\(\left\{{}\begin{matrix}3x-2y=9\\x-3y=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3x-2y=9\\3x-9y=30\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}7y=-21\\x-3y=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}y=-3\\x-3\cdot\left(-3\right)=10\end{matrix}\right.\)

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

Vậy: Hệ phương trình có nghiệm duy nhất \(\left(x;y\right)=\left(1;-3\right).\)

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

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

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

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

\(\Leftrightarrow\left\{{}\begin{matrix}y=1-\left(m-1\right)x\\\left(m^2-3\right)x=m+4\left(I\right)\end{matrix}\right.\)

Xét phương trình \(\left(I\right):\)

Nếu \(m=\pm\sqrt{3}\) thì phương trình vô nghiệm, dẫn đến hệ phương trình vô nghiệm.

Nếu \(m\ne\pm\sqrt{3}\), phương trình \(\left(I\right)\) có nghiệm duy nhất \(x=\dfrac{m+4}{m^2-3}\), suy ra được \(y=1-\dfrac{\left(m-1\right)\left(m+4\right)}{m^2-3}=\dfrac{1-3m}{m^2-3}\)

Vậy: Nếu \(m=\pm\sqrt{3}\), hệ phương trình vô nghiệm, nếu \(m\ne\pm\sqrt{3}\), hệ phương trình có nghiệm duy nhất \(\left(x;y\right)=\left(\dfrac{m+4}{m^2-3};\dfrac{1-3m}{m^2-3}\right)\)

1. creative

2. careless

3. beautifully

4. favourite

5. classmate

6. interesting

7. regularly

8. healthy

9. uninteresting

10. unhealthy

11. regularly

12. sunburned

(a) Gọi \(t\) là thời gian chuyển động của hai xe đến khi gặp nhau.

Hai xe chuyển động ngược chiều nên: \(s_1+s_2=100\)

\(\Leftrightarrow v_1t+v_2t=100\Leftrightarrow t=\dfrac{100}{v_1+v_2}=\dfrac{100}{60+40}=1\left(h\right)\)

\(\Rightarrow\left\{{}\begin{matrix}s_1=v_1t=60\cdot1=60\left(km\right)\\s_2=s-s_1=100-60=40\left(km\right)\end{matrix}\right.\)

Vậy: Lúc 7 + 1 = 8 giờ thì hai xe gặp nhau tại điểm cách A 60km hay cách B 40km.

(b) Không vẽ được đồ thị trên máy tính ạ:).
(c) Xét trong khoảng thời gian chưa gặp nhau, tức \(t_1< t=1\) với \(t_1\) là thời gian đi từ đầu đến khi hai xe cách nhau 50km.

Ta có: \(s_1'+s_2'+50=s\)

\(\Leftrightarrow v_1t_1+v_2t_2+50=100\Leftrightarrow t_2=\dfrac{50}{v_1+v_2}=\dfrac{50}{60+40}=0,5\left(h\right)=30\left(phút\right)\)

\(\Rightarrow\left\{{}\begin{matrix}s_1'=v_1t_1=60\cdot0,5=30\left(km\right)\\s_2'=s-s_1'-50=100-30-50=20\left(km\right)\end{matrix}\right.\)

Xét trong khoảng thời gian đã gặp nhau, tức \(t_2>t=1\) với \(t_2\) là thời gian đi từ điểm gặp nhau đến khi cách nhau 50km.

Ta có được: \(s_1''+s_2''=50\)

\(\Leftrightarrow v_1t_2+v_2t_2=50\Leftrightarrow t_2=\dfrac{50}{v_1+v_2}=\dfrac{50}{60+40}=0,5\left(h\right)=30\left(phút\right)\)

\(\Rightarrow\left\{{}\begin{matrix}s_1''=v_1t_2=60\cdot0,5=30\left(km\right)\\s_2''=50-s_1''=50-30=20\left(km\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}s_{A1}=s_1+s_1''=60+30=90\left(km\right)\\s_{B2}=s_2+s_2''=40+20=60\left(km\right)\end{matrix}\right.\)

Vậy: Hai xe cách nhau 50km lúc:
+) 7 giờ + 30 phút = 7 giờ 30 phút tại điểm cách A một khoảng \(s_1'=30\left(km\right)\) hay cách B một khoảng \(s_2'=20\left(km\right)\)

+) Hoặc lúc 8 giờ + 30 phút = 8 giờ 30 phút tại vị trí xe 1 cách A một khoảng \(s_{A1}=90\left(km\right)\) hay tại vị trí xe 2 cách B một khoảng \(s_{B2}=60\left(km\right).\)

