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P425/1
PURE MATHEMATICS
Paper one
July / August 2013
3 hours
INTERNAL MOCK EXAMINATIONS 2013
Uganda Advanced Certificate of Education
PURE MATHEMATICS
Paper one
3 hours
INSTRUCTIONS TO CANDIDATES:
Answer all the eight questions in section A and five questions from section B.
Any additional question(s) answered will not be marked.
All working must be shown clearly.
Begin each question on a fresh page.
Silent, non-programmable scientific calculators and mathematical tables with a
list of formulae may be used.
Turn over
© 2013 UMSSN Department of Mathematics
Page 1
SECTION A: (40 MARKS)
Answer all the questions in this section.
1.
2.
3.
4.
The roots of the equation x 2  17 x  16  0 are  4 and 4 . Form a
quadratic equation whose roots are α and β.
(05 marks)




Solve the equation 3 sin  2 x    cos 2 x    2 for 0  x  2 .
6
6


(05 marks)
The points A, B and C have position vectors a = 3i – j + 4k, b = j – 4k and
c = 6i + 4j + 5k respectively. Find the position vector of the point R on
BC such that AR is perpendicular to BC.
(05 marks)
Find the equation of the normal to the curve
where y   .
5.
6.
y
 3 at the point
x  sin y
(05 marks)
The sum of n terms of an A.P is 8n 2  n . Find the number of the term
which has a value of 263.
(05 marks)
Determine the Maclaurin’s series expansion of cos ecInx up to and
including the term in x2.
(05 marks)
7.
Find the equations of the two circles of radius 5units which pass through
the origin and whose centres lie on the line x  y  1  0 .
(05 marks)
8.
Integrate 2
3 x 1
with respect to x.
© 2013 UMSSN Department of Mathematics
(05 marks)
Page 2
SECTION B: (60 MARKS)
Answer any five questions from this section. All questions carry equal marks.
9.
(a) Prove by induction that 7 n  4 n  1n is always divisible by 6 for all
positive integral values of n.
(06 marks)
(b) Find the first four terms for the binomial expansion of 4
Hence evaluate 4
10.
2
.
16  5 x
1
correct to 3 significant figures.
261
(a) Prove that; 2 cot
A
A
 tan A  tan A cot2
2
2
(06 marks)
(05 marks)
(b) Solve the simultaneous equations tan 2 x  tan 2 y  0 and
tan x  tan y  3  0 for 0   x, y   180  .
11.
(07 marks)
(a) Investigate the stationary points of the curve y  sin 3 x cos x for
0  x  2 and distinguish between them.
12.
(b) Prove that the volume of the segment of height h cut from a sphere of
h

radius R is h 2  R   .
3

(12 marks)
(a) Show that the locus of the mid-point of the line joining the parabola
y 2  8 x and the point (8, 0) is also a parabola.
(b) (i)
Determine coordinates of the two points at which lines from the
focus of the new parabola are at right angles to the parabola
y 2  8x .
(ii)
Hence find the equations of the tangents to the points in b(i)
above.
(12 marks)
© 2013 UMSSN Department of Mathematics
Page 3
2
13.
(b) Find In3
14.
8x  6
 2x  1 x  2 dx
(a) Evaluate:
2
3
1
3x
3
2x
2
1
dx
1
(12 marks)
(a) The complex number w is such that Re (w) > 0 and w + 3w* = iw2,
where w* denotes the complex conjugate of w. Find w, giving your
answer in the form x + iy, where x and y are real.
(06 marks)
(b) On a sketch of an Argand diagram, shade the region whose points
represent complex numbers z which satisfy both the inequalities
⃓z -2i⃓≤ 2 and 0 ≤ arg(z + 2)≤ π/4. Calculate the greatest value of
⃓z⃓for points in this region, giving your answer correct to 2 decimal
places.
(06 marks)
The points A and B have position vectors 2i − 3j + 2k and 5i − 2j + k
respectively. The plane p has equation x + y = 5.
(i)Find the position vector of the point of intersection of the line through
A and B and the plane p.
(ii)A second plane q has an equation of the form x + by +cz = d, where b, c
and d are constants. The plane q contains the line AB, and the acute
angle between the planes p and q is 60o. Find the equation of q.
(12 marks)
4 y  2 x dx  x  y dy  0 given that y1  3
16. (a) Solve:
15.
(06 marks)
(b) In the recent IAAF marathon race, the Ugandan Golden boy was
observed running at a rate proportional to the remaining distance to be
covered. Assuming the observation was made at the red square a distance
5000m away from the finishing point where Kiprotich was running at a
rate of 5ms-1. How long did the Golden boy take to cover three-fifth of
the distance from the square?
(06 marks)
END
© 2013 UMSSN Department of Mathematics
Page 4