2}} +{x}−\mathrm{12}}\:{dx} \\ $$$$\mathrm{F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\left(\frac{\mathrm{2}{x}+\mathrm{1}}{{x}^{\mathrm{2}} +{x}−\mathrm{12}}−\frac{\mathrm{3}}{{x}^{\mathrm{2}}…

Use-exponential-representation-of-sin-and-cos-to-show-that-a-sin-2-cos-2-1-b-cos-2-sin-2-cos2-c-2-sin-cos-2sin2-

Question Number 86461 by Rio Michael last updated on 28/Mar/20 $$\mathrm{Use}\:\mathrm{exponential}\:\mathrm{representation}\:\mathrm{of}\:\mathrm{sin}\:\theta\:\mathrm{and}\:\mathrm{cos}\:\theta\:\mathrm{to}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left.\mathrm{a}\left.\right)\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:\mathrm{cos}^{\mathrm{2}} \:\theta\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{cos}^{\mathrm{2}} \theta\:−\:\mathrm{sin}^{\mathrm{2}} \theta\:=\:\mathrm{cos2}\theta \\ $$$$\left.\mathrm{c}\right)\:\mathrm{2}\:\mathrm{sin}\theta\:\mathrm{cos}\theta\:=\:\mathrm{2sin2}\theta. \\ $$ Answered by TANMAY PANACEA.…

if-x-y-z-are-natural-numbers-such-that-2x-x-y-y-3z-z-then-find-2021x-2022y-2023z-x-y-z-

Question Number 151988 by mathdanisur last updated on 24/Aug/21 $$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\:\:\mathrm{are}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{2x}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{y}^{\boldsymbol{\mathrm{y}}} \:=\:\mathrm{3z}^{\boldsymbol{\mathrm{z}}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\frac{\mathrm{2021}\boldsymbol{\mathrm{x}}\:+\:\mathrm{2022}\boldsymbol{\mathrm{y}}\:+\:\mathrm{2023}\boldsymbol{\mathrm{z}}}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}}\:=\:? \\ $$ Commented by mathdanisur last updated on 25/Aug/21…

A-body-starts-rotating-about-a-stationary-axis-with-an-angular-acceleration-b-2t-rad-s-2-How-soon-after-the-beginning-of-rotation-will-the-total-acceleration-vector-of-an-arbitrary-point-on-the-bo

Question Number 20916 by Tinkutara last updated on 07/Sep/17 $$\mathrm{A}\:\mathrm{body}\:\mathrm{starts}\:\mathrm{rotating}\:\mathrm{about}\:\mathrm{a} \\ $$$$\mathrm{stationary}\:\mathrm{axis}\:\mathrm{with}\:\mathrm{an}\:\mathrm{angular} \\ $$$$\mathrm{acceleration}\:{b}\:=\:\mathrm{2}{t}\:\mathrm{rad}/\mathrm{s}^{\mathrm{2}} .\:\mathrm{How}\:\mathrm{soon} \\ $$$$\mathrm{after}\:\mathrm{the}\:\mathrm{beginning}\:\mathrm{of}\:\mathrm{rotation}\:\mathrm{will}\:\mathrm{the} \\ $$$$\mathrm{total}\:\mathrm{acceleration}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{an}\:\mathrm{arbitrary} \\ $$$$\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{body}\:\mathrm{forms}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{60}° \\ $$$$\mathrm{with}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{vector}? \\ $$$$\left(\mathrm{1}\right)\:\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{3}}…

Two-shells-are-fired-from-a-canon-with-speed-u-each-at-angles-of-and-respectively-with-the-horizontal-The-time-interval-between-the-shots-is-t-They-collide-in-mid-air-after-time-T-from-the-firs

Question Number 20915 by Tinkutara last updated on 07/Sep/17 $$\mathrm{Two}\:\mathrm{shells}\:\mathrm{are}\:\mathrm{fired}\:\mathrm{from}\:\mathrm{a}\:\mathrm{canon}\:\mathrm{with}\:\mathrm{speed}\:\mathrm{u}\:\mathrm{each},\:\mathrm{at} \\ $$$$\mathrm{angles}\:\mathrm{of}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{respectively}\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{The} \\ $$$$\mathrm{time}\:\mathrm{interval}\:\mathrm{between}\:\mathrm{the}\:\mathrm{shots}\:\mathrm{is}\:{t}.\:\mathrm{They}\:\mathrm{collide}\:\mathrm{in}\:\mathrm{mid} \\ $$$$\mathrm{air}\:\mathrm{after}\:\mathrm{time}\:{T}\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{shot}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{conditions}\:\mathrm{must}\:\mathrm{be}\:\mathrm{satisfied}? \\ $$$$\left({a}\right)\:\alpha\:>\:\beta \\ $$$$\left({b}\right)\:{T}\:\mathrm{cos}\:\alpha\:=\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\beta \\ $$$$\left({c}\right)\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\alpha\:=\:{T}\:\mathrm{cos}\:\beta \\…

The-quadratic-equation-p-x-0-with-real-coefficients-has-purely-imaginary-roots-Then-the-equation-p-p-x-0-has-1-Only-purely-imaginary-roots-2-All-real-roots-3-Two-real-and-two-purely-imag

Question Number 20914 by Tinkutara last updated on 07/Sep/17 $$\mathrm{The}\:\mathrm{quadratic}\:\mathrm{equation}\:{p}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{with} \\ $$$$\mathrm{real}\:\mathrm{coefficients}\:\mathrm{has}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{equation}\:{p}\left({p}\left({x}\right)\right)\:=\:\mathrm{0} \\ $$$$\mathrm{has} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Only}\:\mathrm{purely}\:\mathrm{imaginary}\:\mathrm{roots} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{All}\:\mathrm{real}\:\mathrm{roots} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Two}\:\mathrm{real}\:\mathrm{and}\:\mathrm{two}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots} \\…