Candidate’s Name:

Maths Set: ______Teacher’s Initials:______

BISHOPS

GRADE 12

MATHEMATICS

PAPER 1

Time:3 hoursSeptember 2016

Total:150 marksCSG/MAB

INSTRUCTIONS AND INFORMATION:

Read the following instructions carefully before answering the questions.

1.This paper consists of 12 pages with 10 questions.

2.Answer ALL the questions on the lined paper provided.

3.Number the answers exactly as the questions are numbered.

4.Clearly show ALL calculations, diagrams, graphs etcetera that you have used in

determining your answers.

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

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

non-graphical),unless stated otherwise.

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

8.Diagrams are NOT necessarily drawn to scale.

9.An information sheet with formulae is printed on the back of this page.

10.It is in your interest to write legibly and to present the work neatly.

INFORMATION SHEET: MATHEMATICS Grade 11 and 12 CAPS

; ;

M

In ABC:

P(A or B) = P(A) + P(B) – P(A and B)

QUESTION 1

1.1Solve for x in each of the following:

1.1.1(2)

1.1.2(4)

1.1.3(4)

1.2Solve for xand yif

(6)

1.3Given

1.3.1Solve for x if (4)

1.3.2Determine for which values of x, will be undefined.(2)

1.4Three graphs with the formulae are drawn below.

  1. 2.3.

1.4.1Match the statements below to the graphs drawn. Write only the numbers

1, 2 or 3 next to each question number on your answer sheet.

i)(1)

ii)(1)

iii)(1)

1.4.2Choose one of the elements A to D given below which is true of

i) Graph 1(1)

ii) Graph 3(1)

A) a < 0 and b < 0 B) a < 0 and b0

C)a 0 and b < 0 D)a 0 and b 0 [27]

QUESTION 2

2.1Prove that for any geometric series where the first term is a and the constant

ratio is r, the sum to n terms is .(4)

2.2Given the geometric sequence 7 + 3½ + 1¾ + ………

2.2.1Determine the sum to infinity.(2)

2.2.2Show that the sum to n terms of the sequence can be written as

(3)

2.2.3Hence calculate the smallest value of n for which (4)

2.3Determine (4)

2.4Determine the value(s) of x for which the series

will converge.(2)

2.5A quadratic pattern has T1 = T3 = 0 and T4= –3.

2.5.1Determine the value of the second difference of this pattern.(4)

2.5.2Determine T5(2)

[25]

QUESTION 3

The diagram shows the graphs of and . P is the

turning point of the parabola.Both f(x) and g(x) pass through the point (0; –9).

g(x) passes through Q(4,5; 0)

3.1Write down the equation of the axis of symmetry of f.(1)

3.2Write down the coordinates of the point which is a reflection of the point (0; –9)

in the axis of symmetry of f.(2)

3.3Determine the values of a, b and c.(5)

3.4Determine the length of DR in terms of x if D is on f and R is on g and DR

is parallel to the line x = 0.(2)

3.5Determine the value(s) of x for which DR is a maximum.(2)

[12]

QUESTION 4

The diagram shows the hyperboladefined by .

The asymptotes ofg cut both the x and y-axes at 1.

4.1Write down the values of r and t.(2)

4.2Write down the equation of the axis of symmetry with a negative gradient.(2)

4.3Write down the equation of the vertical asymptote of g(x+4).(2)

[6]

QUESTION 5

5.1Given

5.1.1Write down the equation of (1)

5.1.2Using the axes on the diagram sheet, sketch the graph of ,

showing at least two points, which must include any intercepts with the

axes. Any asymptotes must also be clearly shown.(3)

5.1.3If the point G (4; a) lies on , determine the value of a.(1)

5.1.4For which values of x is 2?(2)

5.1.5Give the equation of h(x), the reflection of g(x) in the line x = 0.(1)

5.2The graph of is sketched below. Points K(0; –2) and R(1; –4)

are on the curve. Determine the value(s) of a,b and q.(4) [12]

QUESTION 6

6.1Determine the rate of interest per annum, compounded quarterly, for

R240000 to accrue to R374522,21over 5 years. (4)

6.2Nicholas is planning to buy a car, advertised at R210000,00. He is able to

make a 10% deposit and takes a loan for the balance, to be repaid over a

period of 6 years in equal monthly instalments at 16% p.a. compounded

monthly. He starts paying the loan one month after the granting of the loan

and continues for the full 6 years.

6.2.1Determine the amount of the loan, after the deposit has been paid.(1)

6.2.2Determine his equal monthly payments.(5)

6.2.3After 2 years, he changes to a different job and his salary increases.

He is now able to increase his monthly payments to R 5 500,00 per

month. Given that the balance on his debt at that stage is

R144661,94how many payments will he still need to make to

settle the loan if the interest does not change?(6)

[16]

QUESTION 7

7.1Determine from first principles if (4)

7.2Determine the derivative of the following, giving answers with positive

exponents.

7.2.1(2)

7.2.2(4)

7.3Given

7.3.1Determine the equation of the tangent to g atx = –1(4)

7.3.2Determine the value(s) for x for which gis concave up.(3) [17]

QUESTION 8

The graphs of and are drawn below. The graphs

intersect at P and Q. R is a turning point of f(x). f(x)has x-intercepts at Q and

T (–1; 0).

8.1Determine the coordinates of Q.(2)

8.2Determine the coordinates of R.(3)

8.3For which values of x is > 0?(2)

8.4Determine the point at which the graph of f(x)+ 2 changes concavity.

Give a reason why the concavity changes at this point.(3)

8.5For which values of x is< 0 and > 0 simultaneously?(2) [12]

QUESTION 9

A cylinder with radius rfits neatly into a sphere with radius 10 units.

Volume(cylinder) = πr2h

9.1Show that the volume of the cylinder in terms of h is

(3)

9.2Calculate the height of the cylinder, correct to 2 decimals, so that the volume

is a maximum.(4) [7]

QUESTION 10

10.1R and Q are two events in a sample space where P(R) = 0,4 , P(R or Q) = 0,9

and P(Q) = y. Determine the value of y if:

10.1.1R and Q are mutually exclusive.(2)

10.1.2R and Q are independent.(3)

10.2In the final for a middle distance race, 9 athletes line up; 3 of the athletes are

Kenyan.

10.2.1 In how many different ways can the athletes line up?(2)

10.2.2If the Kenyan athletes all need to stand next to each other in the

line up, determine in how many ways the athletes can line up.(3)

10.3Olaf has been recording the punctuality of the buses on a particular route for

thepast 160 days. There has been rainon 72 days and buses were delayed

on 56 of those days. The buses were also delayed on 18 ofthe days when

there was no rain. The latest weather report predictsa 65% probability of

rain this Friday. Using this information, estimate the probabilitythat the

buses will be delayed this Friday. (6)

[16]

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DIAGRAM SHEET

5.1.2