First Exam
1. Let 
be a linear transformation expressed by a matrix 
on the 
plane. Answer the following questions:(1) Prove that there exists 2 lines passing through the origin 
such that all points of the lines are mapped to the same lines, then find the equation of the lines.
(2) Find the area
of the figure enclosed by the lines obtained in (1) and the curve 
.
(3) Find
be a linear transformation expressed by a matrix on the plane. Answer the following questions:(1) Prove that there exists 2 lines passing through the origin such that all points of the lines are mapped to the same lines, then find the equation of the lines.(2) Find the area
of the figure enclosed by the lines obtained in (1) and the curve .(3) Find
2. For a real number 
, let 
.
(1) Find the minimum value of
.
(2) Evaluate
.
3.For constant
, 2 points 
move on the part of the first quadrant of the line, which passes through 
and is perpendicular to the axis, satisfying 
. Let a circle with radius 1 centered on the origin intersect with line segments 
at 
respectively. Express the maximum area of 
in terms of 
.
4. On a plane, given a square
with side length 1 and a line which intersects with . For the solid obtained by a rotation of about the line as the axis, answer the following questions:
(1) Suppose that the line
on a plane the same with isn't parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with . Note that the line as axis of rotation is parallel to .
(2) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with.
Second Exam
1. Consider a curve
on the -
plane expressed by 
.
For a constant
, let the line pass through the point 
and is perpendicular to the -axis,intersects with the curve at 
. Denote by 
the area of the figure bounded by the curve , the -axis, the -axis and the line , and denote by 
the area of . Find 
2. For a positive real number
, in the coordiante space, consider 4 points 
.
Let
be the radius of the sphere 
which is inscribed to all faces of the tetrahedron 
.
When moves, find the maximum value of 
, let .(1) Find the minimum value of
.(2) Evaluate
.3.For constant
, 2 points move on the part of the first quadrant of the line, which passes through and is perpendicular to the axis, satisfying . Let a circle with radius 1 centered on the origin intersect with line segments at respectively. Express the maximum area of in terms of .4. On a plane, given a square
with side length 1 and a line which intersects with . For the solid obtained by a rotation of about the line as the axis, answer the following questions:(1) Suppose that the line
on a plane the same with isn't parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with . Note that the line as axis of rotation is parallel to .(2) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with
.Second Exam
1. Consider a curve
on the - plane expressed by .For a constant
, let the line pass through the point and is perpendicular to the -axis,intersects with the curve at . Denote by the area of the figure bounded by the curve , the -axis, the -axis and the line , and denote by the area of . Find 2. For a positive real number
, in the coordiante space, consider 4 points . Let
be the radius of the sphere which is inscribed to all faces of the tetrahedron .When
moves, find the maximum value of 
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