Anderson Junior College
JC1 Promotional Examination 2008
H2 Mathematics
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Answer ALL questions.
Differentiate sin cos (3 x) , leaving your answer in the simplest form.
1
1 (a)
[2]
(b) A curve C has parametric equations
x sin , y 1 cos , 0 2 .
1
At the point on the curve with parameter , the gradient is .
2
Show that 2sin cos 1 .
[3]
1
2. Find the first three non-zero terms of the expansion of 4 x 2 in ascending powers of x and
state the range of values of x for which the expansion is valid.
[3]
1
By substituting x , find an approximation to 11 , giving your answer to three decimal
25
places.
[2]
5 2
3. Solve the inequality , giving your answer in exact form. [3]
x 2 3x 4
5x 2x
Hence find the solution for the inequality . [3]
1 2x 3 4x
dy
4. Show that the differential equation sin 2 2 x y 3 may be reduced by the substitution
dx
du
u 2 x y to the equation cos 2 u . Hence find the equation of the solution curve that
dx
passes through the origin, giving your answer in the form y f x .
[6]
5. (a) Alice has 4800 sweets to be divided into n bags. She puts 25 sweets into the first bag.
For each subsequent bag, she puts 5 more sweets than the previous bag. She continues
to fill the bags until there are not enough sweets to fill the next bag in the same manner.
Find the number of sweets left behind.
[3]
(b) The terms of a sequence a1 , a2 , a3 ,..... form an arithmetic progression with common
difference 2.
Page 2
a
Another sequence b1 , b2 , b3 ,..... is such that bn for n = 1, 2, 3, …
1 n
e
(i) Prove that the terms of the sequence b1 , b2 , b3 ,..... form a geometric progression. [2]
(ii) Determine, giving a reason, whether the sum to infinity of the geometric progression
exists.
[2]
The function f is defined by f : x e x 1 , x , x k .
2
6. (a)
(i) For k = 3, explain why the inverse function f 1 does not exist.
[2]
(ii) State the largest value of k so that the inverse function f 1 exists. Hence for this value
of k, define f 1 in a similar form.
[3]
(b) Functions g and h are defined as follows:
g : x ln(2 x) , x , x 2
1
h:x , x , x 0
1 x2
(i) Explain why hg does not exist.
[1]
(ii) Given that g1 is a restriction of g such that hg1 is a function, find the largest possible
domain for g1.
[2]
(iii) Hence find the range of hg1.
[1]
7. (a) The diagram below shows (not to scale) the region R which is bounded by the curve
3
y , the line y 5 x 1 and the y-axis. Find the exact area of R.
1 4 x2
[4]
y
y 5x 1
3
y
1 4 x2
R
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3
(b) Another region S is bounded by the curve y , the lines y 3 x , x = 1 and
1 4 x2
the x-axis. Find the volume generated when S is rotated through 2 about the y-axis,
giving your answer to 3 significant figures.
[5]
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8
1 x
8. (a) By using the substitution u 1 3 x , evaluate 5
3
x 1 3x
dx , leaving your answer in
exact form.
[4]
dy
(i) Given that y sin x e , find
sin x
(b) .
dx
Hence, or otherwise, show that (sin 2 x)e dx 2(sin x)e 2e C , where
sin x sin x sin x
C is an arbitrary constant.
[3]
(ii) Find (sin x)(sin 2 x)e
sin x
dx . [3]
x2 2x
9. The curve C has the equation y , where a is a constant such that a 0 and a 2 .
xa
(i) Find the equations of the asymptotes of C.
[3]
dy
(ii) Find . Hence find the set of values of a for which C has no stationary points. [3]
dx
(iii) Sketch C for a 1 , showing clearly the axial intercepts, asymptotes and stationary
points (if any).
[2]
Hence, or otherwise, find the range of values of k such that the equation
x 3 2 x 2 kx x 1 has exactly one real root.
[2]
10. The graph of y f (x) is as shown in the diagram below. Sketch, on separate diagrams,
the following graphs, indicating clearly the asymptotes and coordinates of the points
corresponding to the points A, B, C and D where appropriate.
1
(i) y f ( x 1)
2
[3]
2
(ii) y
f ( x)
[3]
(iii) y 2 f ( x ) , given that the gradient of the curve y f (x) at C (0, 0) is 2.
[4] y
y =f (x) x=1
y=½
A (-2,0)
+ + x
C
D (2, 0)
+
B (-1, -½ )
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n
n
1
11. (a) Prove by mathematical induction that r2
r 1
r
2 (2 n) for n .
2
2n
Hence, find r 2 r 2 4 .
r 2
[7]
1 1
(b) (i) Simplify 4r 2 2 .
4 r 2
[1]
(ii) Hence obtain the sum Sn of the first n terms of the series
2 3 4 5
2 2 2 2 2 2 ......
1 3 2 4 3 5 4 6
2 2
[4]
(iii) Determine nlim
Sn .
[1]
2
12. (a) Sketch the graph of y 2 . [2]
2x 1
(b) A sequence x0 , x1 , x2 ,..... of real numbers is defined by the recurrence relation
2 1
xn 1 2 where x0
2 xn 1 2
(i) The sequence converges to either one of 2 possible limits, and where .
By using your graph in part (a) or otherwise, find the values of and .
[2]
(ii) Show that if 0 xn , then
xn xn 1 .
[5]
(iii) It can also be shown that if xn , then xn 1 xn (do not show).
1
For the case where x0 0 ( x0 ), state the limit of the sequence.
2
[1]
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