Bộ đề thi học sinh giỏi cấp tỉnh các môn Khối 11 - Năm học 2020-2021 - Sở GD và ĐT Lạng Sơn (Có đáp án)

Nội dung text: Bộ đề thi học sinh giỏi cấp tỉnh các môn Khối 11 - Năm học 2020-2021 - Sở GD và ĐT Lạng Sơn (Có đáp án)

1. SỞ GIÁO DỤC VÀ ĐÀO TẠO KỲ THI CHỌN HỌC SINH GIỎI CẤP TỈNH LẠNG SƠN LỚP 11 NĂM HỌC 2020 - 2021 Môn thi: Toán lớp 11 THPT ĐỀ THI CHÍNH THỨC Thời gian: 180 phút (không kể thời gian giao đề) Ngày thi: 18/3/2021 (Đề thi gồm 01 trang, 05 câu) Câu 1 (6,0 điểm). a) Giải phương trình sin 4x cos x 1 sin3 x 2 cos 2 x . 4 y 1 x y x y 1 y x 2 b) Giải hệ phương trình xy , . 22 x y 8 y x 8 8 Câu 2 (3,0 điểm). n 2 n * a) Giả sử P x 1 3 x a0 a 1 x a 2 x an x , với n . Biết rằng a23 a 405 n 1 , tính giá trị của a6 . b) Cho tập hợp A 0;1;2;3;4;5;6;7. Gọi S là tập hợp các số tự nhiên gồm 8 chữ số đôi một khác nhau lấy từ A. Tính xác suất để lấy được số tự nhiên mà tổng 4 chữ số đầu bằng tổng chữ số cuối. u1 2021 Câu 3 (3,0 điểm). Cho dãy số un được xác định bởi . unn 1 u 2021 n 1 1 1 1 Đặt Sn . Tính limSn . uuuuuuuu1 2 2 1 2 3 3 2 uuun n 1 n 1 u n Câu 4 (6,0 điểm). Cho hình chóp tam giác S. ABC có đáy là tam giác đều cạnh bằng 2a . Đường thẳng SA vuông góc với mặt phẳng ABC . Gọi M là trung điểm của AB , H là hình chiếu vuông góc của C lên SB và góc tạo bởi đường thẳng AB và mặt phẳng HCM bằng 600 . a) Tính diện tích tam giác HCM . b) Tính sin của góc tạo bởi MH và SC . 1 ab Câu 5 (2,0 điểm). Cho hai số thực dương ab, thỏa mãn ab và 22. Tìm giá ba 11 ab22 trị nhỏ nhất của biểu thức P . a2 ab Hết Họ và tên thí sinh: Số báo danh: Chữ kí giám thị số 1: Chữ kí giám thị số 2:
2. SỞ GIÁO DỤC VÀ ĐÀO TẠO KỲ THI CHỌN HỌC SINH GIỎI CẤP TỈNH LẠNG SƠN LỚP 11 NĂM HỌC 2020 - 2021 HƯỚNG DẪN CHẤM THI MÔN TOÁN LỚP 11 THPT (Hướng dẫn chấm gồm 04 trang) Chú ý: Những cách giải khác HDC mà đúng thì cho điểm theo thang điểm đã định. Câu Nội dung Điểm 1 (6,0 đ) a) Giải phương trình sin 4x cos x 1 sin3 x 2 cos 2 x 4 PT sin4x cos x 1 sin3 x cos2 x sin2 x 0,5 2sin3x cos x sin3 x cos x 1 2cos2 x 1 0 0,5 sin3x 2cos x 1 cos x 2cos2 x 0 0,5 2cosx 1 sin3 x cos x 0 0,5 1 Với cosx x k 2 , k 0,5 23 xk 82 Với sin3x cos x sin3 x sin x , k 0,5 2 xk 4 y 1 x y x y 1 y x 2 1 b) 22 x y 8 y x 8 8 2 Điều kiện x y 0, y 0, x2 8 0 Phương trình 1 y 1 x y x y 1 y y 1 x y 1 0,5 y 1 x y 1 x y 1 y 1 0 y 1 x y 1 x y 1 y 1 0 0,5 x y 11 y 11 y 1 x y 1 0 x y 11 y 0,5 yy 1 0 1 11 (do 0) x y 11 y x x y 11 y Với y 1 thế vào 2 ta được: 8 2 x 0,5 3x x 8 8 3 x 3 22 9x 48 x 64 x 8
