Individual Home Task

Variant 3

1. For the triangle with the vertices А (2;5), В (–6;-4), С (6;–3) find:

a) length of the side АВ;

b) equation of the line АМ which parallel to the side ВС;

c) the equation of the hieght ВF;

d) the equation of the median AD;

e) the interior angle ;

f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;

g) area of the treangular АВС.

2. There are given the points М ₁(1;–2;0), M ₂(2;–3;2), of the plane Р ₁: 2x+3y-4z-2=0,

Р ₂: 2x-y+z+9=0 and lines лінії L ₁: ; L ₂: .

а) Write down the equation of the straight line passing through the points М ₁ and M ₂;

b) Find the acute angle between the lines L ₁ and L ₂;

c) Through the point M ₂ draw the plane parallel to the plane Р ₂;

d) Find the acute angle between the planes Р ₁ and Р ₂;

e) Find the distance from the point М ₁ to the plane Р ₂;

f) Through the point M ₂ draw the line parallel to the line L ₂;

g) Find the acute angle between the line L ₂ and the plane Р ₂;

h) Through the point М ₁ draw the line perpendicular to the plane Р ₁;

i) Write down the equation of the plane passing through the point М ₁ and perpendicular to the planes Р ₁ and Р ₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р ₁ and Р ₂.

3. Calculate the limits: а) ; b) ;

c) ; d) .

4. Calculate the limits: а) ; b). .

5. Investigate the function for continuity: .

Module 2. Analitic geometry. Introduction to Calculus

Individual Home Task

Variant 4

1. For the triangle with the vertices А (-5;-2), В (7;6), С (5;-4) find:

a) length of the side АВ;

b) equation of the line АМ which parallel to the side ВС;

c) the equation of the hieght ВF;

d) the equation of the median AD;

e) the interior angle ;

f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;

g) area of the treangular АВС.

3. There are given the points М ₁(1;-2;-4), M ₂(2;–1;3), of the plane Р ₁: 5x+3y-4z+8=0,
Р ₂: x-y+z+5=0 and лінії L ₁: ; L ₂: .

а) Write down the equation of the straight line passing through the points М ₁ and M ₂;

b) Find the acute angle between the lines L ₁ and L ₂;

c) Through the point M ₂ draw the plane parallel to the plane Р ₂;

d) Find the acute angle between the planes Р ₁ and Р ₂;

e) Find the distance from the point М ₁ to the plane Р ₂;

f) Through the point M ₂ draw the line parallel to the line L ₂;

g) Find the acute angle between the line L ₂ and the plane Р ₂;

h) Through the point М ₁ draw the line perpendicular to the plane Р ₁;

i) Write down the equation of the plane passing through the point М ₁ and perpendicular to the planes Р ₁ and Р ₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р ₁ and Р ₂.

3. Calculate the limits: а) ; b) ;

c) ; d) .

4. Calculate the limits: а) ; b) .

5. Investigate the function for continuity:

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