Revision P1 06

Revision Exercise 4037/1 ADDITIONAL MATHEMATICS PAPER 1 Form 5 ‘O’ – Level 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.

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

5.

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

[3] [2]

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.

[5]

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)

6.

x 2 − 8 x + 32 at the point

the value of k, the equation of the curve

Evaluate

(x

(i)

∫

(ii)

∫

(iii) (iv)

)

+1 x4 1

2

2

[2]

dx

dx 5−x 4 ⎛ 10 ⎞ 3 ∫2 ⎜⎝ x − x 2 ⎟⎠ dx

∫ (7 + 6 x − x ) 0

−1

[2] [4]

2

dx

[2] [2] [2]

2

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)

10.

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) y = (x − 5 ) 2 x + 5

3x 2 − 4x

[5] [4]

[2] [2]

(ii)

y=

(iii)

y = (x − 5)(2 x + 5) 2

(iv)

y=

x

1

(v)

11.

1 − 2x , x ≠ −2 x+2 3 4 y= − 2 x x

[2] [2] [2]

The figure shows part of the curve y = x 3 − 12 x + k . Given that this curve has a minimum points at A on the x-axis, find

(i) (ii)

the value on k, the area of the shaded region.

[7] 2

3

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. 3