3. Với yx 1 thế vào 2 ta được: x x22 2 x 9 x 1 x 8 8 2 2 2 2 0,5 x x 2 x 9 x 1 x 8 8 2 2x2 2 x 7 2 x 1 x 2 8 0 x 1 x 2 8 0 x 1 9 x2 81 x x 2 2 xx 81 2 97 Với xy 0,5 22  97 Vậy nghiệm của hệ phương trình là xy;  3;1 , ;  22 2 n a) Ta có P x Ck3 k . x k a C k .3 k (3,0 đ)  n k n k 0 23 Theo giả thiết ta có a23 a 405 n 1 9 Cnn 27 C 405 n 1 Điều kiện nn 3, 0,5 99 n n 1 n n 1 n 2 405 n 1 22 2 n 10 nn 90 0 nL 9( ) 0,5 66 Khi đó aC6 10.3 b) Số tự nhiên gồm 8 chữ số đôi một khác nhau lấy từ A là nS 8! 7! 0,25 Ta có 0 1 2 3 4 5 6 7 28 nên để tổng của 4 chữ số đầu bằng tổng 4 chữ số cuối thì tổng của 4 chữ số phải bằng 14 Ta lập 4 bộ số có tổng bằng có chứa chữ số 0 là: 0,1,6,7 , 0,2,5,7 , 0,3,4,7 , 0,3,5,6 0,25 Với mỗi bộ số có chứa chữ số 0 trên tương ứng với bộ còn lại không chứa chữ số 0 và có tổng bằng TH1: Bộ có chữ số đứng trước: có 4 bộ có chữ số 0, ứng với mỗi bộ có: +) Xếp số đầu có 3.3! cách. 0,5 +) Xếp số cuối có 4! cách. Áp dụng qui tắc nhân có 4.3.3!.4! 1728 số. TH2: Bộ có chữ số đứng sau: có 4 bộ có chữ số , ứng với mỗi bộ có: +) Xếp bộ không có chữ số đứng trước có cách. 0,5 +) Xếp bộ có chữ số đứng sau có cách. Áp dụng qui tắc nhân có 4.4!.4! 2304 số. Gọi B là biến cố mà số tự nhiên mà tổng 4 chữ số đầu bằng tổng 0,5 chữ số cuối nên nB 1728 2304 4032 số thỏa mãn yêu cầu
4. bài toán. nB 4 Vậy xác suất biến cố B là: PB nS 35 3 Ta có un 11 u n 2021 u n u n 2021 0 nên un là dãy số (3,0 đ) giảm 1,0 Giả sử un bị chặn dưới tức là tồn tại limuan 2021 Qua giới hạn hai vế ta được aa 2021 (vô lý) tức là limun 11 Ta có u u u u 0,5 n n 11 n n un u n 11 u n u n uu 1 1 1 nn 1 0,5 2021 uu un u n 11 u n u n nn 1 1 1 1 1 1 1 Khi đó S n 0,5 2021 u1 unn 1 2021 2021 u 1 1 Vậy limS 0,5 n 2021 2021 4 a) Tính diện tích tam giác HCM . (6,0 đ) CM SA Ta có CM  SAB CM  SB 1 CM AB 0,5 Mặt khác CH SB 2 . Từ 1 và 2 SB CMH AB HCM M 0 Lại có AB, HCM BMH 60 0,5 BH HCM Do CM SAB CM  MH hay tam giác HCM vuông tại M 0,5 a Có CM a 3 ; BM a MH MB.cos600 0,5 2 13a2 Vậy diện tích tam giác HCM là S CM. MH 0,5 HCM 24 b) Tính sin của góc tạo bởi MH và SC .
