Có bao nhiêu cặp số nguyên \((x, y)\) thỏa mãn điều kiện \(0 \leq y \leq 100\) và \(x^{6}+6 x^{4} y+12 x^{2} y^{2}-19 y^{3}+3 x^{2}-3 y=0 ?\)
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Phương pháp giải
Sử dụng hàm đặc trưng
Lời giải
\(x^{6}+6 x^{4} y+12 x^{2} y^{2}-19 y^{3}+3 x^{2}-3 y=0\)
\(\Leftrightarrow x^{6}+6 x^{4} y+12 x^{2} y^{2}+8 y^{3}-27 y^{3}+3 x^{2}-3 y=0\)
\(\Leftrightarrow x^{6}+6 x^{4} y+12 x^{2} y^{2}+8 y^{3}+3 x^{2}+6 y=27 y^{3}+9 y\)
\(\Leftrightarrow\left(x^{2}+2 y\right)^{3}+3\left(x^{2}+2 y\right)=(3 y)^{3}+3.3 y\left(^{*}\right)\)
Xét hàm số: \(f(t)=t^{3}+3 t\)
Ta có : \(f^{\prime}(t)=3 t^{2}+3>0 \forall t \in \mathbb{R}\)
\(\Rightarrow f(t)\) là hàm đồng biến trên \(\mathbb{R}\)
Vì vậy \(\left({ }^{*}\right) \Leftrightarrow f\left(x^{2}+2 y\right)=f(3 y) \Leftrightarrow x^{2}+2 y=3 y \Leftrightarrow x^{2}=y\)
Theo giả thiết ta có : \(0 \leq y \leq 100 \Leftrightarrow 0 \leq x^{2} \leq 100 \Leftrightarrow-10 \leq x \leq 10\)
Vì \(x\) nguyên nên \(x \in\{-10 ;-9 ;-8 ; \ldots ; 8 ; 9 ; 10\}\), với mỗi \(x\) xác định duy nhất giá trị \(y=x^{2}\).
Vậy có 21 cặp \((x ; y)\) thỏa mãn bài toán.
Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 2 mang đến cho thí sinh một trải nghiệm thi cử mới mẻ, bám sát chương trình GDPT 2018, kiểm tra khả năng Giải Quyết Vấn Đề, Tư Duy Sáng Tạo, Giao Tiếp và Hợp Tác. Với thời lượng 195 phút, bài thi bao gồm ba phần: Toán Học Và Xử Lí Số Liệu/Tư Duy Định Lượng, Văn Học - Ngôn Ngữ/Tư Duy Định Tính và Khoa Học/Tiếng Anh. Mỗi phần đều có dạng thức câu hỏi phong phú như trắc nghiệm nhiều phương án lựa chọn và điền đáp án, đảm bảo đánh giá toàn diện năng lực học sinh trên nhiều khía cạnh học thuật khác nhau.
Câu hỏi liên quan
Read the following passage about food addiction and mark the letter A, B, C, or D on your answer sheet to indicate the best answer to each of the following questions from 31 to 40.
Few people would disagree that one of the world's most popular dishes is pizza. Twenty-two-year-old Zack certainly wouldn't. He would eat a variety of pizza-based items, including homemade pizza, precooked microwave pizza, and even pizza leftovers from the night before. He was consuming 800kg of pizza a year. As a child, Zack had eaten a wide variety of food. But he became addicted to pizza when he started secondary school. American school meals have often been criticised for the large amount of fast food present on the menu, and pizza is the star dish. Zack even spent all his pocket money on slices of pizza.
Naturally, Zack's family worried about his high-calorie, low-vitamin diet. So did Zack. In an attempt to break his addiction, Zack agreed to appear on a TV programme called Freaky Eaters. Its aim was to encourage food addicts to beat their addictions with the help of a psychologist and nutritionist. It wasn't easy. He had to eat a fish dish, while his brother sat in front of him enjoying... a pizza! However, Zack met the challenge and has never eaten pizza again.
