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SEKOLAH MENENGAH PEHIN DATU SERI MAHARAJA, MENTIRI MID YEAR EXAMINATION 2005 Form 5 ‘O’ - Level

4037/1 ADDITIONAL MATHEMATICS PAPER 1 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.

2.

Find the coordinates of the points on the curve y = x 3 − 3 x 2 − 9 x + 6 where the gradient is 0.

[4]

Find the values of m for which the line y = mx − 9 is a tangent to the curve x 2 = 4 y .

[4]

3.

The surface area of a sphere of radius r cm is given by the formula A = 4 π r 2 . By using calculus, find the approximate change in the surface area of the sphere when the radius increases from 5 cm to 5.1 cm. [4]

4.

The velocity v ms-1 of a particle after t s is given by v = 6t 2 + 2t + 1 . After 1 s, it distance from its initial position is 6 m. (i) (ii)

5.

6.

Find the equation of the normal to the curve y = where x = 1.

[3] [2]

x 2 − 8 x + 32 at the point [5]

dy 3 x = − 2 , where k is constant. Given that the dx k 1 1 gradient of the normal at the point (−2 , ) on the curve is − , find 2 2 The gradient function of the curve is

(i) (ii)

7.

How far has it traveled in the first 3 s? Find its acceleration when t = 5.

the value of k, the equation of the curve

Evaluate (i)

∫

(ii)

∫

(x

)

2

+1 dx x4 1 dx 5−x 2

[2] [4]

[2] [2]

π

(iii)

∫

(iv)

ex −1 ∫o e 2 x

2

0 1

2 cos 3 x dx dx

[2] [2]

2

3

8.

(

)

Find the stationary points of the curve y = x 3 2 x 2 + x − 2 and determine if they are maximum, minimum or points of inflexion. [8]

9.

In the diagram, the inverted right circular cone, full of water, has a height of 12 cm and a radius of 6cm at the top. Water is poured into this container at a rate of 3 cm 3 s −1 and leaks through a small hole at the vertex at a rate of 5 cm 3 s −1 . Calculate, at the instant when the depth of the water is 3 cm, (i) (ii)

9.

the rate of change of the height of the water level, the rate of change the area of the horizontal surface of the water.

Differentiate the following with respect to x: (i) (x − 5 ) 2 x + 5

3x − 4x

[5] [4]

[2]

2

(ii) (iii) (iv)

11.

[2]

x ⎛ x −2⎞ ln⎜ ⎟ ⎝ x + 2⎠ x sin x

[2] [2]

A curve has the equation y = xe 2 x . (i) (ii)

Find the x-coordinate of the turning point of the curve. Find the value of k for which

d 2y = ke 2 x (1 + x ) 2 dx (iii)

Determine whether the turning point is a maximum or a minimum.

[4] [3] [2]

3

4

12. Answer only ONE of the following two alternatives. [ EITHER ]

30 cm

r cm

h cm

12 cm

The diagram shows the cross-section of a hallow cone of height 30 cm and base radius 12 cm and a solid cylinder of radius r cm and height h cm. Both stand on a horizontal surface with the cylinder inside the cone. The upper circular edge of the cylinder is in contact with the cone. (i)

Express h in terms of r and hence show that the volume, V cm3, of the cylinder is

⎛ ⎝

given by V = π ⎜ 30r 2 −

5 3⎞ r ⎟. 2 ⎠

[4]

Given that r can vary, (ii) find the volume of the largest cylinder which can stand inside the cone and show 4 that, in this case, the cylinder occupies 9 of the volume of the cone. [6] [ The volume, V, of a cone of height H and radius R is given by V =

1 2 πR H .] 3

[ OR ] The volume V cm 3 , of water in a hemispherical bowl, when the depth of water is h cm, is given by V =

1 π h 2 (30 − h ) . 3

Given that water is poured into the bowl at the rate of 100 cm 3 s −1 , find the rate of increase in the depth of water when h = 5 cm, leaving your answer in terms of π . [4] The depth of the water increases by a small amount, p cm, from 5 cm to (5 + p ) cm. Using calculus, find, in terms of p, the approximate increase in V. Hence find the approximate percentage increase in V. [6] …the end. 4