Ta có : \(P=3A+2B\)

\(=\dfrac{2\sqrt{x}}{\sqrt{x}+2}+\dfrac{3}{\sqrt{x}+2}=\dfrac{2\sqrt{x}+3}{\sqrt{x}+2}.\)

\(\Rightarrow P=\dfrac{2\left(\sqrt{x}+2\right)-1}{\sqrt{x}+2}=2-\dfrac{1}{\sqrt{x}+2}\)

Do \(x\ge0\Rightarrow\sqrt{x}+2\ge0\)

\(\Rightarrow-\dfrac{1}{\sqrt{x}+2}\ge-1\)

\(\Rightarrow P=2-\dfrac{1}{\sqrt{x}+2}\ge-1+2=1.\)

Vậy : \(MinP=1.\) Dấu đẳng thức xảy ra khi và chỉ khi \(x=0.\)

1. The movie, which we went to, was interesting.

2. I met the people whom you told me about.

3. I couldn't understand the woman whom I talked to on the phone.

4. I want to tell you about the party which I went to last night.

5. The music, which we listened to last night, was gentle.

6. Alice likes the foreign family with whom she is living.

7. The market, which I usually go to, has refresh vegetables.

8. The man whom I told you about is over there.

9. The film, which they are talking about, is fantastic.

10. She's the nurse to whom we gave the flowers.

11. The man about whom we are talking was our new history teacher.

12. The teacher with whom we studied last year is Mr Pike.

13. We are very interested in the problem which has been discussed in class.

14. Lung cancer which million of people died of is very dangerous.

15. I like standing at the window, from which I can see the park.

16. The woman whom you gave place to on the bus lives next door to me.

17. The examination, which I was successful in, lasted 2 days.

18. This is the result of our work, which I'm pleased with.

19. Mr. Brown is the man for whom I'm responsible for my work.

20. This snake, which I'm afraid of, is dangerous.

21. There are a number of factors over which we have no control.

22. The picture which she was looking at was beautiful.

23. That's the woman to whom John used to be married.

24. My sister has 2 daughters, whom I often have to look after.

25. This is the book which I'm looking for.

26. The boy fell in love with the girl whom he was left by after a few weeks.

1. Manchester, where my brother lives, is the largest city in USA.

2. I have recently gone back to the town where I was born in.

3. Do you know the name of the hotel which John is staying at ?

4. We enjoy the city where we spent our vacation.

5. We enjoy this city which has been renewed a lot after the war.

6. 4^th June is the day when I was born on.

7. This is the house which has a big garden.

8. This is the house where I was born in.

9. May Day is the day when people hold a meeting on.

10. Ha Noi is the place where I'd like to come to.

11. Summer is the time when the weather is very hot.

12. I don't like summer when it's often hot.

13. The bed which I slept it last night was comfortable.

14. I never forget the park where we met each other for the first time.

15. I do not know the reason why she left school for.

16. We have not decided the day when I'll go to London.

17. Our school has a good laboratory where the students practice chemistry in.

18. Our school has a good laboratory which Mr John designed in 2000.

19. 7:30 is the time when my plane will arrive.

20. I love my parents' house where I was born in.

21. The book is a telephone directory where we can look up tel number.

22. We first met in 1945 when the revolution took place.

23. This is the house where we have lived for five years.

24. We often visit our friends in Brostol where is only 30 miles away.

(a) Điều kiện : \(x\ne-1.\)

Ta có : \(P=\dfrac{x^4+x}{x^2-x+1}+1-\dfrac{2x^2+3x+1}{x+1}\)

\(=\dfrac{x\left(x^3+1\right)}{x^2-x+1}+1-\dfrac{\left(2x+1\right)\left(x+1\right)}{x+1}\)

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

\(=x\left(x+1\right)+1-2x-1\)

\(=x^2-x.\)

Vậy : Với mọi \(x\ne-1\) thì \(P=x^2-x.\)

(b) Ta có : \(P=x^2-x\)

\(=\left[x^2-2\cdot x\cdot\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2\right]-\left(\dfrac{1}{2}\right)^2\)

\(=\left(x-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\ge-\dfrac{1}{4}\)

Vậy : \(MinP=-\dfrac{1}{4}.\) Dấu đẳng thức xảy ra khi và chỉ khi \(x=\dfrac{1}{2}.\)

\(D=\left\{1;2;3;4;5;...\right\}\)