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BRIDGE HOUSE

PRE-PRIMARY • PREPARATORY • COLLEGE

GRADE 12

SEPTEMBEREXAMINATIONS 2013

MATHEMATICS PAPER 1

Time: 3 hours Total: 150

Read the following instructions carefully:

  1. This question paper consists of 9pages and 2 separate Formula sheets. Please check that your question paper is complete.
  1. Read the questions carefully.
  2. Number your answers exactly as the questions are numbered.
  3. All the necessary working details must be clearly shown.
  4. Approved non-programmable calculators may be used unless otherwise stated.
  5. Answers should be rounded off to two decimal digits where necessary, unless otherwise stated.
  6. It is in your own interest to write legibly and to present your work neatly.
  7. Detach the Answer Sheet and staple it to your answer script.

SECTION A:

Question 1:

1.1.Simplify: (4)

1.2.Solve for x:

  1. (to 1 decimal place)(2)
  2. (4)
  3. (4)
  4. (3)

1.3.Solve for x and y:

(5)

[22]

Question 2:

2.1.The following sequence of numbers forms a quadratic sequence:

  1. Determine an expression for the term of the quadratic sequence.(4)
  2. Explain why the sequence of numbers will never contain a positive term.(1)

2.2.Fifty-five round water pipes are stacked as shown on the figure. Use an applicable formula to determine the number of pipes that must be placed in the bottom layer in order to have one pipe in the top layer. (6)

[11]

Question 3:

3.1.Differentiate from first principles: (5)

3.2.Evaluate if (3)

3.3.If

  1. Determine the gradient of the tangent to the curve of at the point where . (3)
  2. At which point on the curve of will the gradient be (3)

3.4.Consider . Calculate the average rate of change of the function in the interval to . Approximate the answer correct to six decimal places. (4)

[18]

Question 4:

4.1.Sendra received an inheritance of .

  1. If Sendra invests this money at p.a. compounded annually for years, determine the total that she will accumulate. (2)
  2. Sendra has been told that the car that she is keen to buy will cost in years’ time. Determine the rate of interest (to two decimal places) that she needs to negotiate in order to be able to get the car. (4)

4.2.Thando dreams of winning one million rand (on the Lottery and living off the money. Her standard of living requires per month. An interest rate of p.a. compounded monthly is available. She hopes to start her new life style as soon as she deposits her winnings, so she will draw out her first immediately. Determine how long she could live off her winnings. (8)

[14]

Question 5:

If

5.1.Draw a sketch graph of the curve of on the axes provided on the answer sheet. Clearly show the intercept(s) with the axes and also show one other point on the curve of . (3)

5.2.Indicate on the graph, using the letter , where the value of could be read off.(2)

5.3.Write down the equation of , the inverse of , in the form (2)

5.4.Sketch the graph of on the same set of axes as .(3)

[10]

SECTION B:

Question 6:

6.1.Consider the graph of .

  1. Is the graph of a one-to-one function? Explain.(2)
  2. Write down the range of .(2)
  3. On the graph provided on the answer sheet, draw the inverse of .(3)
  4. Explain why the inverse of is not a function.(2)
  5. Write down a possible restriction for the domain of so that the inverse of the graph of will now be a function. (2)

6.2.In the figure and are the turning points of the graph of . Determine the values of in each of the following:

  1. (2)
  2. (2)
  3. (2)
  4. (3)

[9]

[20]

Question 7:

7.1.For what values of will converge?(4)

7.2.A ladder has rungs. The bottom rung is long. Each rung is shorter than the rung beneath it. Determine the total length of wood required to make rungs. (5)

7.3.The term and the term of a geometric sequence are and respectively.

  1. Find the first term and the constant ratio. (5)
  2. Find the number of terms if the last term is .(3)

[17]

Question 8:

8.1.The derivative of is shown by the following graph.

  1. For which values of is increasing?(2)
  2. Give the value(s) of the turning point(s) of and state whether it is a maximum or minimum. (4)
  3. Determine the equation of .(5)

8.2.A tourist travels in a car over a mountainous pass during his trip. The height above sea level of the car, after minutes, is given as metres, where .

  1. How high is the car above sea level when it starts its journey on the mountainous pass? (2)
  2. Calculate the car’s rate of change of height above sea level with respect to time, minutes after starting the journey on the mountainous pass. (3)
  3. Interpret your answer to b.(2)
  4. How many minutes after the journey has started will the rate of change of height with respect to time be a minimum? (3)

[21]

Question 9:

While preparing for the World Cup in South Africa, plans were drawn up in Cape Town to build a series of small hotels. In each hotel there was to be at least one single and one double room. In one of the hotels, the contractors found that the builders could build at least bedrooms altogether, but not more than . Furthermore, the number of double bedrooms was to be at least twice the number of single bedrooms. They could not build more than double bedrooms. The rate for a single bedroom is per night and that of a double bedroom is per night.

Let be the number of single bedrooms and the number of double bedrooms.

9.1.Write down the constraints as a system of inequalities.(5)

9.2.Draw the system of inequalities on the axes provided on the answer sheet, indicating clearly the feasible region. (5)

9.3.According to these constraints, could this hotel have double and single bedrooms? Give a reason for your answer. (1)

9.4.How many rooms of each type must the contractors build in order that the hotel earns the largest income per night? (The calculation should be made assuming that all the rooms in the hotel are fully occupied) (3)

[14]

Question 10:

Calculate the value of: [3]

TOTAL FOR PAPER: 150

ANSWER SHEET:

NAME: ______

Question 5.1:

Question 6:

Question 9:

Bridge House CollegeSeptember Examinations 2013Grade 12 Paper 1