Zadanie 1. [latex]a_n =a_1+(n-1) cdot r \ egin{cases} a_2 = 8\a_5=20 end{cases} \ egin{cases} a_1+(2-1) cdot r = 8\a_a+(5-1) cdot r=20 end{cases} \ egin{cases} a_1+ r = 8\a_1+4r=20 end{cases} \ egin{cases} a_1= 8-r\8-r+4r=20 end{cases} \ egin{cases} a_1= 8-r\3r=20-8 end{cases} \ egin{cases} a_1= 8-r\3r=12 /:3 end{cases} \ egin{cases} a_1= 8-r\r=4 end{cases} \ egin{cases} a_1= 8-4\r=4 end{cases} \ egin{cases} a_1=4\r=4 end{cases} \ [/latex] [latex]S_{30} = frac{a_1+a_{30}}{2} cdot 30 \ a_{30} = a_1 +(30-1) cdot r \ a_{30} = 4 +29 cdot 4 \ a_{30} = 4 + 116\ a_{30} = 120\ S_{30} = frac{4+120}{2} cdot 30 \ S_{30} = frac{124}{2} cdot 30 \ S_{30} = 62 cdot 30 \ S_{30} = 1860[/latex] Zadanie 2. [latex]a_n = a_1 cdot q^{n-1} \ egin{cases}a_3 = 3\a_6 = 81end{cases} \ egin{cases}a_1 cdot q^{3-1} = 3\a_1 cdot q^{6-1}= 81end{cases} \ egin{cases}a_1 cdot q^{2} = 3\a_1 cdot q^{5}= 81end{cases} \ egin{cases}a_1 = frac{3}{q^2} \a_1cdot q^{5}= 81end{cases} \ egin{cases}a_1 = frac{3}{q^2} \ frac{3}{q^2} cdot q^{5}= 81end{cases} \ egin{cases}a_1 = frac{3}{q^2} \ 3cdot q^3 = 81end{cases} /:3 \ [/latex] [latex]egin{cases}a_1 = frac{3}{q^2} \ 3cdot q^3 = 81 /:3end{cases} \ egin{cases}a_1 = frac{3}{q^2} \ q^3 = 27 end{cases} \ egin{cases}a_1 = frac{3}{q^2} \ q = 3end{cases} \ egin{cases}a_1 = frac{3}{3^2} \ q = 3end{cases} \ egin{cases}a_1 = frac{3}{9} \ q = 3end{cases} \ egin{cases}a_1 = frac{1}{3} \ q = 3end{cases} \ \ a_4 = a_3 cdot q \ a_4 = 3 cdot 3 \ a_4 = 9 \ a_7 = a_6 cdot q \ a_7 = 81 cdot 3 \ a_7 = 243[/latex] Zadanie 3. S = (2,5) , r=3 [latex](x-2)^2 +(y-3)^2 = 3^2 \ (x-2)^2 +(y-3)^2 =9 \ egin{cases} (x-2)^2 +(y-3)^2 =9 \y=4end{cases} \ egin{cases} (x-2)^2 +(4-3)^2 =9 \y=4end{cases} \ egin{cases} (x-2)^2 +1 =9 \y=4end{cases} \ egin{cases} (x-2)^2 =8 \y=4end{cases} \ egin{cases} x^2 -4x+4-8=0 \y=4end{cases} \ egin{cases} x^2 -4x-4=0 \y=4end{cases} \ delta = (-4)^2 -4 cdot 1 cdot (-4)\ delta = 16+16\ delta =32 \ sqrt{delta} = 4sqrt2\ x_1 = frac{4-4sqrt2}{2} = 2-2sqrt2 \ [/latex] [latex]x_2 = frac{4+4sqrt2}{2} = 2+2sqrt2 \[/latex] MA dwa punkty wspólne: [latex]P_1 = (2-2sqrt2 ;4) \ P_2 =(2+2sqrt2 ;4) \[/latex] Zadanie 4. a) [latex]alpha = 80 \ eta = 0,5 alpha \ eta = 40 \ frac{1}{2}alpha + frac{1}{3}eta =frac{1}{2} cdot 80+frac{1}{3} cdot 40 = 40+frac{40}{3} = 40+ 13frac{1}{3} = 53frac{1}{3} [/latex] b) [latex]eta =33 \ alpha = 2eta \ alpha=66\ [/latex] [latex]frac{1}{2}alpha + frac{1}{3}eta = frac{1}{2}cdot 66 + frac{1}{3} cdot 33 = 33+11=44[/latex] Zadanie 5. a - bok trójkata [latex]P = frac{a^2 sqrt3}{4}\ P = frac{12^2 sqrt3}{4}\ P = frac{144 sqrt3}{4}\ P = 36sqrt3 \ [/latex] R - promień okręgu opisanego r - promień okręgu wpisanego h- wysokośc trójkata [latex]r = frac{1}{3}h \ h = frac{asqrt3}{2} \ h = frac{12sqrt3}{2} \ h = 6 sqrt3 \ r = frac{1}{3} cdot 6sqrt3 \ r=2 sqrt3\ R = frac{2}{3}h\ R= frac{2}{3} cdot 6sqrt3 \ R = 4 sqrt3\ Pk = pi R^2\ Pk = (4sqrt3)^2 pi\ Pk = 48pi[/latex]