5. 3. ___ he failed in the entrance exam caused his family much sadness. A. Which B. That C. The thing D. What 4. Mike was not feeling well, so we gave him ___ to do than you. A. less work B. fewer workers C. more work D. least work 5. The old woman came in, ___ by a young man. A. accompany B. to be accompanied C. accompanying D. accompanied 6. I’ve applied for the job I saw___ in the newspaper last month. A. advertised B. advertising C. be advertised D. was advertised 7. No, I didn’t know his number; ___, I’d have phoned him. A. otherwise B. so C. therefore D. unless 8. ___ the two sisters, Mary is___. A. Of/ the prettier B. Between/ the prettiest C. Of/ prettier D. Among/ the prettier 9. John: Do you think it will rain tonight? Mary: ___. I am attending the evening class. A. I don’t hope so B. I hope so C. I hope not D. I am not hoping 10. Ninety___ the maximum length of time allowed for the entrance exam to this school. A. minute is B. minutes are C. minute are D. minutes is 11. I really appreciate___ to help me, but I am sure I’ll be able to manage by myself. A. you to offer B. you are offering C. your offering D. which you offer 12. They never came to class late. ___. A. So did we B. Neither did we C. We did either D. Neither we did 13. When Tet holiday comes, Vietnamese people often feel inclined to ___ their houses. A. do up B. do in C. do through D. do over 14. It is necessary that an employee ___his work on time. A. finish B. to finish C. finishes D. finishing 15. The knee is ___ most other joints in the body because it cannot twist without injury. A. to be damaged more than likely B. more likely to be damaged than C. likely to be more than damaged D. more than likely to be damaged 16. All the sale people I met always try to ___ buying their products. A. persuade me for B. force me to C. talk me into D. help me with 17. The flight was delayed and the hotel was awful; ___, we still had a good time. A. on top of that B. nonetheless C. for all that D. by the same token 18. A: Why didn't you go to work yesterday? B: I was feeling a bit ___. A. under the weather B. full of hot air C. breeze D. under pressure 19. The High Street is so narrow that the council has decided to ___ it. A. extend B. widen C. increase D. lengthen 20. Students these days are ___ with too much unnecessary information. A. helped B. given C. covered D. overloaded 3
6. Your answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Part II: Use the word given in capitals to form a word that fits in the space of each sentence. Write your answers in the corresponding numbered boxes. 1. It was thanks to the ___of the medical staff that she recovered from her injuries. (DEDICATE) 2. ___, the hole in the ozone layer has doubled in size this year. (ALARM) 3. Her son is always mischievous and ___, which annoys her very much. (OBEY) 4. The fans’ bad behavior has resulted in the ___ of their football team from the championship. (QUALITY) 5. They frequently ___ the traffic as they march through the streets. (MOBILE) Your answers: 1. 2. 3. 4. 5. Part III: Choose and correct the underlined part that needs correction in each of the following sentences. Write the letter A, B, C or D and the correction in the numbered boxes. 1. The production of different kinds of artificial materials are essential to the conservation of A B C our natural resources. D 2. Half of the people inviting to the party didn't turn up. A B C D 3. They had to call on the match as the ground was too wet to play on. A B C D 4. For more than 450 years, Mexico City has been the economic, culture, and political center A B of the Mexican people. C D 5. Two men, neither of them I had ever seen before, came into my office yesterday. A B C D Questions Answers Corrections 1 2 3 4 5 4
