1.
The fundamental objects of study in Algebraic Geometry are
Correct Answer
A. Algebraic Varieties
Explanation
Algebraic varieties are the fundamental objects of study in Algebraic Geometry. They are sets of points defined by polynomial equations in several variables. These objects are important in understanding the geometry and properties of solutions to polynomial equations. By studying algebraic varieties, mathematicians can gain insights into the relationships between the solutions of equations and the geometric shapes they represent.
2.
A point of the place belongs to an Algebraic curve if it's coordinates Satisfy a given
Correct Answer
A. Polynomial Equation
Explanation
The correct answer is "Polynomial Equation." A point of a place belongs to an Algebraic curve if its coordinates satisfy a given polynomial equation. This means that the coordinates of the point can be plugged into the polynomial equation, and the equation will hold true. Algebraic curves are defined by polynomial equations, and the points on the curve are the solutions to these equations. Therefore, a point belonging to an Algebraic curve must satisfy a polynomial equation.
3.
A natural class of functions on an Algebraic set is called
Correct Answer
C. Regular functions
Explanation
Regular functions are a natural class of functions on an Algebraic set. These functions are defined and continuous on the entire set, and they can be expressed as a quotient of two polynomials. Regular functions are important in algebraic geometry as they provide a way to study the properties of algebraic sets by examining their regular functions.
4.
Just as with the Regular functions on Affine space, the regular functions on V forms a ring called
Correct Answer
D. Coordinate Ring
Explanation
The regular functions on V form a ring called the coordinate ring. This ring consists of all the functions that are defined on the variety V and can be expressed as a polynomial in the coordinates of V. The coordinate ring is an important concept in algebraic geometry as it allows us to study the geometric properties of V through algebraic techniques.
5.
As regular maps make the collection of all Affine algebraic set into a category this regular maps are known as
Correct Answer
B. Morphisms
Explanation
Regular maps are known as morphisms because they map affine algebraic sets to other affine algebraic sets. In the category of affine algebraic sets, morphisms are the arrows that connect objects (affine algebraic sets) and preserve the structure of the sets. Therefore, regular maps can be considered as morphisms in this category.
6.
If there are two rational functions between two Affine Varieties which are inverse to each other in the region where they are both define. It is said to be
Correct Answer
D. Birationally Equivalent
Explanation
If there are two rational functions between two Affine Varieties that are inverse to each other in the region where they are both defined, it means that these varieties can be transformed into each other by a series of rational transformations. This implies that the two varieties are birationally equivalent.
7.
An Affine Variety is a rational Variety if it is birationally Equivalent to
Correct Answer
B. Affine space
Explanation
An affine variety is considered rational if it is birationally equivalent to an affine space. This means that there exists a rational map between the affine variety and the affine space that is defined by polynomial equations. The concept of rationality in algebraic geometry is closely related to the existence of a birational map between two varieties, which essentially allows for a one-to-one correspondence between points on the varieties. In this case, the correct answer is "Affine space" because it represents the type of variety that an affine variety can be birationally equivalent to in order to be considered rational.
8.
Expressions containing a number of like terms can be simplified by
Correct Answer
D. Grouping them together
Explanation
When we have an expression with a number of like terms, grouping them together means combining the terms that have the same variable or exponent. This simplifies the expression by reducing the number of terms and making it easier to perform operations on them. By grouping like terms together, we can easily combine or cancel out terms with the same variable or exponent, resulting in a simplified expression.
9.
Algebraic Expression can be simplified by Grouping like terms, Removing Brackets and
Correct Answer
A. Expanding Bracket
Explanation
Expanding the bracket is a method used to simplify algebraic expressions. It involves multiplying each term inside the bracket by the term outside the bracket. This is done by using the distributive property. By expanding the bracket, the expression becomes less complex and easier to work with.
10.
What is important when substituting values into an Algebraic?
Correct Answer
A. Order of operation
Explanation
When substituting values into an algebraic expression, it is important to follow the order of operations. This means performing operations inside parentheses first, then doing any multiplication or division from left to right, and finally performing any addition or subtraction from left to right. By following the order of operations, we ensure that the expression is evaluated correctly and the correct answer is obtained.