Obtain the points where both functions intersect.
Draw their graphs in the same coordinate system.
At what interval are the function values of the function f greater than the value of the function g?

Find the angle of slope of the line:

The equation for a circle is given:

Write down the coordinates of the centre of the circle and its radius.
At what angle does the circle intersect the abscissa axis at the point ?

What is the length of the chord that the parabola cuts from the line ?

Find the angle of inclination of the line:

At what angle does the parabola and ellipse intersect?

Determine the acute angle at which the function intersects the abscissa axis.

Given a circle with radius 6 m and centre O. The points and are drawn on the circle so that the angles and are right angles (as shown in the diagram below for one of the possible drawn situations). The size of the angle depends on the selection of the positions of the points and on the circle.

Prove that for any points and drawn in this way, the area of the shaded area as a function of the size of the angle is equal to , if the angle is expressed in radians.
For which obtuse angle is the area of the shaded area the largest? Round the angle to the nearest tenth of a degree.
The diagram above also shows the cross-section of a tunnel with a plane perpendicular to the direction of travel. The circle has a radius of , the width of the tunnel is , and its length is . The curved ceiling and walls of the tunnel will be painted over. For this purpose, 400 l of paint was bought. Will there be enough paint if they use exactly 5 l of paint for ? Justify your answer.

10.

What is the length of the chord that the parabola cuts off from the line ?

11.

Calculate the angle between and .

12.

Draw the points A, B, C and D in the coordinate system below and obtain the equation of the triangle of the outlined circle.
Calculate the area of quadrilateral ABCD
Rotate the triangle ABC by around the side AC to get the geometric shape G. Prove that the volume of the solid G is , where and .
Calculate the abscissa of the point T on the x-axis so that the sum of the squares of the distances from the point T to the point A and from the point T to the point C is minimal.

13.

At what points of the curve does the tangent have an inclination angle ? Hence, calculate the equation of normal to curve at this point.

14.

Given four trapeziums ABCD where the bases and , and the length of the side .

In the first trapezium, the non-parallel sides intersect at the point E. Calculate the length of the distance DE.
In the second trapezium, the size of the angle is equal to . Calculate the size of the angle . Round the result to the nearest minute.
In the third trapezium, the diagonal BD measures . Calculate the length of .
In the fourth trapezium, the size of the angle is equal to . There is a point T on the side AD and a point V on the side AB such that . Calculate at what length the area of the triangle VBT will be maximum.

15.

Calculate the angle at which the function intersects the ordinate axis.

16.

A quadrilateral is orthodiagonal if its diagonals intersect at right angles. The area of any orthodiagonal quadrilateral with diagonal e and f is given as .

Let be such points on the unit circle that the quadrilateral ABCD is a trapezium with bases AB and CD. The diagonals of the trapezium intersect at the point P.

Let O be the origin of the coordinate system and be the size of the angle AOB.

Prove that trapezium ABCD is isosceles. Draw a sketch.
Calculate the sizes of the angles and and prove that the trapezium ABCD is orthodiagonal.
Express the coordinates of the point B and the length of the diagonal in term of .
Express the area of the trapezium ABCD in term of . Which point B on the unit circle will give the maximum area of the trapezium ABCD? Justify your answer.

17.

Determine whether the line is a tangent to the parabola .

18.

Determine the angle between the curves and .

19.

At what angle does the curve intersects the abscissa axis?

20.

At what angle does the graph of the rational function:

intersect the abscissa axis? Round up the result to the nearest hundredths of degrees.

21.

Determine whether the line is a tangent to the parabola .

22.

At what angle does the curve with the equation intersect the abscissa axis?

23.

Determine whether the line is a tangent to the parabola .

24.

What is the angle of inclination of the tangent to the function at the point where the abscissa is 2?

25.

Do the line and the parabola have a contact point?

26.

Find the point of intersection between the circle and the line .

27.

At what angle does the graph of the function intersect the ordinate axis?

28.

Do the line and the parabola have any point of contact?

29.

Calculate the point of intersection and the angle between the lines and .

30.

Do the line and the parabola have a contact point?

31.

Determine the equation of the line on which the common chord of the circles and lies.

32.

Calculate the point of intersection and the angle between the functions and .

33.

Do the line and the parabola have a contact point?

34.

Given the function , obtain the zeros, the intersection with the ordinate axis and local maxima.

35.

Calculate the area of a triangle that has vertices at the foci of the hyperbola and the coordinate origin.

