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9000123105

Level:

Find the all values of the real parameter

p

which ensure that the line

q : y = x - 1

is a tangent to the parabola

x^{2} = 2 p y .

p = 2

p \in {0; 2}

p = - 2

p \in {- 2; 0}

9000123104

Level:

In the following list identify a line which is a tangent to the ellipse

(x - 2)^{2} + \frac{y^{2}}{9} = 1

\begin{aligned} p : x & = 3 + t, \\ y & = 3; t \in R \end{aligned}

p : x = 2

p : y = 3 x

p : y = - x - 2

9000123107

Level:

In the following list identify a line such that the line has a unique intersection with the hyperbola

x^{2} - y^{2} = 5

but the line is not the tangent to this hyperbola.

p : \frac{x}{5} + \frac{y}{5} = 1

p : y = 5 x

p : 2 x + y = 5

\begin{aligned} p : x & = 1 \\ y & = - 1 + t; t \in R \end{aligned}

9000123103

Level:

The ellipse

5 x^{2} + 9 y^{2} = 45

has tangent

2 x + 3 y = 9

. Find the values of the real parameter

k

which ensure that the line

y = k x + 3

is a secant for the ellipse.

k \in (- \infty; - \frac{2}{3}) \cup (\frac{2}{3}; \infty)

k \in [- \frac{2}{3}; \frac{2}{3}]

k \in (- \frac{2}{3}; \frac{2}{3})

k \in (- \infty; - \frac{2}{3}] \cup [\frac{2}{3}; \infty)

9000124503

Level:

A tall radio mast is attached by several cables. The length of each cable is

30 m

and all cables are attached

2 m

under the top of the mast. The second end of the cable is anchored to the ground. The cable is in the height

6 m

if measured directly above the point which is in the distance

8 m

from the point where the cable is anchored to the ground. Find the height of the mast.

20 m

24 m

22.5 m

24.5 m

9000123101

Level:

Find the values of the real parameter

q

which ensure that the line

y = q

is a tangent to the circle

x^{2} + y^{2} + 4 x - 8 y + 4 = 0.

{0; 8}

{- 6; 2}

{- 8; 0}

{- 2; 6}

9000123102

Level:

Find a true statement about the ellipse

x^{2} + 4 y^{2} - 8 y = 0.

The tangent to the ellipse can pass through any point on the line

y = - 1

The tangent to the ellipse can pass through any point on the line

x = 1

The tangent to the ellipse can pass through the point

[- 1; 1]

The tangent to the ellipse can pass through any point on the line

y = 1

9000123106

Level:

Find the tangent line

q

to the parabola

4 (y - 2) = (x + 1)^{2}

, so that the tangent

q

is parallel to the line

p : 4 x - 5 y + 17 = 0.

q : 20 x - 25 y + 54 = 0

q : 20 x - 25 y - 27 = 0

q : 4 x - 5 y + 27 = 0

q : 4 x - 5 y - 17 = 0

9000123108

Level:

Find all the tangents to the hyperbola

x^{2} - 2 y^{2} = 8

such that the angle between each tangent and the

x

-axis is

45^{\circ}

y = x + 2, y = x - 2, y = - x + 2, y = - x - 2

y = x + 2, y = x - 2

y = x + 2, y = - x + 2

y = x + 2

9000124502

Level:

A rectangle-shaped land has dimensions

3 \times 5 cm

on a map with scale

1 : 2 000

. The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions

4 \times 5 cm

on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.

40 m

20 m

80 m

10 m

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