Example Exam Wiskunde B

Transcription

1 CENTRALE COMMISSIE VOORTENTAMEN WISKUNDE Date: Autumn 2018 Time: 3 Hours Questions: 6 Example Exam Wiskunde B Please read the instructions below carefully before answering the questions. Failing to comply with these instructions may result in reduction of points. Make sure your name is clearly written on every answer sheet. Take a new answer sheet for every question. Show all your calculations clearly. Illegible answers and answers without a calculation or an explanation of the use of your calculator are invalid. Write your answers in ink. Do not use a pencil, except when drawing graphs. Do not use correction fluid. You can use a basic scientific calculator. Other equipment, like a graphing calculator, a calculator with the option of computing integrals, a formula chart, BINAS or a book with tables is NOT permitted. On the last page of this exam you will find a list of formulas. You can use a dictionary if it is approved by the invigilator. Please switch off your mobile telephone and put in your bag. Points that can be scored for each question: Question a b c d 7 Total You will pass the exam if your grade is at least 5.5.

2 Question 1 Source: Centraal Examen vwo 2018 tijdvak 2 The function is given by. Also given is the function by For we have. 4pt a Prove that for we indeed have Furthermore, the function. is given by. There is a straight line k that for coincides with the graph of. In figuur 1 the graphs of and are shown. Point is the perforation of both graphs. In figuur 2 the graph of, line k and their intersection P are shown. We have: The graphs of and are perpendicular in their perforation P if the graph of and line k are perpendicular in their intersection P. 6pt b Prove that the graphs of and are perpendicular in their perforation P. Example exam 2 Wiskunde B Autumn 2018 page 2

3 Question 2 Source: CCVX entrance exam wiskunde B July 2018 In the figure below, the graphs are shown of the functions and. The vertical line intersects the graph of in point P and the graph of in point Q. The distance between point P and point Q equals. 4pt a Use an exact computation to find the value of. The graph of intersects the y-axis in point R. The graph of intersects the y-axis in point S. The tangent to the graph of in point R and the tangent to the graph of in point S intersect in point T. 7pt b Compute exactly the x-coordinate of point T and simplify the answer. Furthermore, the function is given by. 5pt c Compute exactly the coordinates of the intersection(s) of the graphs of and. V is the bounded region enclosed by the graph of, the x-axis and the y-axis. 7pt d Compute exactly the volume of the solid of revolution that is formed by revolving V around the y-axis. Example exam 2 Wiskunde B Autumn 2018 page 3

4 Question 3 Source: Centraal Examen vwo 2018 tijdvak 2 Given is the square with vertices,, en. Point is on side. See figuur 1. The points, and are on one circle. 6pt a Find an equation for this circle. A point is moving along line segment (from to ). There is a position of point for which line segment is perpendicular to line segment. See figuur 2. 6pt b Compute exactly the coordinates of point when it is in this position. In figuur 3 triangle is shaded. There is a position of point for which the area of triangle is one third part of the area of square. 6pt c Compute exactly the coordinates of point when it is in this position. Example exam 2 Wiskunde B Autumn 2018 page 4

5 Question 4 Source: CCVX entrance exam wiskunde B July 2018 (adjusted) For each, the function is given by. In the figure below, the graph is shown of the function. As you can see, the graph of has two points of inflection. 5pt a Compute exactly the x-coordinates of these two points of inflection. Like, all functions have a minimum in (0,0). They also have a maximum. The point on the graph where has a maximum is denoted by. The points are all on the same parabola. 6pt b Use an exact computation to find an equation for this parabola. There are two tangent lines to the graph of with an equation of the form, Therefore, these tangent lines pass through the origin (0,0). 7pt c Use an exact computation to find equations for these two tangent lines. Example exam 2 Wiskunde B Autumn 2018 page 5

6 Question 5 Source: Centraal Examen vwo 2018 tijdvak 2 For, the position of a moving point is given by the parametric equations { The path of the moving point is shown in figuur 1. At and, the moving point is in O. This situation is disregarded in the entire question. is the position of the moving point at time t. We have: The line through and is vertical for every value of that is possible in this situation. 4pt a Show that this line is indeed vertical. There are several times at which the distance of of to the y-axis. to the x-axis is twice the distance 5pt b Compute exactly the fourth time at which this is the case. For each value of t, the velocity vector from point and the vector can be drawn. In figuur 2, point, vector and vector are drawn for. 5pt c Show that for we have: Example exam 2 Wiskunde B Autumn 2018 page 6

7 Question 6 Source: New The family of functions is given by For, the graph of the function intersects the parabola with equation in the point (2,0). 5pt a Use an exact computation to determine the value(s) of for which this parabola and the graph of are touching in point (2,0). In the figure below the graph is shown of the function. This graph has two asymptotes. 5pt b Use an exact computation to determine the equations of the two asymptotes of the graph of. Extra item: What do the graphs of and look like? End of the exam. Is your name on all answer sheets? Example exam 2 Wiskunde B Autumn 2018 page 7

8 Formula list wiskunde B Example exam 2 Wiskunde B Autumn 2018 page 8

9 Extra question 1 Source: CCVX entrance exam wiskunde B July 2018 In the figure below the graph is shown of the function. Line l is the tangent line to the graph of at the intersection with the y-axis. T is the triangle enclosed by line l, the x-axis and the y-axis. 6pt a Compute exactly the area of triangle T. V is the bounded region enclosed by the graph of, the x-axis and the y-axis. 6pt b Compute exactly the area of region V. Furthermore, the function is given by. 5pt c Solve exactly the equation. Extra question 2 Source: CCVX entrance exam wiskunde B July 2018 The functions and are given by and. 7pt a Solve the equation exactly and find all solutions in the interval [ ]. 7pt b Use the derivative function to find the minimal and the maximal value of exactly. The function is given by, 4pt c Show that can be written as. 4pt d Compute exactly: Example exam 2 Wiskunde B Autumn 2018 page 9