Seventeen-year-old Stacy Irvine's friends visit her and discover that she has been rewarded thousands of free toys from fast food restaurants as a regular customer. Stacy's mother often gave her two-year-old girl a portion of fried chicken in a McDonald's restaurant. [I] Since then, Stacy has refused to eat anything else except for pieces of fried chicken, and as a result, her diet has created serious health. She has been warned that if she doesn't change her diet, she will die. [II] Stacy once collapsed due to lack of vitamins and nutrients and needed to be hospitalised. [III] She insisted that Stacy adopt a healthier lifestyle. However, worryingly, Stacy's mother said that while Stacy was beginning to understand the seriousness of her situation, she couldn't eat anything else but chicken nuggets. [IV]
Twenty-year-old Hanna Little was keen on chips. The only thing that Hanna consumed for fifteen years was one plate of chips after another. Not surprisingly, a diet that contained few nutrients and an excess of fat and carbohydrates could only have negative consequences. And it did because not only did Hanna develop health problems, but she also had to leave her job after collapsing at work. At five, Hanna became addicted to chips and refused to eat fruits and vegetables despite her mother's efforts. Hanna says that she was terrified of tasting the flavour of different types of food and she would feel anxious just at the idea of doing it. Parties, picnics and barbecues with friends and family used to be a nightmare. Furthermore, Hanna was soon seen as 'odd' by her friends' parents.
It wasn't until the age of sixteen that Hanna realised she had Selective Eating Disorder (SED). Despite this new knowledge, no solution was found. It wasn't until she lost her job before she started eating normally. Hanna sought therapy from psychologist Felix Economakis, who used hypnosis, a treatment that none of the doctors that she had seen before had mentioned, to get her to eat fruit after one one-hour session, and since then she has been willing to eat anything.... including chips!
(Adapted from Friends Global)
According to paragraph 1, which one is NOT MENTIONED as pizza-based items?
Hệ thống khởi động bằng động cơ điện dùng loại động cơ nào?
Số 62010 khi biểu diễn sang hệ nhị phân cần bao nhiêu byte?
Tập xác định của hàm số \(y=\frac{1}{x}+\sqrt{3-x}\) là
Read the following passage about education in Nepal and mark the letter A, B, C or D on your answer sheet to indicate the best answer to each of the following questions from 31 to 40.
Fashion in the 1960s reflected the cultural and political upheaval of the time. The hippie and counterculture movement gave rise to accessories like Pocahontas-style headbands and medallions worn around the neck. Hippies wore long, loosely fitting clothing, often in natural fabrics. Hemlines rose as Mary Quant popularized the miniskirt and hot pants in her lines for JC Penney. And mod fashion, with brightly colored space-age geometric patterns, shift dresses and colored tights became the mainstream. Both miniskirts and mod-shift dresses were worn with knee-high go-go boots.
[A] During the 1970s, the loose-fitting hippie clothes of the prior decade gave way to exotic fabrics and bell-bottom jeans for men and women. Hemlines continued to rise and hot pants continued their popularity, reflecting the flashy style of the decade. However, the 1970s also saw the popularity of a longer hemline, from mid-calf to ankle-length. [B] Women wore peasant-style clothing, especially blouses with off-the-shoulder necklines and lace trim, with their jeans or skirts. [C] Disco fashion emerged, featuring bright patterns and tightly fitting clothing meant to show off the body. Platform shoes elevated their wearer, male or female, anywhere from two to four inches or more. [D]
The 1980s saw another fashion upheaval. Angst-ridden punks wore Dr. Marten and steel-toed army boots, chains, tartan patterns, and bondage pants. Punks focused more on their hair during this period, dying, teasing and chopping their hair into asymmetric styles that seemed to defy gravity. Women—and men—experimented with heavy eyeliner in a variety of colors. Women wore neon colors, jelly shoes, tight jeans, leggings, leg warmers and oversized sweatshirts. Menswear-inspired looks also became popular for both sexes. Men wore ‘power suits,’ named so for their price, which made these suits a reflection of the wealth of the wearer. Women’s jackets featured big shoulder pads that made the shoulders appear higher and larger, similar to the male silhouette.
1990’s fashion was far more subdued than the garish, wild styles of the 1980s. Bands like Nirvana influenced the grunge look, which featured rock-concert tee shirts under plaid flannel shirts, jeans and long, greasy hair. Overalls were worn by both sexes, either with one or both shoulder straps unhooked. The punk style of the 1870s and 1980s evolved into the goth fashion trend, which featured black clothing, black boots and studded bracelets in the style of bands like Nine Inch Nails and Marilyn Manson.
Which of the following does the passage mainly discuss?
