Question

Dau coroana
VA ROG VARIANTA 1​

Answer 1

$\displaystyle\it\\1.~a)~18-2\cdot8=18-16=\boxed{\it 2}.\\----------------\\1.~b)~a=3,1(23),~iar~b=3,1(2),~scriem~numerele~desfasurat.\\a=3,12323...\\b=3,12222...\\de~aici~rezulta~evident~ca~\boxed{\it a>b}~,~pentru~ca~3,123>3,122,~se~compara\\ultima~zecimala~cu~ultima~zecimala.\\----------------\\2.~a)~\boxed{\itmedia~geometrica~a~n~numere~este~:mg=\sqrt[n]{\Pi_{i=1}^n x_i }}.$

$\displaystyle\it\\mg=\sqrt{\sqrt{288}\cdot\sqrt{162}}=\sqrt{6\sqrt{8}\cdot3\sqrt{6}}=\\\sqrt{12\sqrt{2}\cdot 9\sqrt{2}}=\sqrt{216}=\boxed{\it6\sqrt{6}}~.\\----------------\\2.~b)~\mathbb{Z}-\mathbb{N}=\left\{...,-2,-1,0,1,2,...\right\}-\left\{0,1,2,...\right\}=\\\left\{...,-2,-1\right\},~cel~mai~mare~numar~este~\boxed{\it -1}~.\\----------------\\ 3.~a)~\measuredangle A+\measuredangle B=180^{\circ} \implies \\\boxed{\it\measuredangle{B}=110^{\circ}}~.\\----------------$

$\displaystyle\it\\3.~b)~AM=MD=BM=MC=5 \implies diagonalele~se~injumatatesc \implies\\patrulaterul~ABCD~poate~fi~\boxed{\it cel~putin~paralelogram}.\\----------------\\4.~a)~(x-1)^2=4,~extragem~radacina~patrata\implies |x-1|=2 \implies\\x-1=2 \implies x=3~sau~x-1=-2 \implies x=-1.\\suma~elementelor~multimii~este~-1+3=\boxed{\it 2}.\\----------------\\4.~b)~\sqrt{48}:\sqrt{3}+\frac{5}{\sqrt{5}}\cdot\sqrt{20}=\sqrt{16}+5\cdot2=4+10=\boxed{\it 14}.\\----------------$

$\displaystyle\it\\5.~din~paralelism~rezulta~ca~MC~este~perpendiculara~pe~AB.\\asadar,~de~aici~rezulta~ca~patrulaterul~AMCD~este~patrat\\sau~dreptunghi~\implies m(\measuredangle DCM)=90^{\circ}\implies\\m(\measuredangle DCM)+m(\measuredangle BCM)=m(\measuredangle C) \implies m(\measuredangle MCB)=45^{\circ} \implies\\triunghiul~BCM~este~dreptunghic~isoscel,~CM=MB=10.\\MB=10 \implies AM=10,~din~teorema~lui~pitagora~aplicata\\in~triunghiurile~AMC~si~ADC\implies DC=MC=AM=AD=10.$

$\displaystyle\it\\m(\measuredangle ACM)=45^{\circ} \implies m(\measuredangle ACB)=m(\measuredangle ACM )+m(\measuredangle MCB)=90^{\circ} \implies\\\boxed{\it AC~perpendicular~pe~BC}.\\AC~si~MD~sunt~diagonale~in~patrat \implies \boxed{\it AC~perpendicular~pe~MD}.$