Grade 12 Mathematics

Grade 12 Mathematics

Prelim Exam

Paper 2

02/08/2011

Examiner: J.F. vd Merwe

Moderator: D. v Schalkwyk

Instructions:

This paper consist of 10 pages and 13 questions.
Answer all the questions and show all calculations.
Non programmable and non graphics calculators are allowed
A formula sheet is attached.

Section A

Question 1

B(2 ; 4)

0 θ x

A(x ; y)

C(0 ; 6)

In the diagram, A, B and C are vertices of ΔABC in the Cartesian plane. The equation of the straight line AB is given by y = -x + 8. Use analytical methods to answer the following questions:

1.1)Calculate the length of BC in the simplest surd form.(3)

1.2)Determine the coordinates of M, the midpoint of BC.(2)

1.3)Give the magnitude of θ, the angle of inclination of BC.(2)

1.4)Determine the equation of the line AM which is perpendicular to BC at M.(4)

1.5)Show that the x-coordinate of A is 11, and then find the y-coordinate of A.(5)

(16)

Question 2

2.1)If and , determine using a sketch, the value of

, in terms of a andb.(4)

2.2)Simplify the following to one trigonometric identity:

(8)

2.3)If sin 23° = p, determine each of the following in terms of p.

2.4.1)cos 23°(1)

2.4.2)cos 67°(2)

2.4.3)sin 46°(3)

2.4)Given the equation:sin 2x – cos x = 0

2.5.1)Determine the general solution for this equation.(4)

2.5.2)Find the values of (3)

2.5)Simplify the following as far as possible WITHOUT a calculator:

(5)

(30)

Question 3

B( 120°;2 )

A( -60°;-2 )

Sketched is the graph of . Where A andB are turning points.

Using the graph, find the values of:

3.1)a and b(2)

3.2)coordinates of C and D.(4)

3.3)period of f(x)(1)

3.4)range of f(x)(2)

(9)

Question 4

The data below shows the total monthly rainfall(in millimetres) at Cape Town International Airport for the year 2008

Jan / Feb / Mrt / Apr / May / Jun / Jul / Aug / Sept / Oct / Nov / Dec
44.2 / 14.3 / 9.5 / 24 / 70.6 / 79.1 / 95.1 / 66 / 23.2 / 26.5 / 22.3 / 10.2

4.1)Determine the mean monthly rainfall for 2008.(2)

4.2)Write down the five number summary for the data.(5)

4.3)Draw a box-and-whisker diagram for the data on the answer sheet.(3)

4.4)By making reference to the box-and-whisker diagram, comment on the spread of the rainfall for the year. (2)

4.5)Calculate the standard deviation of the data.(3)

(15)

Question 5

Let be any point on a Cartesian plane.

Match column A to column B. (eg. 2 d)

1)Rotation through 90° clockwise / a)
b)
2)Reflection about the x-axis / c)
d)
3)Reflection about the y = x / e)
f)
4)Rotation through 180° / g)
h)
5)Enlargement by a factor of 2 through the origin / i)
j)

(5)

Section B

Question 6

The equation of a straight line ABis .

The equation of a straight line CDis .

Find the value of p in each case, if:

6.1)(2)

6.2)AB cuts CD at (2)

6.3)The angle of inclination of CD is 120°(2)

(6)

Question 7

and are two points on the circumference of the circle centre M, in a Cartesian plane. M lies on AB. DA is a tangent to the circle at A. The coordinates of D are and the coordinates of Care . Points C and D are joined. K is the point . CTD is a straight line.

7.1)Show that the coordinates of M, the midpoint of AB, are (1)

7.2)Determine the equation of the tangent AD.(4)

7.3)Determine the length of AM.(3)

7.4)Determine the equation of the circle with centre M in the form:

(4)

7.5)Quadrilateral ACKD is one of the following: parallelogram, kite, rhombus or rectangle. Which one is it? Justify your answer. (4)

7.6)Find the angle of .(2)

(18)

Question 8

8.1)Given that: and .

Evaluate: , where and A is an obtuse angle.(5)

8.2) B

F E

C D

A rectangular greeting card has a height of y cm and a with of x cm. The card is opened in order to read the message inside, and placed on a table so that the angle between the front and the back covers 2θ.

8.2.1)Show that (5)

8.2.2)Prove that (3)

8.2.3)If x = 12 cm, y = 20 cm and θ = 63°, determine a possible size of (3)

(16)

Question 9

The ogive (cumulative frequency graph) represents the finishing times of 590 runners who completed a 10km race.

9.1)Estimate in how many minutes a runner would have to complete the race in order to place at the 15th percentile or better. (2)

9.2)If a silver medal is awarded to all runners completing the race in under 45 minutes, estimate the number of runners who would have received a silver medal. (2)

9.3)Construct a frequency table for the information given in the ogive.(3)

(7)

Question 10

10.1)A point is reflected around the y-axis, then it is rotated 90° anti-clockwise around the origin, and finally translated 1 unit left and 3 units up. Write down the rule for this transformation. (Show all steps). (4)

10.2)Calculate,without the use of a calculator, the coordinates of the image of A(4;8) if A is rotated in an anti clockwise direction around the origin through an angle of 60°. Give the answer in simplest surd form. (6)

(10)

Question 11

Consider the expression :

11.1)Prove that (2)

11.2)Hence determine the maximum value of (2)

11.3)What is the corresponding value(s) of x for the maximum value in (12.2) if . (2)

(6)

Question 12

The depth,d metres, of water in a harbour on a certain day is given by

Where t is the number of hours after 12 (midnight).

Calculate the times on this day when the depth of the water in the harbour was 7 metres.(7)

Question 13

The figure shows an equilateral triangle with a square fitted inside it. All 4 vertices of the square lie on the sides of the triangle. Determine the value of .

(5)

Total 150