Advanced Programme Mathematics:Algebra and Calculus

Advanced Programme Mathematics:Algebra and Calculus

GRADE 12 FET
PRELIMINARY EXAMINATION 2016

BRIDGE HOUSE

MATHEMATICS DEPARTMENT

Advanced Programme Mathematics:Algebra and Calculus

Time: 2 hours200 marks

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

  1. This question paper consists of 7 pages and 11 questions. Please check that your paper is complete.
  2. Please make sure you get a separate formula sheet and answer sheet.
  3. Read the questions carefully.
  4. Answer all the questions.
  5. Number youranswers exactly as the questions are numbered.
  6. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator.USE RADIAN MEASURE!
  7. Round youranswer to two decimal digits where necessary.
  8. All the necessary working details must be clearly shown.
  9. It is in your own interest to write legibly and to present your work neatly.

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GRADE 12: APM: Algebra and Calculus
PRELIMINARY EXAMINATIONS 2016Page 1 of 7

QUESTION 1

Given , prove by mathematical induction that:

[14]

QUESTION 2

(Showing all relevant working, answers rounded off to the second decimal place.)

  1. Solve for in (5)
  2. Hence solve the equation (6)
  3. Solve for in the following equation.
    (9)
  4. Solve the equation correct to 3 decimal places.(7)
  5. A curve has equation . Find the equation of the tangent to the curve at the point where the line intersects the curve. (11)

[38]

QUESTION 3

It is given that is a root to the equation , . Showing your working, find the value of , and write down the other complex root of this equation.

[9]

QUESTION 4 (USE RADIANS)

The diagram shows the curve for .

  1. Show that .(11)
  2. Find the exact value of the area of the region enclosed by this part of the curve and the . Show your working. (14)

[25]

QUESTION 5

Given the split function

Determine the value(s) of and if is differentiable at .

[9]

QUESTION 6

Consider the following diagram showing and inscribed cone with base radius and height and sphere with P the centre.

Answer the questions below in order to find the volume of the largest right circular cone that can be inscribed in a sphere of radius . The radius (r) should be treated as an constant.

  1. First give in terms of and .(4)
  2. Now determine the maximum height in terms of . Let be a constant value.(7)
  3. Using your answer in (b), determine the maximum volume in terms of .(6)

[17]

QUESTION 7

Given:

  1. Calculate the and intercepts of this function.(5)
  2. Find the equations of all the asymptotes.(8)
  3. Determine .(4)
  4. Determine the of the turning points of .(6)

[23]

QUESTION 8 (RADIANS!)

The diagram shows a circle with centre A and radius . Diameters CAD and BAE are perpendicular to each other. A larger circle has centre B and passes through C and D.

  1. Show that the radius of the larger circle with centre B is .(4)
  2. Find the area of the shaded region in terms of .(13)

[17]

QUESTION 9

Consider the equation:

  1. By sketching a suitable pair of graphs, show that the equation has exactly one real root, . Showing intercepts with axes. (7)
  2. Using your calculator solve for . Rounding your answer off to two decimal places. (2)
  3. Now using the Newton-Raphson iterative method, showing all working, solve this equation to 4 decimal places. Given that . (8)

[17]

QUESTION 10

Given the functions and .

  1. Given the-coodrinates of intercepts at and .
  2. Determine the shaded area between the curves using a Riemann Sum. Given that: and . [12]

QUESTION 11

The diagram shows part of the curve and point lying on the curve. The line intersects the at .

  1. Show that is a normal to the curve.(9)
  2. Determine the equation of the line .(3)
  3. Now, showing all working, find the exact volume of revolution obtained when the shaded region is rotated about the . (7)

[19]

TOTAL FOR THIS PAPER: 200 MARKS

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