Candidate’s Name: Maths Set: ______ Teacher’s Initials:_______

BISHOPSGRADE 12

MATHEMATICSPAPER 1

Time: 3 hours September 2016Total: 150 marks CSG/MABINSTRUCTIONS 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 et cetera that you have used indetermining 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.

PLEASE PUT YOUR NAME, SET NO. AND TEACHER’S INITIALS AT THE TOPOF THIS PAGE – YOU WILL BE HANDING IN THE QUESTION PAPER

WITH YOUR SCRIPT.

2016 September | Grade 12 | Paper 1 | Page 2

INFORMATION SHEET: MATHEMATICS Grade 11 and 12 CAPS

; ;

M

In ABC:

P(A or B) = P(A) + P(B) – P(A and B)

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QUESTION 1

1.1 Solve for x in each of the following:

1.1.1 (2)

1.1.2 (4)

1.1.3 (4)

1.2 Solve for x and y if

(6)

1.3 Given

1.3.1 Solve for x if (4)

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

1.4 Three graphs with the formulae are drawn below.

1. 2. 3.

1.4.1 Match 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)

1.4.2 Choose 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 b > 0

2016 September | Grade 12 | Paper 1 | Page 4

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

QUESTION 2

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

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

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

2.2.1 Determine the sum to infinity. (2)

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

(3)

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

2.3 Determine (4)

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

will converge. (2)

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

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

2.5.2 Determine T5 (2)

[25]

2016 September | Grade 12 | Paper 1 | Page 5

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)

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

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

in the axis of symmetry of f. (2)

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

3.4 Determine 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)

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

[12]

2016 September | Grade 12 | Paper 1 | Page 6

QUESTION 4

The diagram shows the hyperbola defined by .

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

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

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

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

[6]

2016 September | Grade 12 | Paper 1 | Page 7

QUESTION 5

5.1 Given

5.1.1 Write down the equation of (1)

5.1.2 Using 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)

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

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

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

5.2 The 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]

2016 September | Grade 12 | Paper 1 | Page 8

QUESTION 6

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

R 240 000 to accrue to R 374 522,21 over 5 years. (4)

6.2 Nicholas is planning to buy a car, advertised at R 210 000,00. 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.

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

6.2.2 Determine his equal monthly payments. (5)

6.2.3 After 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

R 144 661,94 how many payments will he still need to make to

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

[16]

2016 September | Grade 12 | Paper 1 | Page 9

QUESTION 7

7.1 Determine from first principles if (4)

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

exponents.

7.2.1 (2)

7.2.2 (4)

7.3 Given

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

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

2016 September | Grade 12 | Paper 1 | Page 10

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).

8.1 Determine the coordinates of Q.

(2)

8.2 Determine the coordinates of R. (3)

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

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

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

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

2016 September | Grade 12 | Paper 1 | Page 11

QUESTION 9

A cylinder with radius r fits neatly into a sphere with radius 10 units.Volume(cylinder) = πr2h

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

(3)

9.2 Calculate the height of the cylinder, correct to 2 decimals, so that the volume is a maximum. (4)

[7]

2016 September | Grade 12 | Paper 1 | Page 12

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QUESTION 10

10.1 R 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:

10.1.1 R and Q are mutually exclusive.(2)

10.1.2 R and Q are independent.(3)

10.2 In 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)

10.2.2 If 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)

10.3 Olaf has been recording the punctuality of the buses on a particular route for the past 160 days. There has been rain on 72 days and buses were delayed on 56 of those days. The buses were also delayed on 18 of the days when there was no rain. The latest weather report predicts a 65% probability of rain this Friday. Using this information, estimate the probability that thebuses will be delayed this Friday. (6)

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DIAGRAM SHEET5.1.2