Xét ΔABC có AD là phân giác

nên \(\frac{AB}{BD}=\frac{AC}{CD}\)

=>\(\frac{10}{CD}=\frac86=\frac43\)

=>\(CD=10\cdot\frac34=7,5\)

Xét ΔOAC có OA=OC=AC(=5cm)

nên ΔAOC đều

=>\(\hat{AOC}=60^0\)

Ta có: \(\hat{AOC}+\hat{BOC}=180^0\) (hai góc kề bù)

=>\(\hat{BOC}=180^0-60^0=120^0\)

Độ dài cung AC nhỏ là:

\(l=\frac{\pi\cdot R\cdot n}{180}=\frac{\pi\cdot60\cdot5}{180}=\frac{5\pi}{3}\)

Độ dài cung BC nhỏ là:

\(l=\frac{\pi\cdot R\cdot n}{180}=\frac{\pi\cdot120\cdot5}{180}=\frac{600\pi}{180}=\frac{10\pi}{3}\)

Câu 9:

a: \(u_1-u_3+u_5=15\)

=>\(u_1-u_1-2d+u_1+4d=15\)

=>\(u_1+2d=15\)

=>\(2u_1+4d=30\)

\(u_1+u_6=27\)

=>\(u_1+u_1+5d=27\)

=>\(2u_1+5d=27\)

=>\(2u_1+5d-2u_1-4d=27-30\)

=>d=-3

\(u_1=15-2d=15-2\cdot\left(-3\right)=21\)

=>Đúng

b: Sai

c: \(u_{11}=u_1+10d=21+10\cdot\left(-3\right)=21-30=-9\)

=>Đúng

d: \(u_{2024}=u_1+2023d=21+2023\cdot\left(-3\right)=-6048\)

=>Đúng

Câu 2:

a: Sai

b: \(u_8=u_1+7d=\frac32+\frac72=\frac{10}{2}=5\)

=>Đúng

c: Đặt \(\frac{15}{4}=u_1+n\cdot d\)

=>\(\frac12n+\frac32=\frac{15}{4}\)

=>\(\frac{n}{2}=\frac{15}{4}-\frac32=\frac{15}{4}-\frac64=\frac94\)

=>n=9/2(loại)

=>Sai

d: \(S_{100}=\frac{100\cdot\left\lbrack2\cdot u_1+99\cdot d\right\rbrack}{2}=50\cdot\left(2\cdot u_1+99\cdot d\right)\)

\(=50\left(2\cdot\frac32+99\cdot\frac12\right)=50\cdot\left(3+49,5\right)=50\cdot52,5=2625\)

=>Sai

Câu 3:

a: \(u_6=u_1+5d\)

=>\(5d=u_6-u_1=27-\left(-3\right)=30\)

=>d=6

=>sai

b: \(u_{85}=u_1+84d=-3+84\cdot6=501\)

=>Đúng

c: \(u_{10}=u_1+9d=-3+9\cdot6=-3+54=51\)

=>Sai

d: \(S_{85}=\frac{85\cdot\left\lbrack2\cdot u_1+84\cdot d\right\rbrack}{2}=85\cdot\left\lbrack u_1+42\cdot d\right\rbrack=85\cdot\left\lbrack-3+42\cdot6\right\rbrack=85\cdot249=21165\)

=>Đúng

Câu 4:

a: \(S_{50}=5150\)

=>\(50\cdot\frac{\left\lbrack2\cdot u_1+49\cdot d\right\rbrack}{2}=5150\)

=>\(25\cdot\left(2\cdot u_1+49\cdot d\right)=5150\)

=>\(2u_1+49d=\frac{5150}{25}=206\)

=>10+49d=206

=>49d=196

=>d=4

=>Sai

b: \(u_{85}=u_1+84d=5+84\cdot4=5+336=341\)

=>Đúng

c: \(u_{10}=u_1+9d=5+9\cdot4=5+36=41\)

=>Sai

d: \(S_{85}=85\cdot\frac{\left\lbrack2\cdot u_1+84d\right\rbrack}{2}=85\cdot\left(u_1+42d\right)\)

\(=85\cdot\left(5+42\cdot4\right)=85\cdot173=14705\)

=>Đúng

Câu 9: \(u_8=u_1+7d\)

=>\(7d=26-\frac13=\frac{77}{3}\)

=>\(d=\frac{11}{3}\)

=>Chọn A

Câu 10: \(S_{n}=\frac{n\cdot\left\lbrack2u_1+\left(n-1\right)\cdot d\right\rbrack}{2}\)

=>\(n\cdot\frac{\left\lbrack2\cdot3+4\left(n-1\right)\right\rbrack}{2}=253\)

=>n(3+2n-2)=253

=>n(2n+1)=253

=>\(2n^2+n-253=0\)

=>\(2n^2-22n+23n-253=0\)

=>(n-11)(2n+23)=0

=>\(\left[\begin{array}{l}n=11\left(nhận\right)\\ n=-\frac{23}{2}\left(loại\right)\end{array}\right.\)

=>n=11

=>Chọn B
Câu 11: A

Câu 12: \(u_6=u_1+5d\)

=>\(5d=27-\left(-3\right)=27+3=30\)

=>d=6

=>Chọn D