Class

2

Bendemeer Secondary School AM 2011 Prelims Paper 1Mathematical Formulae1. ALGEBRA

Quadratic Equation

For the equation ,

Binomial Theorem

,

where n is a positive integer and

2. TRIGONOMETRYIdentities

Formulae for ABC

1Given that the roots of the quadratic equation are and , find the[6]

equation whose roots are and .

2(i)Find the coordinates of all the points at which the graph of

meets the coordinate axes.[3]

(ii)Sketch the graph of .[3]

(iii)Hence or otherwise, solve the equation

[2]

3The expression has remainders 2 and 3 when divided by

and respectively.

(a)Find the values of and .[4]

(b)With these values of and , find the remainder when the expression is [2]

divided by .

4(a)Solve the equation

[4]

(b)Solve the following equation[4]

5Given that , solve the equations

(a)

[3]

(b)

[3]

6(i)Prove the identity .[3]

(ii)Hence, solve the equation for .[2]

7(a)Find the range of values of x for which[4]

(b)Show that is always positive for all values of .[2]

8(a)Given that , evaluate .[3]

(b)Given that ,

(i)

show that can be written in the form and state the value

of and of .[4]

(ii)Hence, evaluate .[3]

9Answer the whole of this question on the graph paper provided.

Variables and are related by the equation , where and are

constants. The table below shows measured values of and .

0.250.500.751.001.251.50

0.180.420.731.151.792.81

(i)Using a scale of 4 cm to 1 unit for both axes, draw the graph of against .[4]

(ii)Use your graph [4]

(a)to estimate the value of and of .

(b)the value of when .

10(i)Write down the coordinates of the centre and the radius of the circle

whose equation is .[3]

(ii)Another circle has centre and passes through the centre[3]

of circle . Find the equation of .

11

The diagram shows a solid body made up of a cylinder of length cm and a

hemispherical cap of radius cm. If the total surface area, cm2, of the solid is

cm2, show that the total volume of the solid cm3, is given by .[4]

Hence find

(a)the value of for which has a stationary value[3]

(b)the value of and of corresponding to this value of .[2]

Determine whether the stationary value of is a maximum or minimum.[2]

-------- End of Paper --------

1

Or

7(a)

2(i), and

7(b)proving

2(ii)

8(a)

2(iii)

8(b)(i),

3(a),

8(b)(ii)

3(b)

9(i)Graph

4(a)

9(ii)(a),

4(b)

9(ii)(b)

5(a)

10(i)Centre

Radius

5(b)

10(ii)

6(a)proof11(a)

6(b)

11(b)

maximum.

1Given that the roots of the quadratic equation are and , find the[6]

equation whose roots are and .

M1

Sum of new roots :

M1

M1

M1

Product of new roots :

M1

Hence, equation :

A1

Or

[-1m for not writing equation]

2(i)Find the coordinates of all the points at which the graph of

meets the coordinate axes.[3]

When ,

M

When ,

M1

Hence, the graph cuts the axes at , and

A1

[-1m for coordinates not given]

(ii)Sketch the graph of .[3]

Let

M1

When ,

Shape 1, points 1

(iii)Hence or otherwise, solve the equation

[2]

From Graph,

A2

Or

3The expression has remainders 2 and 3 when divided by

and respectively.

(a)Find the values of and .[4]

Let

M1

-----(1)

M1

-----(2)

(1) (2) :

A1

Sub into (1)

A1

(b)

With these values of and , find the remainder when the expression is [2]

divided by .

M1, A1

4(a)Solve the equation

[4]

or

Let

Let

M1

M1

M1

A1

(b)Solve the following equation[4]

M1

M1

M1

A1

5Given that , solve the equations

(a)

[3]

M1

A2

[1 m awarded for when is cancelled]

(b)

[3]

M1

M1

A1

6(i)Prove the identity .[3]

M1

M1

A1

(ii)Hence, solve the equation for .[2]

M1

or

A1

7(a)Find the range of values of x for which[4]

and

M1

M1

M1

Hence,

A1

(b)Show that is always positive for all values of .[2]

M1

shown

A1

Or

Discriminant

M1

Since coefficient of and discriminant ,

A1

[ m deducted for first step assuming D