Vit

1. The parabola P has equation y2 = 2 p x. A variable point P has coordinates (p/2,t). The parameter is the real number t greater than p . Calculate the the tangent of the sharp angle between the tangent lines through P.

From the theory about the tangent lines through a given point P(xo,yo), we know that the slopes are given by ______________ | 2 yo + \| yo - 2 p xo m1 = --------------------2 xo ______________ | 2 yo - \| yo - 2 p xo m2 = --------------------2 xo In our case here, xo = p/2 and yo = t. So, the slopes are ______________ | 2 2 t + \| t - p m1 = --------------------p ______________ | 2 2 t - \| t - p m2 = --------------------p The tangent of the angle between the lines is given by m1 - m2 ----------1 + m1 m2 Now,

__________ | 2 2 2 \| t - p m1 - m2 = -------------and m1.m2 = 1 p Then the tangent of the angle between the lines is __________ | 2 2 \| t - p ------------p 2.Calculate the cartesian equation of the plane containing the point A(1,2,3) and

parallel to the lines b and c b: 4x = 3y ; z = 2

and c:

-5x + 3y + 2=0 ; x + z = 4

First, we calculate direction numbers of the lines b and c. Take two points on b. B(0,0,2) and B'(3,4,2). Direction numbers of b are (3,4,0). Take two points on c. C(4,6,0) and C'(1,1,3). Direction numbers of c are (-3,-5,3). The cartesian equation of the plane is | x-1 | 3

y-2 4

z-3 | 0 |

=

0

<=> 4x - 3y - z + 5 = 0

3.Calculate the orthogonal projection A' of point A(1,2,3) on the plane 3x-y+4z = 0. The normal direction to the plane is (3,-1,4). The line from A orthogonal to the plane is / x = 1 | y = 2 \ z = 3

+ r.3 + r.(-1) + r.4

The variable point P(1 + r.3, 2 + r.(-1),3 + r.4) of that line is in the plane if and only if 3(1 <=>

+ r.3)-(2

+ r.(-1))+4(3

+ r.4) = 0

r = -1/2

The point A' is (-1/2, 5/2, 1). 4.Take a plane x + y - z = 1 and point A(1,2,-3). A line l has equations / x = 1 + r.3 | y = 2 + r.(-1) \ z = 3 + r.4 Calculate the coordinates of a point B of line l, such that AB is parallel to the plane.

Consider B(1+3r,2-r,3+4r) as a variable point of line l. The direction numbers of AB are (3r,-r,6+4r). The normal direction to the plane is (1,1,-1). AB is parallel to the plane

<=> AB is normal to the direction <=> 3r -r-6-4r = 0 <=> r = -3 So, B is point (-8,5,-9)

(1,1,-1)

5. Take a point A(1,2,0). A line l has equations / x = 1 | y = 2 \ z = 1

+ r - r + r

Calculate the coordinates of the points B of line l, such that |AB| is sqrt(6). Consider B(1+r,2-r,1+r) as a variable point of line l. The coordinates of vector AB are (r,-r,1+r). Now, ||AB|| must be sqrt(6). <=> 3r.r +2r +1 = 6 <=> r = 1 or r = -5/3 The points are (2,1,2) and (-2/3, 11/3, -2/3) 6. Define all real values of m such that the asymptotes of the curve 2(m-1)x - m + 1 y = ---------------(m+3)x + m intersect in a point above the line y = 2x-1

-m The vertical asymptote is x = ------m+3 The horizontal asymptote is Now, we must have 2(m-1) ------ > m+3

<=>

2(m-1) y = ---------m+3

y > 2x - 1 -2m ---- -1 m+3

<=> ... <=>

5m + 1 --------- > 0 m + 3

<=> (5m + 1)(m + 3) > 0 The m values are defined by

( m <-3

or m > 1/5 )

7.Find real values of the number a for which a.i is a solution of the polynomial equation

z4 - 2z3 + 7z2 - 4z + 10 = 0. Then find all roots of this equation. Since a.i is a solution of the equation, we have <=> <=> <=> <=>

(a.i)4 - 2(a.i)3 + 7(a.i)2 - 4(a.i) + 10 = 0 a4 + 2.i.a3 - 7a2 - 4.i.a + 10 = 0 a4 - 7a2 + 10 = 0

and 2a3 - 4a = 0

a2 = 2 a = sqrt(2) or a = - sqrt(2)

Now, we know that sqrt(2).i and - sqrt(2).i are roots of z4 - 2z3 + 7z2 - 4z + 10 = 0 . This means that z4 - 2z3 + 7z2 - 4z + 10 is divisible by (z - sqrt(2).i)(z + sqrt(2).i) = z2 - 2. The quotient is (z2 - 2 z + 5) . The roots of this polynomial are 1 + 2 i and 1 2 i. The four roots of the given equation are sqrt(2).i -sqrt(2).i 1 + 2 i 1 - 2 i. 8. a, b, c are real numbers in the polynomial p(z) = 2 z4 + a z3 + b z2 + c z + 3 . Find a such that the numbers 2 and i are roots of p(z) = 0.

Since all the coefficients of p(z) are real, -i is a root of p(z) = 0. Let 2, i, -i, w be all the roots. The sum of the roots

= 2 + w = -a/2.

The product of the roots = 2w = 3/2. From this we find w = 3/4 and a = -11/2 9.