Revision P2 05

SEKOLAH MENENGAH PEHIN DATU SERI MAHARAJA, MENTIRI MID YEAR EXAMINATION 2005

4037/2 ADDITIONAL MATHEMATICS PAPER 2 FORM 5 ‘O’ – Level

Time: 2 hours

Instructions to candidates:

Write your name and class on top of every answer papers you have used. Answer ALL questions. Write your answers on the separate answer paper provided. If you use more than one sheet of paper, fasten the sheets together. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of degree of accuracy is not specified in the question.

Information for Candidates

The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.

2

1. A curve x 2 + y 2 = 10 and a line 2 x + y = 5 intersect at two points A and B. Find the coordinates of A and B. [4]

2. Given that

a+b 3 =

13 4+ 3

, where a and b are integers, find, without using a

calculator, the value of a and of b

[4]

3. Given that ξ = {students in a collage} ,

A = {students who are over 180 cm tall} , B = {students who are vegetarian}, C = {students who are cyslists},

express in words each of the following (i) (ii) A∩B ≠φ ,

A ⊂ C′

[2]

Express in set notation the statement (iii)

all students who are both vegetarians and cyclists are not over 180 cm tall. [2]

4.

π⎞ ⎛ y = 6 sin⎜ 3 x + ⎟ 4⎠ ⎝

⎛ ⎝

The diagram shows part of the curve y = 6 sin⎜ 3 x +

π⎞

⎟ . Find the area of the shaded 4⎠

region bounded by the curve and the coordinate axes.

5.

(i) (ii)

Differentiate x sin x with respect to x. Hence evaluate

∫

[6]

[2]

π 2

0

x cos x dx

[4]

3

6.

7.

8.

Find the nature of the roots of the following quadratic equations: a. 5x2 + 1 = 7x b. 4x2 = a (4x – a) Given a. b. c.

y = 6 − x − x 2 , find the range of values of x for which y is positive, the maximum value of y Sketch the graph y = 6 − x − x 2

A rectangular box, with a lid, is made of thin metal. Its length is 2x cm and it width is x cm. If the box must have a volume of 72 cubic metres. a. show that the area A sq. metres of the metal used is given by

A = 4x 2 +

10.

[2] [2] [3]

[4]

216 x

b. find the value of x so that A is minimum.

9.

[2] [3]

The function f is defined as f : x a

[4]

20 . ax + b

-1 a. Find f ( x ) in term of a and b,

[2]

-1 b. Given that f (1) = 4 and f ( −4 ) = − 4 , find the value of a and of b,

[4]

c. From (ii) what will be the value of x for which f(x) = x.

[4]

A spherical balloon is expanded so that its surface area is increasing at the rate of 2 m2/min. a. At what rate is its radius increasing when the radius is 4 m? [4] b. At what rate is its volume increasing at the same instant? [4]

4 [ Surface areas, s = 4π r 2 ; volume, v = π r 3 ] 3

11.

2 2 The diagram shows part of the graphs of the curves, y = 2 x , x = 2y and the line y = 4. Find the areas of the shaded region.

[8]

4

12. Answer only ONE of the following two alternatives. [ EITHER ] A particle moves in a straight line so that, t s after leaving a fixed points O, its velocity,

⎛ ⎜ ⎝

− 1t

⎞ ⎟ ⎠

v ms-1 , is given by v = 10⎜1 − e 2 ⎟ .

(i) Find the acceleration of the particle when v = 8.

[4]

(ii) Calculate, to the nearest metre, the displacement of the particle from O when t = 6. [4] (iii) Sketch the velocity-time graph for the motion of the particle. [2]

[OR]

2 The diagram shows part of the curve y = x ln x , crossing the x-axis at Q and having a minimum point at P.

(i) Find the value of

dy at Q. dx

(ii) Show that the x-coordinate of P is

(iii) Find the value of

d 2y dx 2

at P.

[4]

1 e

.

[3]

[3]

… The end.