Question Number 150864 by puissant last updated on 16/Aug/21 Answered by ajfour last updated on 16/Aug/21 $$\mathrm{70}°=\theta,\:\mathrm{30}°=\alpha,\:\mathrm{40}°=\beta,\:\mathrm{6}{cm}={r} \\ $$$${A}_{\mathrm{1}} =\left({r}−\mathrm{2}{r}\mathrm{cos}\:\theta\right){r}\mathrm{sin}\:\beta \\ $$$${A}_{\mathrm{2}} =\frac{{r}\left(\mathrm{1}−\mathrm{2cos}\:\theta\right)\left\{{r}\mathrm{sin}\:\theta−{r}\mathrm{sin}\:\beta\right\}}{\mathrm{2}} \\ $$$${A}_{\mathrm{3}}…
Question Number 19794 by Tinkutara last updated on 15/Aug/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC}\:\mathrm{with}\:\angle{BCA}\:=\:\mathrm{90}°, \\ $$$$\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of}\:{AB} \\ $$$$\mathrm{intersects}\:\mathrm{segments}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{at}\:{X} \\ $$$$\mathrm{and}\:{Y},\:\mathrm{respectively}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{quadrilateral}\:{BXYC}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{13}\::\:\mathrm{18}\:\mathrm{and} \\ $$$${BC}\:=\:\mathrm{12}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:{AC}? \\ $$ Terms…
Question Number 19792 by Tinkutara last updated on 15/Aug/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{with}\:\angle{DAB}\:=\:\angle{BDC}\:=\:\mathrm{90}°.\:\mathrm{Let}\:\mathrm{the} \\ $$$$\mathrm{incircles}\:\mathrm{of}\:\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BCD} \\ $$$$\mathrm{touch}\:{BD}\:\mathrm{at}\:{P}\:\mathrm{and}\:{Q},\:\mathrm{respectively}, \\ $$$$\mathrm{with}\:{P}\:\mathrm{lying}\:\mathrm{in}\:\mathrm{between}\:{B}\:\mathrm{and}\:{Q}.\:\mathrm{If} \\ $$$${AD}\:=\:\mathrm{999}\:\mathrm{and}\:{PQ}\:=\:\mathrm{200}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{incircles}\:\mathrm{of} \\ $$$$\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BDC}? \\…
Question Number 150861 by naka3546 last updated on 16/Aug/21 $${Find}\:\:{the}\:\:{solution}\:\:{of}\:\:: \\ $$$$\left\{_{{x}^{\mathrm{2}} +\mathrm{3}{xy}+\mathrm{2}{y}^{\mathrm{2}} −\mathrm{4}\:=\:\mathrm{0}} ^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{3}{y}^{\mathrm{2}} +\mathrm{7}\:=\:\mathrm{0}} \right. \\ $$$${Please}\:\:{show}\:\:{your}\:\:{working}… \\ $$ Answered by EDWIN88…
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Question Number 85322 by mathocean1 last updated on 20/Mar/20 $${please}\:{help}\:{me}\:{to}\:{solve}\:{this}\:{in}\:\mathbb{N} \\ $$$${A}_{{n}} ^{\mathrm{2}} −\mathrm{3}{C}_{{n}} ^{\:{n}−\mathrm{2}} +{n}=−\mathrm{20} \\ $$ Commented by mr W last updated on…
Question Number 85323 by sahnaz last updated on 20/Mar/20 $$\int\frac{\mathrm{z}−\mathrm{3}}{\mathrm{5z}−\mathrm{10}}\mathrm{dz} \\ $$ Answered by MJS last updated on 20/Mar/20 $$\int\frac{{z}−\mathrm{3}}{\mathrm{5}{z}−\mathrm{10}}{dz}=\frac{\mathrm{1}}{\mathrm{5}}\int\frac{{z}−\mathrm{3}}{{z}−\mathrm{2}}{dz}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{5}}\int{dz}−\frac{\mathrm{1}}{\mathrm{5}}\int\frac{{dz}}{{z}−\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{5}}{z}−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{ln}\:\mid{z}−\mathrm{2}\mid\:+{C} \\ $$ Terms…
Question Number 19786 by Tinkutara last updated on 15/Aug/17 $$\mathrm{For}\:\mathrm{natural}\:\mathrm{numbers}\:{x}\:\mathrm{and}\:{y},\:\mathrm{let}\:\left({x},\:{y}\right) \\ $$$$\mathrm{denote}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{common}\:\mathrm{divisor}\:\mathrm{of} \\ $$$${x}\:\mathrm{and}\:{y}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:{x}\:\mathrm{and}\:{y}\:\mathrm{with}\:{x}\:\leqslant\:{y}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{xy}\:=\:{x}\:+\:{y}\:+\:\left({x},\:{y}\right)? \\ $$ Answered by mrW1 last updated…
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Question Number 19785 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{x}^{\left({x}^{\mathrm{4}} \right)} \:=\:\mathrm{4},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}^{\left({x}^{\mathrm{2}} \right)} \:+\:{x}^{\left({x}^{\mathrm{8}} \right)} ? \\ $$ Answered by Tinkutara last updated…