a: AC=4cm

AH=2,4cm

BH=1,8cm

CH=3,2cm

a: \(A=4-\sin^245^0+2\cdot cos^260^0-3\cdot\cot^345^0\)

\(=4-\left(\frac{\sqrt2}{2}\right)^2+2\cdot\left(\frac12\right)^2-3\cdot1^3\)

\(=4-\frac12+2\cdot\frac14-3=4-3=1\)

b: \(B=tan45^0\cdot cos30^0\cdot\cot30^0\)

\(=1\cdot\frac{\sqrt3}{2}\cdot\sqrt3=\frac32\)

c: \(C=cos^215^0+cos^225^0+\cdots+cos^275^0\)

\(=\left(cos^215^0+cos^275^0\right)+\left(cos^225^0+cos^265^0\right)+\left(cos^235^0+cos^255^0\right)+cos^245^0\)

\(=1+1+1+\left(\frac{\sqrt2}{2}\right)^2=3+\frac12=\frac72\)

d: \(D=\sin^210^0+\sin^220^0+\cdots+\sin^280^0\)

\(=\left(\sin^210^0+\sin^280^0\right)+\left(\sin^220^0+\sin^270^0\right)+\left(\sin^230^0+\sin^260^0\right)+\left(\sin^240^0+\sin^250^0\right)\)

=1+1+1+1

=4

Đặt \(\left\{{}\begin{matrix}\sqrt[3]{3x-1}=a\\\sqrt[3]{x+1}=b\\\sqrt[3]{-2x}=c\end{matrix}\right.\) ta được hệ:

\(\left\{{}\begin{matrix}a+b=c\\a^3+b^3=-2c^3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\\left(a+b\right)^3-3ab\left(a+b\right)=-2c^3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c^3-3abc=-2c^3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left(c^2-ab\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left[\left(a+b\right)^2-ab\right]=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=c\\c\left[\left(a+\dfrac{b}{2}\right)^2+\dfrac{3b^2}{4}\right]=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}c=0\\a=b=0\left(vn\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt[3]{-2x}=0\Leftrightarrow x=0\)

a: Xét tứ giác MAOC có góc MAO+góc MCO=180 độ

nên MAOC là tứ giác nội tiếp

b: ΔOBC cân tại O

mà OK là trung tuyến

nên OK là đường phân giác và OK vuông góc với BC

Xét ΔOBN và ΔOCN có

OB=OC

góc BON=góc CON

ON chung

Do đó: ΔOBN=ΔOCN

=>NB=NC

=>NB^2=NC^2=NK*NO

36B

37C

38D

39B

40D

41A

42B

43B

44A

45B

46B

47A

48C

50B

51B

52B

53D

54C

55D

56C

a: Xét (O) có

ΔADB nội tiếp

AB là đường kính

Do đó: ΔADB vuông tại D

=>AD⊥BC tại D

Xét tứ giác AHDC có \(\hat{AHC}=\hat{ADC}=90^0\)

nên AHDC là tứ giác nội tiếp

b: AHDC nội tiếp

=>\(\hat{AHD}+\hat{ACD}=180^0\)

mà \(\hat{AHD}+\hat{MHD}=180^0\) (hai góc kề bù)

nên \(\hat{MHD}=\hat{ACD}=\hat{ACB}\)

Xét ΔOAC vuông tại A có AH là đường cao

nên \(OH\cdot OC=OA^2\)

=>\(OH\cdot OC=OB^2\)

=>\(\frac{OH}{OB}=\frac{OB}{OC}\)

Xét ΔOHB và ΔOBC có

\(\frac{OH}{OB}=\frac{OB}{OC}\)

góc HOB chung

Do đó: ΔOHB~ΔOBC

=>\(\hat{OHB}=\hat{OBC}=\hat{ABC}\)

mà \(\hat{OHB}+\hat{MHB}=\hat{OHM}=90^0\) và \(\hat{ABC}+\hat{ACB}=90^0\) (ΔABC vuông tại A)

nên \(\hat{MHB}=\hat{ACB}\)

=>\(\hat{MHB}=\hat{DHM}\)