36.

Calculate the point of intersection and the angle between the curves:

37.

Given the function , obtain the zeros, intersection with ordinate axis, the stationary points and hence draw the graph.

38.

Determine the tangent point between the line and the hyperbola .

39.

Given the function , obtain the zeros, intersection with the ordinate axis, the stationary points and sketch the graph of the given function.

40.

Determine the equation of a hyperbola that has foci and , and its imaginary pole measures b = 5.

41.

Given the function . Obtain the zeros, domain, intersection with the ordinate axis, asymptote and the stationary points.

42.

Given the function obtain the zeros, domain, intersection with ordinate axis, asymptotes and the turning points.

43.

A hyperbola and the ellipse have common foci. Determine the equation of the hyperbola.

44.

Given the function obtain the zeros, points of discontinuity, intersection with ordinate axis, asymptotes and the turning points.

45.

What is the angle of inclination of the function at the point of intersection with the ordinate axis?

46.

At what point and at what angle does the function intersects the abscissa axis?

47.

Calculate the intersections of the parabola and the line .

48.

Given the function , obtain the zeros, the intersection with the ordinate axis and the turning point.

49.

Given the parabola

and the line

Accurately draw the parabola and the line in the cartesian coordinate system.
Calculate the abscissas of the intersections of the parabola and the line.
For which x does the line lie above the parabola?

50.

A quadratic function is given

Calculate the zeros, y-intercept, vertex of the function and plot its graph in the given coordinate system.
Obtain the equation of the tangent to the graph of the function at the point .
Calculate the distance of point from the coordinate origin. Round the result to two significant figures.

51.

Calculate the intersection of the parabola and the line . Also calculate the angle at which the line and the parabola intersect in the first quadrant. Round the result to the nearest hundredth of a degree.

52.

The tangent to the graph of the function:

is parallel to the line:

which touches the function at two points. Calculate the exact coordinates of the points of intersection and .

53.

The picture shows a part of the graph of the derivable function f, which asymptotically approaches the abscissa axis and is symmetrical with respect to the ordinate axis. The function f has no zeros.

Copy and complete the table below:

Domain
Range
Coordinates of the intersection of the graph of the function f with the ordinate axis
Value of function f at
Is function f even or odd? Justify your answer.
Record the value of

54.

Given the parabolas and .

Calculate the coordinates of point P of the intersection of the parabolas.
Obtain the equations of the tangents to the parabola at their point of intersection.
Calculate the angle between the parabolas.

55.

Obtain the equation of a circle centred on the line , the lines and are the tangents of the circle .

56.

A line touches the circle at a point . Obtain the equation of this line.
What is the relative position of the line and the circle ? Justify your answer.
Compute the real number a such that the x axis is a tangent to the circle .
Calculate the real number b such that the center of the circle lies on the line .

57.

A circle K and a straight line p are drawn in the picture.

Write down the equations of the circle and the line.
Calculate the area of the minor segment bounded by the line p and the circle K.

58.

The line with the equation intersects the x axis at C. Write the equation of the circle with centre C passing through the point .

59.

Circle K has a centre at the point and touches the ordinate axis (axis y). The line with the equation intersects the circle K at the points A and B.

Write down the equation of the circle K and calculate the coordinates of the points A and B.
Show that the angle ASB is a right angle.
From the origin of the coordinate system we draw two tangents on the circle K. One of the tangents is the ordinate axis. Write the equation of the second tangent and calculate the coordinates of it to the circle K.

60.

In the picture, a line p and a circle k are drawn. The line p passes through the points A and B, the diameter of the circle k is the distance AB.

Write down the equation of the line p.
Write down the equation of the circle k.
Calculate the sum of the areas of the shaded sections in the figure. Solve the task without using a computer.

61.

Find the points of intersections of the parabola and the line .

62.

Given the functions

find the points of intersections of both functions.

63.

Calculate the point of the intersections on the x-axis of the parabola and straight line .

64.

For which value of n is the line of the tangent of the parabola ? Write the tangent of the line and the parabola .

65.

Find the intersections of the parabola and the line given respectively, as:

and

66.

For what values of k will the line touch the parabola ? Obtain the points of contact of tangent of the line and the parabola.

67.

Find the intersections between the parabolas and .

68.

The function is given. Find the equation of the tangent to the parabola that intersects the ordinate axis at , and write the coordinate point of intersection of the tangent.