1
1. a. Show that ( A B)' (A B) = B' where A and B are any two
sets . [3]
b. Given that 13(4 ) 7(6 ) and 3(8 ) 25(6 )x y x y+= = , show
that 25 18752 and 656 21952
x y= = ,hence
find the value of x correct to 2 decimal places [5] 2. a Let a
be a positive real number. Find, in terms of a , the range of
values of x for which the
inequality 1
12 +
> x
xax
is satisfied. [4]
b. Hence, deduce the solution to the inequality 1
12 +
> x
xax
. [3]
3. a. Use the substitution xu = , show that 223lnd
1
9
4+=
x
xx . [4]
b. Hence, find xx
x d )1ln( 9
4 exactly. [4]
4. Write down the first three terms in ascending powers of x of
the expansion of 2)1( x .
State the coefficient of nx and the range of values of x for
which the expansion is valid.
Hence, or otherwise, find the value of
=1 2n nn . [6]
5. The diagram below shows the cross-section of a cylinder of
radius x that is inscribed in a sphere of
fixed internal radius R. Show that 2 2 2 2 216 ( )A x R x= ,
where A is the curved surface area of the
cylinder. Prove that, as x varies, the maximum value of A is
obtained when the height of the
cylinder is equal to its diameter. [9]
R
x
3
10. If A and B are squares matrices such that BA = pI , where p
is a constant and I is the identity
matrix, show that -1 1 A Bp
= .
Let A = 3 2 01 0 01 3 3
, find 2B=7I A [6]
11. By using the trapezium rule with 5 ordinates , obtain an
approximation for the integral, giving your answer correct to two
decimal places.
4
2
21
x dxx
[8]
12. If y = 2
21 ,4
x x Rx+
, show that y has no values between 1 and 14
Find the stationary point of y Hence, sketch the graph of y and
label all its asymptotes. Use the graph to deduce the number of
roots if 2 2( 4) 32 32x x x = + [11] END OF QUESTIONS PAPER