=>HM là phân giác của góc DHB

7. Ta có: \(\left(x+\sqrt{x^2+3}\right)\left(\sqrt{x^2+3}-x\right)=x^2+3-x=3\)

\(\Rightarrow\sqrt{x^2+3}-x=y+\sqrt{y^2+3}\Rightarrow x+y=\sqrt{x^2+3}-\sqrt{y^2+3}\left(1\right)\)

Lại có \(\left(y+\sqrt{y^2+3}\right)\left(\sqrt{y^2+3}-y\right)=y^2+3-y=3\)

\(\Rightarrow\sqrt{x^2+3}+x=\sqrt{y^2+3}-y\Rightarrow x+y=\sqrt{y^2+3}-\sqrt{x^2+3}\left(2\right)\)

Lấy \(\left(1\right)+\left(2\right)\Rightarrow2\left(x+y\right)=0\Rightarrow x+y=0\)

9. Ta có: \(\sqrt{55+\sqrt{109}}-\sqrt{55-\sqrt{109}}\)

\(=\sqrt{\dfrac{110+2\sqrt{109}}{2}}-\sqrt{\dfrac{110-2\sqrt{109}}{2}}\)

\(=\sqrt{\dfrac{\left(\sqrt{109}+1\right)^2}{2}}-\sqrt{\dfrac{\left(\sqrt{109}-1\right)^2}{2}}=\dfrac{\sqrt{109}+1}{\sqrt{2}}-\dfrac{\sqrt{109}-1}{\sqrt{2}}\)

\(=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)

Lại có: \(\dfrac{\sqrt{2-\sqrt{4y-y^2}}}{y-2}.\sqrt{4+2\sqrt{4y-y^2}}\)

\(=\dfrac{\sqrt{4-2\sqrt{y\left(4-y\right)}}}{\sqrt{2}\left(y-2\right)}.\sqrt{\left(\sqrt{y}\right)^2+2\sqrt{y\left(4-y\right)}+\left(\sqrt{4-y}\right)^2}\)

\(\dfrac{\sqrt{\left(\sqrt{y}\right)^2-2\sqrt{y\left(4-y\right)}+\left(\sqrt{4-y}\right)^2}}{\sqrt{2}\left(y-2\right)}.\sqrt{\left(\sqrt{y}+\sqrt{4-y}\right)^2}\)

\(=\dfrac{\sqrt{\left(\sqrt{y}-\sqrt{4-y}\right)^2}}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|=\dfrac{\left|\sqrt{y}-\sqrt{4-y}\right|}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|\)

Vì \(y>2\Rightarrow\left\{{}\begin{matrix}\sqrt{y}>\sqrt{2}\\\sqrt{4-y}< \sqrt{2}\end{matrix}\right.\Rightarrow\sqrt{y}-\sqrt{4-y}>0\)

\(\Rightarrow\dfrac{\left|\sqrt{y}-\sqrt{4-y}\right|}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|=\dfrac{\left(\sqrt{y}-\sqrt{4-y}\right)\left(\sqrt{y}+\sqrt{4+y}\right)}{\sqrt{2}\left(y-2\right)}\)

\(=\dfrac{y-\left(4-y\right)}{\sqrt{2}\left(y-2\right)}=\dfrac{2y-4}{\sqrt{2}\left(y-2\right)}=\dfrac{2\left(y-2\right)}{\sqrt{2}\left(y-2\right)}=\sqrt{2}\)

\(\Rightarrow\dfrac{\sqrt{2-\sqrt{4y-y^2}}}{y-2}.\sqrt{4+2\sqrt{4y-y^2}}=\sqrt{55+\sqrt{109}}-\sqrt{55-\sqrt{109}}\)

a: Δ=(m-2)^2-4(m-4)

=m^2-4m+4-4m+16

=m^2-8m+20

=m^2-8m+16+4

=(m-2)^2+4>=4>0

=>Phương trình luôn có 2 nghiệm pb

b: x1^2+x2^2

=(x1+x2)^2-2x1x2

=(m-2)^2-2(m-4)

=m^2-4m+4-2m+8

=m^2-6m+12

=(m-3)^2+3>=3

Dấu = xảy ra khi m=3