7. READING (5.0 POINTS) Part I: Read the passage below and choose the correct word or phrase that best fits each of the numbered blanks. Write your answers in the corresponding numbered boxes. RENEWABLE ENERGY COMES OF AGE The British Wind Energy Association was founded 30 years ago by a group of scientists. At that time, the term “alternative energy” was used to describe the generation of wind and solar power. These days, we tend to (1)___ to them as “renewable energy” and the use of this name (2)___ a real change in their status. These sources of energy, apart from being alternative, have now become mainstream and are (3)___ to make significant contribution to the country’s energy needs in the future. Two closely linked developments (4)___ behind this (5)___ in status. Firstly, over the past decade or so, the price of oil and gas has been rising (6)___ reflecting the extent to which reserves of these fossil fuels are becoming (7)___. However, price is only part of the explanation. (8)___ as important is the growing consensus that carbon emissions must be curbed. The scientific evidence for climate change is now irrefutable, and both policy makers and the (9)___ public are finally in agreement that doing nothing about the prospect of global warming is no longer a viable option. Renewable energy represents one real way of (10)___. 1. A. consider B. refer C. mention D. regard 2. A. regards B. reproduces C. reminds D. reflects 3. A. set B. held C. put D. stood 4. A. sit B. reside C. lie D. recline 5. A. move B. shift C. switch D. jump 6. A. equally B. serenely C. habitually D. steadily 7. A. depleted B. decreased C. depressed D. debased 8. A. Just B. Still C. Much D. Yet 9. A. deeper B. greater C. bigger D. wider 10. A. copying B. engaging C. addressing D. dealing Your answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Part II: Fill in each of the blanks of the following passage with ONE suitable word. Write your answers in the corresponding numbered boxes. Vitamins are substances required for the (1)___ functioning of the body. In this century, thirteen vitamins have been discovered. A lack of any vitamins in a person’s body can cause illness. In some cases, an excess of vitamins can also (2)___ to illness. For example, sailors in the past were prone to suffer from scurvy that is a disease (3)___ from the lack of vitamin C. It causes bleeding of the gum, loss of teeth and skin rashes. Sailors suffer from scurvy because they did not eat fruits and vegetables. Fruits and vegetables (4)___ vitamin C which is necessary for good (5)___ 5
8. Vitamin B complex is composed of eight different vitamins. A lack of any of these vitamins will lead to different (6)___. For instance, a person who has too little vitamin B1 will suffer from beriberi, a disease that causes heart problems and mental (7)___. A lack of vitamin B2 results in eye and skin problems while deficiency of vitamin B6 causes problems of the nervous system. Too little vitamin B12 will cause anemia. The (8)___ that vitamin deficiencies caused certain diseases led doctors to cure people suffering from these illnesses by giving them doses of the (9)___ vitamins. Today, vitamins are (10)___ in the form of pills and can easily be bought at any pharmacy. Your answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Part III: Read the following passage and choose the correct answer (A, B, C or D) to each of the questions below. Write your answers in the corresponding numbered boxes. Modern surgery The need for a surgical operation, especially an emergency operation, almost always comes as a severe shock to the patient and his family. Despite modern advances, most people still have an irrational fear of hospitals and anesthetics. Patients do not often believe they really need surgery – cutting into a part of the body – as opposed to treatment with drugs. In the early years of the 20th century there was little specialization in surgery. A good surgeon was capable of performing almost every operation that had been advised up to that time. Today the situation is different. Operations are now being carried out that were not even dreamed of fifty years ago. The heart can be safely opened and its valves repaired. Clogged blood vessels can be cleaned out, and broken ones mended or replaced. A lung, the whole stomach, or even part of the brain can be removed and still permit the patient to live a comfortable and satisfactory life. However, not every surgeon wants to, or is qualified to carry out every type of modern operation. The scope of surgery has increased remarkably in the past