WRP Examination Mathematics 2012

2012

Preliminary Course

FINAL EXAMINATION

Mathematics

General Instructions

o Reading time – 5 minutes

o Working time – 2 hours

o Write using black or blue pen

o Board-approved calculators may be used

o All necessary working should be shown

in every question

Total Marks 70

10 marks

o Attempt Questions 1-10

o Answer on the Multiple Choice

answer sheet provided.

o Allow about 15 minutes for this

section.

60 marks

o Attempt questions 11 – 16

o Answer on the blank paper provided,

unless otherwise instructed. Start a

new sheet for each question.

o Allow about 1¾ hours for this

section

Section II

Section I

2012 Preliminary Final Examination Mathematics

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Section I

10 marks

Attempt Questions 1-10

Allow about 15 minutes for this section

Use the multiple choice answer sheet for questions 1 – 10.

1. Evaluate 3.23 4.96

2

3.45 + 1.22

correct to 3 significant figures.

(A) 16·2 (B) 16·3

(C) 24·4 (D) 24·5

2. If 2 2

2 2 – 3= a – b then the values of a and b are:

(A) 4 and 12 (B) 4 and 72

(C) 8and 12 (D) 8 and 72

3. The solutions to the equation 2x2– 7x – 2 = 0are:

(A) –7± 334

(B) –7± 654

(C) 7± 334

(D) 7± 654

4. The domain and range for the function y = 7 – x is

(A) x 7 ; y 0 (B) 0 x 7 ; y 0

(C) x 0 ; y 0 (D) All real x, All real y

5. A function is defined by f (x) = x2– 4x + 5 .

The point on y = f (x) which has a gradient of 2 is:

(A) (3, 1) (B) (2, 1)

(C) (3, 2) (D) (2, 3)


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6. The derivative of2

x is :

(A) –2

x (B) –

1

x

(C) –2

x3

(D) –1

x3

7. The calculation which would be used to find the value of xis:

(A) 15 tan 36° (B) 15 cos 36°

(C) 15

tan 36° (D)

15

cos 36°

8. The values of k for which the expression – x2

4– x – k is negative definite is:

(A) k < 1 (B) k > 1

(C) k < 4 (D) k > 4

9.

The value of x in the above diagram is:

(A) 42 (B) 69 (C) 111 (D) 138

10. The equation x2 + y

2 – 6x + 4y + 6 = 0 represents a circle. The coordinates of the

centre of the circle are:

(A) (3, 2)

(B) (3, 2)

(C) (3, 2)

(D) (3, 2)

End of Section I


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Section II

Total Marks (60)

Attempt Questions 11 - 16

Allow about 1¾ hours for this section.

Answer all questions, starting each question on a new sheet of paper, with your name

and question number at the top of the page. Do not write on the back of sheets.

All necessary working should be shown in every question.

Question 11 (10 marks) Start a new sheet of writing paper. Marks

(a) Express 2.3.6

.4 as a rational number in simplest form.

2

(b) Solve:

(i) 5x + 3

= 3x

2

(ii) |2x + 1| = 3x + 2 2

(c) Solve the following equations simultaneously:

y = 2x + 4 and y = x2

+ 1.

2

(d) In a right angled triangle, one of the sides adjacent to the right angle is 6cm

longer than the other. Find the length of all three sides of this triangle if its

area is 216cm2.

2

End of Question 11


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(a) An interval AB is formed by the points A(3, 5) and B(2, 3) on the number plane.

(i) Find the gradient of the interval AB. 1

(ii) Find the length of the interval AB. 1

(iii) Show that the equation of the line AB is 8x + 5y – 1 = 0 . 1

(iv) Find the equation of the line through C(5, 8) perpendicular to AB. 2

(v) Calculate the distance of the point C from the line AB. 1

(vi) Calculate the area of the triangle ABC. 1

(vii) Shade the region for which the inequalities 8x + 5y – 20 0and y 0 hold simultaneously.

1

(b) A point P moves so that it is equidistant from the point (5, 3) and the line x = 1.

Find the equation of locus of P.

2

End of Question 12


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(a) Differentiate:

(i) 3x2– 2x – 5

1

(ii) 4 x

3

1

(iii) 5xx

2– 4

4

2

(iv) 2x + 34x – 9

2

(b) Find the equation of the normal to the curve y = 2x2– 4x + 5 at the point

where x = 3.

2

(c) Evaluate limx 4

x2– x – 12x – 4

.

2

End of Question 13


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(a) Simplify 1

cosec2

+1

sec2

+1

cot2

.

2

(b) If sin = 724

and cos < 0 ,find the exact value of sec . 2

(c) A point P is at the foot of the hill which is inclined at 5o to the horizontal. The base

B, of a tower AB, is situated 150 metres up the incline of the hill from P. From the

top of the tower the angle of depression of the point P is 32.

(i) Copy the diagram and mark on it the size ofAPB and of PAB . 1

(ii) Find the height of the tower, correct to the nearest metre. 2

(d) Two ships leave a port P at the same time. Ship A travels at 12 km/h on a bearing

of 213. Ship B travels in a south easterly direction at a speed of 8 km/h.

(i) Draw a diagram to illustrate the relative positions of the two ships after

3 hours. 1

(ii) Calculate the distance between the ships after 3 hours. 2

End of Question 14


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(a) In the diagram, ABCD and AB = AC.

The lines AD and BC are perpendicular.

BCD = 54.

(i) Calculate, giving reasons the size of CAD . 2

(ii) Prove that triangle ACD is isosceles. 1

(b) In the triangle PQR, S and T are points on the sides PR and QR respectively such

that ST || PQ.

(i) Prove that the triangles PQR and STR are similar. 2

(ii) Find the value of y. 2

(c) A function is defined as

x 5 x 0

f(x) = 2 3 < x < 0

2 + x x 3

Find

(i) f(–1) + f(–5) 1

(ii) f a

2 1

(d) Explain why the function f(x) = 3x2– 2x + 6 is neither odd nor even.

1

End of Question 15


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(a) Solve x2– 5x – 24 0

2

(b) If and are the roots of the equation 3x2

+ 5x – 4 = 0 , find

(i) 1

(ii) + 1

(iii) 22

+ 22

1

(iv) 3

+

3

1

(c) Given the equation (k + 1)x2 + (k + 2)x + k = 0 find the value of k if the

equation has equal roots.

2

(d) For the parabola y2– 6y – 4x – 15 = 0 , find:

(i) The coordinates of the vertex. 1

(ii) The coordinates of the focus. 1

End of Examination.


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Preliminary Final Examination –Mathematics 2012 Multiple Choice Answer Sheet

Name ______________________________

Completely fill the response oval representing the most correct answer.

1. A B C D

2. A B C D

3. A B C D

4. A B C D

5. A B C D

6. A B C D

7. A B C D

8. A B C D

9. A B C D

10. A B C D


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Section I

10 marks

Attempt Questions 1-10

Allow about 15 minutes for this section

Use this multiple choice answer sheet for questions 1 – 10.

Select the alternative A, B, C or D that best answers the question. Fill in the response oval

completely.

Sample

2 + 4 = ? (A) 2 (B) 6 (C) 8 (D) 9

A B C D

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new

answer.

A B C D

If you change your mind and have crossed out what you consider to be the correct answer, then

indicate this by writing the word correct and drawing an arrow as follows:

correct

A B C D

WRP Examination Mathematics 2012

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