2011 bdms 4e prelims 2 am paper 1.doc
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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