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Số lượng câu hỏi 2
Số lượng câu trả lời 940
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Điểm SP 1264

Người theo dõi (347)

Đang theo dõi (8)


Chủ đề:

Luyện tập tổng hợp

Câu hỏi:

Babe Didrikson Zaharias was one of the most extraordinary athletes of all time. She was achampion in basketball, track and field, and golf. She also swam, boxed, played baseball, and manyother sports.As the sixth child in her family, Zaharias was called “Baby”. That soon got shortened to“Babe”. One day, when she was in elementary school, her mother sent her to the grocery store tobuy beef for dinner. On the way home, Babe spotted some boys playing baseball. She dropped her purchases and joined the game.An hour later,her angry mother showed up,looking for the groceries. Babe pointed to where she’d left them. A dog was happily eating the beef.Zaharias gained fame at the 1932 Olympic Games. She won gold medals in the javelin throwand the 80­meter hurdles, and set world records in both. In 1935, Zaharias began playing golf andwas soon winning important championships in the sport. She won every major championship inwomen’s golf and helped make the sport popular for women. In 1950, she was named the greatestfemale athlete of the first half of the 20th century. Zaharias died of cancer in 1956, at the age of 45.
31. What sport is not mentioned as one that Babe Didrikson Zaharias did?
A. Running.
B. Basketball.
C. Tennis.
D. Swimming.
32. What does the author imply by giving the example of Babe’s buying beef?
A. Babe was an absent­minded girl.
B. Babe was keen on playing ball.
C. Babe was not a good child.
D. Babe could help her mother with doing some shopping.
33. What word or phrase is similar in meaning to “purchases”?
A. Buys.
B. What is bought.
C. Goods.
D. Food.
34. When did Babe set world records in javelin throw?
A. When she was 30 years old.
B. In 1935.C. When she was also an excellent golfer.
D. At the Olympic Games in 1932.
35. Which of the following sentences is NOT true?
A. Zaharias won important championships in golf before the 1930’s.
B. Zaharias helped make golf popular for women after 1935.
C. Zaharias died of cancer in 1956 when she was younger than 50.
D. Zaharias was named the greatest female athlete of her time

Câu trả lời:

Chứng minh đẳng thức phụ:

\(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+2\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}\right)\)

\(=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+2.\dfrac{a+b+c}{abc}\)

\(\Rightarrow\) Với \(a+b+c=0\). Ta có: \(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)

\(\Leftrightarrow\sqrt{\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}}=\sqrt{\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2}=\left|\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right|\)với \(a+b+c=0\)

Ta có:

\(S=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+.....+\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}\)

Áp dụng đẳng thức phụ trên:

\(\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}=\sqrt{\dfrac{1}{1^2}+\dfrac{1}{1^2}+\dfrac{1}{\left(-2\right)^2}}=1+1-\dfrac{1}{2}\left(>0\right)\)

\(\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}=\sqrt{\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{\left(-3\right)^2}}=1+\dfrac{1}{2}-\dfrac{1}{3}\left(>0\right)\)

\(\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}=\sqrt{\dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{\left(-4\right)^2}}=1+\dfrac{1}{3}-\dfrac{1}{4}\left(>0\right)\)
\(.................\)

\(\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}=\sqrt{\dfrac{1}{1^2}+\dfrac{1}{99^2}+\dfrac{1}{\left(-100\right)^2}}=1+\dfrac{1}{99}-\dfrac{1}{100}\)

Cộng vế với vế các đẳng thức trên, ta có:

\(S=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+........+\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}\)

\(=1+1-\dfrac{1}{2}+1+\dfrac{1}{2}-\dfrac{1}{3}+1+\dfrac{1}{3}-\dfrac{1}{4}+............+1+\dfrac{1}{99}-\dfrac{1}{100}\)

\(=99+1-\dfrac{1}{100}=99+\dfrac{99}{100}=99\dfrac{99}{100}\)