decades. Its safety has increased too. Deaths from most operations are about 20% of what they were in 1910 and surgery has been extended in many directions, for example to certain types of birth defects in new born babies, and, at the other end of the scale, to life-saving operations for the octogenarian. The hospital stay after surgery has been shortened to as little as a week for most major operations. Most patients are out of bed on the day after an operation and may be back at work in two or three weeks. Many developments in modern surgery are almost incredible. They include the replacement of damaged blood vessels with stimulated ones made of plastic; the replacement of heart valves with plastic substitutes; the transplanting of tissues such as the lens of the eye; the invention of the artificial kidney to clean the blood of poisons at regular interval and the development of heart and lung machines to keep patients alive during very long operation. All these things open a hopeful vista for the future of surgery. 6
9. One of the most revolutionary areas of modern surgery is that of organ transplants. Until a few decades ago, no person, except an identical twin, was able to accept into his body the tissues of another person without reacting against them and eventually killing them. Recently, however, it has been discovered that with the use of X-rays and special drugs, it is possible to graft tissues from one person to another which will survive for periods of a year or more. Kidneys have been successfully transplanted between non-identical twins. Heart and lung transplants have also been reasonably successful. “Spare parts” surgery, the simple routine replacement of all worn-out organs by new ones, is still a dream of the future but surgery is ready for such miracles. In the meantime, you can be happy if your doctors say to you, “Yes, I think it is possible to operate on you for this condition.” 1. Most people are afraid of being operated on ___. A. in spite of improvements in modern surgery B. because they think modern drugs are dangerous C. because they do not believe they need anesthetics D. unless it is an emergency operation 2. Surgeons in the early 20th century, compared with modern ones___. A. had less to learn about surgery C. could perform every operation known today B. had more knowledge D. were more trusted by their patients 3. A patient can still live a comfortable life even after the removal of___. A. part of his brain C. all major organs except the stomach or one lung B. his lungs D. part of the stomach or the whole liver 4. The word “clogged” in paragraph 2 is most likely to correspond to___. A. cleaned B. blocked C. covered D. unwashed 5. Today, compared with 1910___. A. 20% fewer of all operation patients die B. 20% of all operation patients recover C. operation deaths have increased by 20% D. five times fewer patients die after being operated on 6. What is the similar meaning to the word “octogenarian” in paragraph 3? A. eighteen-year-old s B. people in their eighties C. people having eighth operation D. eye specialists 7. Some of the more astonishing innovations in modern surgery include___. A. ear, nose and throat transplants B. valve plastics hearts C. plastic heart valves D. leg transplants 8. The main difficulty with organ transplant is___. A. it is difficult to find organs of the same size B. only identical twins can give permission for their organs to be exchanged C. the body’s tendency to reject alien tissues D. the patient is not allowed to use drugs after them. 7
10. 9. Which of the following has the same meaning as “vista” in paragraph 4? A. support B. prospect C. history D. visit 10. You can be happy if your surgeon can operate on you because it means___. A. he thinks your condition may be curable B. he is a good doctor C. you have no other option D. you are getting better already Your answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. WRITING (4.0 POINTS) Part I: Finish the second sentence in each pair in such a way that it means the same as the sentence given before it, using the words given. 1. If Jane hadn’t refused to work overtime, she would have got promotion. - If it hadn’t ___. 