1d Determining the derivative 1 Solving 2 First alternative: Determine the points where the graphs of and meet:

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1 Marking scheme for the VWO Mathematics-B Exam of 3 July 04 Question. a. Determining the coordinates of point Taking the first derivative: Calculating the slope of the tangent line: Determining a formula for the tangent line: b Setting up and rewriting the equation Reducing the equation to: or Solving c Determining the integration limits: Expanding Taking the anti-derivative of Calculating the volume of revolution: d Determining the derivative Solving First alternative: Determine the points where the graphs of and meet: Conclusion: The graphs of and meet at the points where attains its extreme values. extreme values.. Conclusion: the graphs of and meet when attains it

2 Question. a. First alternative: Performing the calculation: Performing the calculation: Subsequently equating and cross multiplying: Simplifying the result: Solving: b. Applying both the quotient and chain rule: Simplifying the numerator: Factorizing the numerator: c Setting up the equation: Solving the equation: Concluding that there is one solution: so there is one extreme value. d Application of the difference and chain rule: Equalizing the denominators: Simplifying: e Determinating the integration limits of V: en Calculating the integral: Calculating the integral: Total surface area

3 Question 3. 3a. First alternative:. (identity). (given fact) 3. (given fact) 4. From,, and 3: (SSS congruence) 5. From 4 follows that and, so bisects both and. From (given fact) follows (isosceles triangle) and from (given fact) follows (isosceles triangle)..,, en (constant angle) 3. From en follow that and, so bisects both and 3b. First alternative:. and (sum of angles in a triangle). From question 3a. follows that and 3. From and we conclude: 4. (cyclic quadrilateral) 5. From 3 and 4 we conclude:. From (given fact) follows (isosceles triangle) ans from (given fact) follows (isosceles triangle). Because (identity) and (identity) the results of and yield 3. (cyclic quadrilateral) 4. The conclusions of and 3 yield: Third alternative:. Let M be the centre of the circle.. From (given fact) follows (central angle) 3. From (given fact) follows (central angle) 4. Because (circle) the conclusions of and 3 yield that 5. From 4 follows (Thales/circumferential angle) Fourth alternative:. bisects both and (question 3a) so and (cyclic quadrilateral) 5. From 3 and 4 follow:

4 3c.. (circumferential angle). (bisector) 3. Conclusions and yield, question 3b gives so: Analogously: 3d. First alternative:. (question 3b.) (Thales). From (question 3c.) and yield 3. (radius circle). 4. and 3 yield that is a square [because the diagonals have the same length and intersect each other midway]. and (constant angle and question 3b.). From question 3c: From we conclude (Thales) 3. Conclusion yields (Thales) and because we conclude 4. angle). 5. Conclusion, and 4 yield that PAQC is a square. Third alternative:. Analogously to question 3c we have: (central and The diagonals AC and PQ therefore intersect perpendicularly at point M.. (radius circle) diagonals and intersect each other midway perpendicularly PAQC is a square. Fourth alternative:. Analogously to question 3c. We have: and. (conclusion and question 3c.) yield (central angle) 3. (conclusion and question 3c.) yield that AC and PQ are diameter lines. Thales theorem now yields. 4. From and 3 we conclude that PAQC is a square 3

5 Question 4. 4a Determining the derivative Solving Evaluating 4b Solving Determining Setting up the equation Determining b: 4c Taking the second derivative: Solving (of equivalent) 4d First alternative: 4e Surface area equals

6 Question 5. 5a Setting up the equation and applying a double-angle formula: Factorizing Solving for t: Calculating the -coordinates: 5b Setting up the equation and applying a double-angle formula: Simplifying: Reducing and solving for t: Determining the -coordinates: en 5c Taking the -derivative of : Taking the -derivative of : Substituting. Note. In thise case the candidate does not have to give the entire formule for v. 5d 5e The -axis is an axis of symmetry of curve K. Explanation: The mirror image of each point of curve K in the -axis is again a point of curve K.