1b) \(C=\sqrt{81a}-\sqrt{144a}+\sqrt{36a}\left(a\ge0\right)=8\sqrt{a}-12\sqrt{a}+6\sqrt{a}=2\sqrt{a}\)

Bài 2:

a),b) \(P=\left(\dfrac{1}{1-\sqrt{a}}-\dfrac{1}{1+\sqrt{a}}\right)\left(\dfrac{1}{\sqrt{a}}+1\right)\left(đk:x>0,x\ne1\right)\)

\(=\dfrac{1+\sqrt{a}-1+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}.\dfrac{\sqrt{a}+1}{\sqrt{a}}=\dfrac{2\sqrt{a}}{1-\sqrt{a}}.\dfrac{1}{\sqrt{a}}=\dfrac{2}{1-\sqrt{a}}\)

c) \(P=\dfrac{2}{1-\sqrt{a}}=\dfrac{2}{1-\sqrt{4}}=\dfrac{2}{1-2}=-2\)

d) \(P=\dfrac{2}{1-\sqrt{a}}=9\)

\(\Rightarrow-9\sqrt{a}+9=2\Rightarrow\sqrt{a}=\dfrac{7}{9}\Rightarrow a=\dfrac{49}{81}\left(tm\right)\)

chép đề là sao?

ghi lại đề thì b tự lm dc mà

Thay m=2 vào HPT ta có: 

\(\left\{{}\begin{matrix}2x+y=1\\x+2y=1\end{matrix}\right.\)

⇔\(\left\{{}\begin{matrix}4x+2y=2\\x+2y=1\end{matrix}\right.\)

⇔\(\left\{{}\begin{matrix}4x+2y=2\\3x=1\end{matrix}\right.\)

⇔\(\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{1}{3}\end{matrix}\right.\)

b) \(\left\{{}\begin{matrix}mx+y=1\\x+my=1\end{matrix}\right.\)

⇔\(\left\{{}\begin{matrix}y=1-mx\\x+m\left(1-mx\right)=1\left(1\right)\end{matrix}\right.\)

(1) ⇔x+m-m²x=1

⇔x(1-m²)=1-m (2)

TH1: 1-m² = 0

⇔m = +- 1

Thay m=1 vào (2) ta có: 0x=0 (Luôn đúng) ⇒m=1 (chọn)

Thay m=-1 vào (2) ta có: 0x=2 (Vô lí) ⇒m=-1 (loại)

TH2: 1-m² ≠0

⇔m≠ +-1

⇒HPT có nghiệm duy nhất:

x=  \(\dfrac{1-m}{1-m^2}\)

⇒y= \(1-m.\dfrac{1-m}{1-m^2}\)

⇔y=\(\dfrac{1-m}{1-m^2}\)

Dễ thấy x=y nên: 

\(\dfrac{1-m}{1-m^2}>0\)

⇔1-m>0

⇔m<1

Vậy m <1 thì Thỏa mãn yêu cầu đề bài.

a) Thay m=2 vào hệ phương trình, ta được:

\(\left\{{}\begin{matrix}2x+y=1\\x+2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+y=1\\2x+4y=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-3y=-1\\2x+y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{1}{3}\\2x=1-y=\dfrac{2}{3}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{1}{3}\end{matrix}\right.\)

Vậy: Khi m=2 thì hệ phương trình có nghiệm duy nhất là \(\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{1}{3}\end{matrix}\right.\)

1.

\(\dfrac{3\pi}{2}< a< 2\pi\Rightarrow sina< 0\)

\(\Rightarrow sin\alpha=-\sqrt{1-cos^2a}=-\dfrac{12}{13}\)

\(\Rightarrow tan2a=\dfrac{sin2a}{cos2a}=\dfrac{2sina.cosa}{cos^2a-sin^2a}=\dfrac{2.\left(-\dfrac{12}{13}\right).\left(\dfrac{5}{13}\right)}{\left(\dfrac{5}{13}\right)^2-\left(-\dfrac{12}{13}\right)^2}=...\)

3.

\(P=\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{x}+\dfrac{4}{4y}\ge\dfrac{\left(1+2\right)^2}{x+4y}=\dfrac{9}{6}=\dfrac{3}{2}\)

\(P_{min}=\dfrac{3}{2}\) khi \(\left(x;y\right)=\left(2;1\right)\)

4.

Lưu ý: hàm \(sinx\) đồng biến khi \(0< x< 90^0\) và nghịch biến khi \(90^0< x< 180^0\), hàm cos nghịch biến khi \(0< x< 90^0\)

Đường tròn (C) tâm \(I\left(1;1\right)\) bán kính \(R=4\) , \(\overrightarrow{IA}=\left(1;-1\right)\Rightarrow IA=\sqrt{2}\)

Theo công thức diện tích tam giác:

\(S_{IMN}=\dfrac{1}{2}IM.IN.sin\widehat{MIN}=\dfrac{1}{2}R^2.sin\widehat{MIN}=8.sin\widehat{MIN}\)

\(\Rightarrow S_{IMN}\) đạt max khi \(sin\widehat{MIN}\) đạt max

Gọi H là trung điểm MN \(\Rightarrow IH\perp MN\Rightarrow IH\le IA\) theo định lý đường xiên - đường vuông góc

\(\Rightarrow cos\widehat{HIM}=\dfrac{IH}{IM}\le\dfrac{IA}{IM}=\dfrac{\sqrt{2}}{4}\Rightarrow\widehat{HIM}>69^0\)

\(\Rightarrow\widehat{MIN}=2\widehat{HIM}>120^0>90^0\)

\(\Rightarrow sin\widehat{MIN}\) đạt max khi \(\widehat{MIN}\) đạt min

\(\Rightarrow\widehat{HIM}=\dfrac{1}{2}\widehat{MIN}\) đạt min

\(\Rightarrow cos\widehat{HIM}\) đạt max

\(\Rightarrow cos\widehat{HIM}=\dfrac{\sqrt{2}}{4}\Leftrightarrow H\) trùng A

Hay đường thẳng MN vuông góc IA \(\Rightarrow\) MN nhận (1;-1) là 1 vtpt

Phương trình MN: \(1\left(x-2\right)-1\left(y-0\right)=0\Leftrightarrow x-y-2=0\)

1.C

2.D

3.C

4.C

5.C

6.C

7.D

8.D

9.C

10.D

11.A

12.A

B. WRITING

I. Circle the letter A, B, C or D under the word/ phrase that needs correcting.

B (if to tell → whether to tell)

C (didn’t have → wouldn’t have)

B (suggest → suggests)

II. Rewrite these sentences begin with the words given.

She asked me how to deal with school pressure.

He advised me to call the helpline because they could give me good advice.

III. Write an email (100 - 120 words) to a friend of yours telling him/her about one city that you would like to visit the most.

Dear [Friend's Name],

I hope you're doing well! I wanted to share with you about a city I’m really excited to visit – Paris, France. I’ve always dreamed of exploring its rich culture, stunning architecture, and delicious cuisine. The Eiffel Tower, Louvre Museum, and Notre-Dame Cathedral are some of the iconic landmarks I can’t wait to see.

What attracts me most is the city’s artistic vibe and romantic atmosphere. I plan to visit during the spring, around April, when the weather is pleasant, and the flowers are in full bloom. I’d love to stroll along the Seine River and enjoy a coffee at a charming café.

I hope to make this trip next year. Let me know if you have any tips or recommendations!

Best,
[Your Name]


=)))