2. Hearing that an earthquake had occurred was a great shock to them. - They were ___. 3. The only way to eliminate world terrorism is by united opposition. - Only by ___. 4. That rumor about the politician and the construction contract is absolutely false. - There is ___. 5. The value of the sterling has fallen considerably in the past week. - There has ___. Part II: Use the word given in brackets and make any necessary additions to write a new sentence in such a way that it is as similar as possible in meaning to the original sentence. Do NOT change the form of the given word. You must use between THREE and SIX words, including the word given. 1. I’m sure it was unbearably hot where you were staying. MUST - The ___ where you were staying 2. “It’s not worth worrying about the past,” I told him. POINT - I told him that there was ___ about the past. 3. I didn’t agree with the idea. FAVOUR - I ___ the idea. 4. I bought it without thinking about it first. SPUR - I bought ___ the moment. 5. “Do you realize how late it is?” Bob asked his wife. IF - Bob asked his wife ___ late it was. Part III: In about 120-150 words write a description of a popular celebration in Vietnam. Your description should include the following main points. - Name of the celebration 8
11. - Time of the celebration - Purpose of the celebration - Main activities of the celebration - Your feeling about the celebration (whether you like it or not and why) 9
12. ___ The end ___ 10
13. SỞ GIÁO DỤC VÀ ĐÀO TẠO KỲ THI CHỌN HỌC SINH GIỎI CẤP TỈNH LẠNG SƠN LỚP 11 NĂM HỌC 2020 - 2021 HƯỚNG DẪN CHẤM THI MÔN TIẾNG ANH (Hướng dẫn chấm gồm 02 trang) Chú ý: Những cách giải khác HDC mà đúng thì cho điểm theo thang điểm đã định. LISTENING (4.0 POINTS) Part I. (2.0 points) 0.2 points for each correct answer 1. hostel 2. Buckleigh 3. PE9 7QT 4. waiter 5. politics 6. cycling 7. cinema 8. disabled 9. 4.30 10. 07788 136711 Part II. (2.0 points) 0.25 points for each correct answer 1. B 2. A 3. C 4. B 5. C 6. A 7. C 8. C PHONETICS (1.0 POINT) 0.2 points for each correct answer 1. A 2. B 3. C 4. B 5. B LEXICO-GRAMMAR (4.0 POINTS) Part I. (2.0 points) 0.1 point for each correct answer. 1. D 2. B 3. B 4. A 5. D 6. A 7. A 8. A 9. C 10.D 11. C 12. B 13. A 14. A 15. B 16. C 17. B 18. A 19. B 20. D Part II. (1.0 point) 0.2 points for each correct answer 1. dedication 2. Alarmingly 3. disobedient 4. disqualication 5. immobilize Part III. (1.0 point) 0.2 points for each correct answer Questions Answers Corrections 1. B are -> is 2. B inviting-> invited 3. A on-> off 4. B culture-> cultural 5. B them-> whom 1
14. READING (5.0 POINTS) Part I (1.0 points) 0.1 point for each correct answer 1.B 2. D 3. A 4. C 5. B 6. D 7. A 8. A 9. D 10.C Part II. (2.0 points) 0.2 points for each correct answer 1. proper/normal 2. lead 3. resulting 4. contain 5. health 6. diseases 7. disorders 8. knowledge 9. necessary 10. available Part III. (2.0 point) 0.2 points for each correct answer 1. A 2. A 3. A 4. B 5. D 6. B 7. C 8. C 9. B 10. A WRITING (4.0 points) Part I. (1.0 point) 0.2 points for each correct answer 1. If it hadn’t been for Jane’s refusal to work overtime, she would have got promotion. 2. They were (greatly/very) shocked to hear/when they heard that an earthquake had occurred. 3. Only by united opposition can/ do we eliminate world terrorism. 4. There is absolutely no truth in that rumor about the politician and the construction contract. 5. There has been a considerable fall in the value of the sterling in the past week . Part II. (1.0 point) 0.2 points for each correct answer 1. heat / temperature / hot weather must have been unbearable. 2. no point (in) worrying 3. wasn’t in favour of 4. it on the spur of 5. if she realised how PART III. Write a paragraph ( 2.0 points) 1. Đúng cấu trúc đoạn văn (câu mở, câu phát 0.25 triển đoạn, câu kết bài) 2. Nêu được các n i dung sau Name of the celebration Time of the celebration Purpose of the celebration 1.0 (0.2 điểm cho m t ý) Main activities of the celebration Your feeling about the celebration (whether you like it or not and the reason(s) 3. Sai ko quá 3 lỗi ngữ pháp hoặc chính tả. 0.25 4. Sử dụng đa dạng cấu trúc, từ vựng. 0.5 2