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P425/1
PURE MATHEMATICS
PAPER 1
22ND JULY, 2012
3 HOURS.
UGANDA ADVANCED CERTIFICATE OF EDUCATION
INTERNAL MOCK EXAMINATIONS
PURE MATHEMATICS
PAPER 1
3 HOURS
INSTRUCTIONS.

Attempt ALL the EIGHT questions in section A and only FIVE from
section B.

Begin each answer on a fresh sheet of paper.

No paper should be given for rough work.

Mathematical tables and squared paper are provided.

Silent, non-programmable calculators may be used.

State the degree of accuracy at the end of each answer attempted a
calculator or table; and indicate CAL for calculator or TAB for
mathematical tables.
© INTERNAL MOCK EXAMINATIONS 2012
1
TURN OVER
SECTION A (40 marks)
1. Given that A is an acute angle such that Sin 2 A  Cos 2 A  2Sin 2 A  0 , find the
exact value of tan A .
(5 marks)
2. The tangent to curve at a point x, y  passes through the point  x,0 . If the
curve passes through the point 1,4 , show that the equation of the curve is
y 2  16 x .
(5 marks)
3. Prove by induction; 1 2  2  2  3  2  ...  n(2
0
1
2
n 1
)  1  (n  1)2 n .
(5 marks)
4. Find  e x ln 1  e x dx .
(5 marks)
5. Find the distance of the point P(4,1,2) from the line;
  3
 5 
 
 
r   4     4
 1 
 3 
 
 
(5 marks)
6. If the line 3x – 4y – 12 = 0 is the tangent to the circle with a centre at (1, 1). Find
the equation of that circle.
(5 marks)
7. Expand
1  4 x  up to the term in x
3
. State the range of values of x for which
the expansion is valid. Taking x  400 , estimate 11 to 3dp . (5 marks)
8. Given that y  e
tan 1 x
, show that
d2y
dy
(1  x ) 2  (2 x  1)
 0.
dx
dx
(5 marks)
2
SECTION B (60 marks)
9. a) Given that log a b c  m, log b a c  n ; prove that log c a 
1 n
.
1  mn
(6 marks)
b) i) Find the number of arrangements of all the letters of the word CALCULUS.
ii) How many of the arrangements in (i) above are the A and S separate?
(6 marks)
2
3
10. a) Given that z  4 z  5 is a factor of z  az  b ; find the values of the real
numbers a, b.
(6 marks)
Z
b) Given that Z  x  iy , and the quantity
is a real number, find and sketch
Z 4
the locus of the point representing Z.
(6 marks)
© INTERNAL MOCK EXAMINATIONS 2012
2
TURN OVER




11. a) Solve the equation Sin   30 0  Cos   60 0  Cos ; for 0 0    360 0 .
b) A and B are points a metres apart on level ground. A tower OT stands,
vertically, at point O in the line AB such that the angles of elevation of T from A
and B are  and  respectively. C is a point in AB such that CB  b , prove that


a
the elevation of T from C is tan 1 

 a  b Cot  bCot 
(6 marks)
12. a) Differentiate with respect to x ;
i)
e 3 x Sin 3 x
ii) x ( x
1  2 x 
2
 Cot 2 x )
(5 marks)
b) A cylinder of height h is inscribed in a right pyramid of square base
12cm 12cm , and of slant length of 9cm, given that the upper rim of the
cylinder touches the slant faces of the pyramid, show that its volume is
(7 marks)
4h3  2h . Hence find the maximum volume.
 x 2  3x  4
does not
x 8
lie, hence determine the coordinates of the stationary points; hence sketch the
curve.
(12 marks)
13. Find the range of values of y within which the curve y 
14. a) Evaluate;


2
Sin 3 2 xdx .
(5 marks)
4
b) A(1,2) is a point on the curve y  x 2  1 , the area bounded by the curve, the
y - axis and the tangent to the curve at A is rotated about the y - axis. Find the
volume generated.
(7 marks)

15. a) Show that the equation of the tangent at P ct , c
 on the hyperbola xy  c is
2
t
x  t 2 y  2ct .
(4 marks)
b) The normal, to the tangent at P , through the origin O (0,0) meets the tangent
at N, and Q is the point such that ONPQ is a rectangle; find;
i)
the coordinates of N and Q in terms of t.
ii)
the loci of N and Q.
(8 marks)
© INTERNAL MOCK EXAMINATIONS 2012
3
TURN OVER
dy
 y 1
dx
b) In a certain type of chemical reaction a substance A is continuously
transformed into a substance B. Throughout the reaction the sum of masses of A
and B remains constant and equal to m. The mass of B present at time t after the
commencement of the reaction is denoted by x . At any instant, the rate of
increase of the mass of B is proportional to the mass of A.
i)
Write down the differential equation relating x and t .
1
ii)
Given that x  0 when t  0 and also x  m when t  In2 , show that at
2
16. a) Find the general solution of the differential equation x( x  1)


time t , x  m 1  e t .
iii)
Hence find the value of t when x 
3
m
4
(12 marks)
END
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