This Question Paper Consists of 9 Pages and 1 Information Sheet

Mathematics/P11NW/September 2014

NSC

MARKS: 150

TIME: 3 hours

This question paper consists of 9 pages and 1 information sheet.

Copyright reservedPlease turn over

Mathematics/P11NW/September 2014

NSC

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1.This question paper consists of 11 questions.

2.Answer ALL the questions.

3.Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining the answers.

4.Answers only will not necessarily be awarded full marks.

5.You may use an approved scientific calculator (non-programmable and

non-graphical), unless stated otherwise.

6.If necessary, round off answers to TWO decimal places, unless stated otherwise.

7.Diagrams are NOT necessarily drawn to scale.

8.An information sheet with formulae is included at the end of this question paper.

9.Number the answers correctly according to the numbering system used in this question paper.

10.Write neatly and legibly.

QUESTION 1

1.1Solve for x,correct to TWO decimal places where necessary:

1.1.1(3)

1.1.2(5)

1.1.3(3)

1.2Given:

1.2.1Show that: (2)

1.2.2Hence, solve for a and b simultaneously if it is further given that

(5)

1.3Determine the value of without using a calculator.(2)

1.4If and , prove that without using a

calculator.(3)

[23]

QUESTION 2

2.1Evaluate:

2.2A water tank contains 216litres of water at the end of day 1. Because of a leak,

the tankloses one-sixth of the previous day’s contentseach day.

How many litres of waterwillbe in the tank by the end of:

2.2.1the 2nd day?(2)

2.2.2the 7th day?(3)

2.3Consider the geometric series:

2.3.1For which valuesof x is the series convergent?(3)

2.3.2Calculate the sum to infinity of the series if .(4)

2.42 ;x ; 12 ; y ; . . . are the firstfour terms of a quadratic sequence. If the second difference is 6, calculate the values of x and y. (5)

[21]

QUESTION 3

Given: for

3.1Draw a neat sketch graph of h in your ANSWER BOOK. Show all intercepts

with the axes and asymptotes.(4)

3.2Write down the equation of k if k is the reflection of h about the x-axis.(3)

[7]

QUESTION 4

4.1Sketched below are the functions: and .

Determine:

4.1.1the coordinates of turning point D.(2)

4.1.2the coordinates of A and B.(3)

4.1.3the value of k .(2)

4.1.4the values of p if has no real roots.(2)

4.1.5for which values of x is .(2)

4.1.6the value of t if is a tangent to f.(4)

4.2Consider the following two functions: and .

4.2.1How will you shift p to become the function r ?(3)

4.2.2Write down the range of p .(1)

[19]

QUESTION 5

The sketch below represents the inverses of and ; .

5.1Write down the coordinates of ONE point through which both f and will pass. (1)

5.2Determine the equationof f.(3)

5.3Calculate x if .(3)

5.4If , for which values of xwill ?(2)

[9]

QUESTION 6

6.1Convert an interest rate of 14% p.a. compounded monthly to an interest rate per

annumcompounded quarterly.(3)

6.2Nelson deposits R3500 into a savings account. Three years later he adds R5700

to the account. The interest for the first two years is 7% p.a. compounded

quarterly. The interest for the last three years is 8% p.a. compounded monthly. Calculate the balance in the savings account at the end of five years. (4)

[7]

QUESTION 7

Mr Daniels wants to take out a loan for a house over twenty years. He has approached

two financial institutions and was offered two different options. The two options are

shown in the table below:

7.1Determine the total cost for Option 1.(7)

7.2Which option is the best? Provide relevant calculations to justify your answer.(3)

[10]

QUESTION 8

8.1Given: . Determine from first principles.(5)

8.2Determine if:

8.2.1(3)

8.2.2(4)

8.3Determine the value of x where the gradient of the tangent to

is equal to .(4)

[16]

QUESTION 9

9.1The function defined by is represented by the sketch below.A and B are the turning points of f, while is a tangent to f at A

which cuts f at point C.

9.1.1Show that and .(7)

9.1.2Determine the coordinates of C.(3)

9.1.3Write down the coordinates of a turning point of k, if

(2)

9.2The turning points of the graph of a cubic polynomial are

and . Draw a sketch graph of the derivative function , clearly showingthe x-intercepts. (3)

[15]

QUESTION 10

PQRS is a rectangle with P on the curve and with the x-axis and the line as boundaries.

10.1Show that the area of rectangle PQRS can be expressed as:

A .(3)

10.2Determine the largest possible area for rectangle PQRS. Show all your

calculations.(4)

[7]

QUESTION 11

11.1The probability of getting the first answer in a quiz correct, is 60%. If the first

answer is correct, the probability of getting the second answer correct, rises to 70%. However, if the first answer is wrong, the probability of getting the second answer correct, is only 40%. Determine the probability of getting the second answer

correct.(4)

11.2A and B are independent events.

Determine the values of x and y. All calculations must be shown.(5)

11.3Ping-Pong balls are numbered 1 to 12 and placed in a hat. Every time a ball is

drawn, it is placed on a rack, one next to the other.

11.3.1How many different arrangements of the 12 balls are possible?(2)

11.3.2Assume that the numbers 8 and 11 must be placed next to each other,

in any order. In how many different ways can the numbers then be

arranged?(3)

11.3.3The numbers 2, 5, 6 and 9 are taken from the group of balls. Two-digit numbers must be formed out of these four numbers. How many two-digit

numbers can be formed?(2)

[16]

TOTAL: 150

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Mathematics/P11NW/September 2014

NSC

INFORMATION SHEET: MATHEMATICS

;;

M

In ABC:

P(A or B) = P(A) + P(B) – P(A and B)

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