Trong không gian , cho mặt phẳng . Vectơ nào sau đây là vectơ pháp tuyến của ?

Read the following passage about and mark the letter A, B, C, D on your answer sheet to indicate the best answer to each of the following questions from 31 to 40.

    Some people look at an equation and see a bunch of complicated numbers and symbols while others see beauty [I]. Now, thanks to a new tool created at Carnegie Mellon University, anyone can now translate the abstractions of mathematics into beautiful and instructive illustrations [II]. This exciting new tool is named Penrose after the mathematician Roger Penrose, who is famous for using diagrams and other drawings to communicate complicated mathematical ideas [III]. Penrose enables users to create diagrams simply by typing mathematical expressions and letting the software do the drawing [IV].

    Unlike a graphing calculator, these aren’t restricted to basic functions, but can be complex relationships from any area of mathematics. “Some mathematicians have a talent for drawing beautiful diagrams by hand, but they vanish as soon as the chalkboard is erased,” said Keenan Crane, an assistant professor of computer science and robotics. “We want to make this expressive power available to anyone.”

    Diagrams are often underused in mathematical communication, since producing high-quality illustrations is beyond the skill of many researchers and requires a great deal of time and effort. Penrose addresses these challenges by letting diagram-drawing experts turn their knowledge about creating diagrams into computer codes so that other users can access this capability using familiar mathematical language and a computer. “We started off by asking: ‘How do people translate mathematical ideas into pictures in their head?’” said Katherine Ye, a Ph.D. student in the Computer Science Department who is involved in the development of Penrose. “The secret sauce of our system is to empower people to easily ‘explain’ this translation process to the computer, so the computer can do all the hard work of actually making the picture.”

    Once the computer learns how the user wants to see a mathematical object visualized – a vector represented by a little arrow, for instance, or a point represented as a dot – it uses these rules to draw several candidate diagrams. Users can then select and edit the diagrams they want from a gallery of possibilities. A special, simple-to-learn programming language was also developed so that they can easily convey the ideas in their minds to the Penrose system, Crane said. “Mathematicians can get very picky about notations,” he explained. “We let them define whatever notation they want, so they can express themselves naturally.”

    The researchers will present Penrose at the SIGGRAPH 2020 Conference on Computer Graphics and Interactive Techniques, which will be held this July. “Our vision is to be able to dust off an old math textbook from the library, drop it into the computer and get a beautifully illustrated book - that way more people understand,” Crane said, noting that Penrose is a first step toward this goal.

(Adapted from sciencedaily.com)

Vào mùa hè, người ta sử dụng các thực phẩm

Read the following passage about music and mark the letter A, B, C, or D to indicate the option that best fits each of the numbered blanks from 31 to 40.

SOUND FROM THE PAST

Much of the music we listen to today is a mixture of styles from various countries and time periods. A lot of music has roots in older traditional songs heard in many different countries around the world. Traditional, or folk, music is collected over decades, if not centuries. Younger generations learn these songs from their elders through practice and repetition.

Since music can tell us a lot about different cultures through its lyrics, melodies, and the instruments used, researchers and music fans see music as an essential part of history. They fear that traditional and older types of music are slowly disappearing, partly because they are less likely to be written down or recorded, or because the format in which they are recorded is no longer in use. Also, younger generations may not find such music very appealing, so once older generations pass away, the music may die out with them. Whole genres of music may go extinct.

There is a growing effort to preserve music in its many forms. Some researchers create field recordings – those made outside of a recording studio – to capture live performances. For example, in the early 1900s, social scientist Frances Densmore made recordings of Native American songs that had been sung for many generations but were in danger of falling by the wayside. Researchers also transcribe old music by listening to old recordings – sometimes the only one of its kind left. They try to write out the music so that it can be studied and played by modern musicians.

[I] Collecting music is another form of music preservation. [II] Some collectors are extremely passionate about their music and will spend a lot of time and money looking for things that have not been produced or sold for many years. [III] Their efforts help to document the music of different cultures, genres, time periods, and places. [IV] There are also associations and societies to preserve and celebrate very specific kinds of music.

Now, modern technology makes it much easier to preserve music. Smartphones can be used to record music, while the Internet lets us share these recordings and connect with like-minded enthusiasts. Soon, losing a piece of music may be a thing of the past.

Which of the following best summarizes the passage?