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H2 Maths Specimen Paper 1
1) Use the method of mathematical induction to prove that the sum of the first n
terms of the series
1 3  5   2  4  6    3  5  7  

1
n  n  1 n  4  n  5 
...
One piece is bent into the form
of a square and the other piece is bent into the form of a circle
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07d units
...

iii) State the sum to infinity of the series in which the rth term is
r 1

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4
iii) The complex number w is represented by the point of intersection of the loci
in parts (i) and (ii)
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[4]

[1]
[2]
[2]

[5]

5) i)

Show that



n 1

n

ln x dx   n  1 ln  n  1  n ln n  1
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The
area of the shades region represents the error when the value of the integral in
part (i) is approximated by using a single trapezium
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2  n

iii)
1
Use a series expansion to show that if n is large enough for 3 and higher
n
1
powers of to be neglected, then the area in part (ii) is approximately equal
n
k
to 2 , where k is a constant to be determined
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i) Sketch the curve
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iii)
1 
The tangent at the point P  , 4  meets the y-axis at T and the normal at P
4 
65
meets the y-axis at N
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256

[4]

[3]

[2]

[6]

[4]

7) a)
b)

c)

8) a)

b)

2x 1
dx
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 2
 x  8 x  17
Find the volume of revolution when the region bounded by the part of the
1
curve y  cos2 x when x  0 and x   and by the x- and y-axes is rotated
2
completely about the x-axis
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The first four terms u1 , u2 , u3 , u4 of an arithmetic progression are such that
u4  u2  15 and 4u3  9u1
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[3]

[5]

[3]

[3]

The positive numbers x n satisfy the relation
1

xn 1   xn  5  2 for x  1, 2,3,

As n  , xn  l
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[3]
2
2
ii) Prove that  x   l  x  l
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[3]
9) The parts of this question may be done in any order
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3x  6
Consider the curve y 

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[1]
ii) State the equations of the asymptotes
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[5]
iv) Draw a sketch of the curve
a)
3x  6
y
x  x  6
b)
3x  6
y
x  x  6
making clear the main relevant features of each curve
...

[5]
1

The function f  x   3   8  x  3 is defined for 0  x  19
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iii) Sketch the graphs of y  f  x  and y  f -1  x  on the same axes, and show the

[3]

geometrical relation between the two graphs and the line y  x
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v)

[2]
1
3

Use the first two terms of the expansion of  8  x  to obtain an
approximation to the x-coordinate of the point intersection
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[1]



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