Cáccấp độ tổ chức của thế giới sống là
Mark the letter A, B, C, or D on your answer sheet to indicate the correct arrangement of the sentences to make a meaningful paragraph/letter in each of the following questions.
a. Also, volunteering helps students connect with the community, making them aware of the needs around them.
b. Additionally, students develop such important skills as teamwork and communication ones while they volunteer.
c. Firstly, students who do voluntary work gain valuable real-world experience which can help in their future careers.
d. Finally, doing volunteer work can be a rewarding experience, as students feel good about helping others.
e. High school students benefit from doing voluntary work in many ways.
Cho tập hợp B={x∈R∣(x2−4).(x2+25)=0}. Tập hợp B viết bằng cách liệt kê các phần tử là
Một người bơi dọc trong bể bơi dài 50 m. Bơi từ đầu bể đến cuối bể hết 20 s, bơi tiếp từ cuối bể quay về đầu bể hết 22 s. Xác định vận tốc trung bình trong trường hợp bơi từ đầu bể đến cuối bể.
Gọi \(S\) là tập hợp các số tự nhiên có 6 chữ số. Chọn ngẫu nhiên một số từ \(S\), xác suất để các chữ số của số đó đôi một khác nhau và phải có mặt chữ số 0 và 1 bằng
Hệ quả nào không phải là của khu vực hóa kinh tế?
Đứng đầu mỗi ủy ban trong Công xã Pari là
Năm 1911, Rơ-đơ-pho (E. Rutherford) và các cộng sự đã dùng các hạt α bắn phá lá vàng mỏng và dùng màn huỳnh quang đặt sau lá vàng để theo dõi đường đi của các hạt α. Kết quả thí nghiệm đã rút ra các kết luận về nguyên tử như sau
(1) Nguyên tử có cấu tạo rỗng.
(2) Hạt nhân nguyên tử có kích thước rất nhỏ so với kích thước nguyên tử.
(3) Hạt nhân nguyên tử mang điện tích âm.
(4) Xung quanh nguyên tử là các electron chuyển động tạo nên lớp vỏ nguyên tử.
Số kết luận sai là:
Một tên lửa khối lượng vỏ 200kg, khối lượng nhiên liệu 100kg, bay thẳng đứng lên nhờ nhiên liệu cháy phụt toàn bộ tức thời ra sau với vận tốc 400m/s. Tìm độ cao mà tên lửa đạt tới, biết sức cản của không khí làm giảm độ cao của tên lửa 5 lần. Lấy đơn vị là mét.
Ý nào dưới đây không phải nhược điểm của cơ chế thị trường?
Read the following passage about beginning a lifelong journey and mark the letter A, B, C, or D on your answer sheet to indicate the best answer to each of the following questions from 23 to 30.
Lifelong learning is the continuous pursuit of knowledge and skills throughout a person's life. In today's world, where technology is evolving rapidly, lifelong learning is essential for both personal growth and career development. But just how does one go about becoming a lifelong learner?
The first step in becoming a lifelong learner is to develop a growth mindset. This means believing that your abilities can be developed through hard work, dedication and commitment. Embrace challenges, view mistakes as opportunities to learn, and be open to new ideas and perspectives. Adopting a growth mindset will help you approach learning with a positive attitude and a desire to improve.
Setting achievable goals is also an important part of lifelong learning. Goals give direction and motivation to your learning journey. First, identify areas of interest and choose specific skills or knowledge that you want to develop. Second, break your goals down into smaller, more manageable tasks and track your progress. Finally, celebrate your accomplishments along the way to stay motivated.
People learn in different ways, so it's also essential that you find the methods that work best for you. You can do this by trying out a variety of different learning methods, such as reading, watching videos, attending lectures and engaging in hands-on practice. At the same time, you should also experiment with different learning tools and techniques until you find what works best for you. Becoming a lifelong learner is a continuous process that requires dedication and effort. Developing a growth mindset, setting achievable goals, and using a variety of learning methods are essential steps to becoming a successful lifelong learner. By embracing learning as a lifelong journey, you will continue to grow, develop new skills, and stay relevant in today's ever-changing world.
(Adapted from Bright)
In which paragraph does the writer mention the strategy to maintain motivation?
Bản vẽ kĩ thuật thi công:
Một ô tô có khối lượng 1500kg chuyển động đều qua đoạn cầu cong vòng lên có bán kính cong là 80m với vận tốc 36km/h. Lấy \(g = 10m/{s^2}\). Áp lực mà xe lên cầu khi qua vị trí cao nhất có giá trị là
Khi thiết kế, yếu tố an toàn thể hiện ở chỗ, màu hổ phách:
kWh (ki-lô-oát-giờ) là đơn vị của