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P425/1
PURE MATHEMATICS
Paper I
Jul/Aug, 2023
3 hours
WAKISO-KAMPALA TEACHERS’ ASSOCIATION (WAKATA)
WAKATA MOCK EXAMINATIONS 2023
Uganda Advanced Certificate of Education
PURE MATHEMATICS
Paper 1
3 hours
INSTRUCTIONS TO CANDIDATES:
Answer all the eight questions in section A. and any five questions from section B
Any additional question(s) answered will not be marked.
‘All necessary working must be clearly shown.
Begin each answer on a fresh sheet of paper.
Silent, non ~ programmable scientific calculators and mathematical tables with alist of formulae
may be used.
Neat work is a must!!
YL
\
a
4 SECTION A (40 MARKS)
Answer all questions in this section.
(05 marks)
Solve the equation: (Z + 1-2)? + 4i =3-
Given that sin(@ + a) = aand sin(@ + B) = b-
— 2b? 05
Show that cos2(a@ — 8) — 4abcos(« — B) = 1— 2a" — 2b (05 marks)
Find 2 if y= gx2+2x, ‘ (05 marks)
A straight line joining the points (2,1, 4) and (a — 1, 4, —1) is parallel to the line
joining points (0, 2, b — 1) and (5, 3,—2). Find the values of @ and Db. (05 marks)
a -
P and Q are two points whose coordinates are (at, 2at), (3,
ea
ar (05 marks)
7 ) respectively and
S is a point (a, 0). Show that $+
6
Z
Evaluate [ (2S
= \(1 - cosx))
) dx
2, (05 marks)
The quadratic equation (P + 1)x? — 6(P + 1)x + 3(P +9) = 0, P # 1h
equal roots, Find the roots of the equation, (05 marks)
Solve the differential equation; 2 +2 = 2x3 (05 marks)
x 10.
12,
13,
SECTION B (60 MARKS)
Answer any five questions from this section, All questions carry equal marks
(a) The polymomial f (x) = ax? + 3x? + bx —3 is exactly divisible by (2x + 3)
and leaves a remainder ~3 when divided by (x + 2). Find the values of @ and .
(07 marks)
(0) Given that (x ~ 2) and (x — =) are factors of ax? + Sx + b
Show that a = b (05 marks)
i 3 ae a
(a) Given that y3 — 3xy’ + 3x?y, Find az (06 marks)
(b) The volume V of. liquid in container is given by V = (3h? + 4)2 — 8 ; where
‘he mis the depth of the liquid. The liquid is leaking from the container. It is
observed that, when the depth of the liquid is 0.6m, the depth is decreasing at a
rate of 0.015m per hour. Find the rate at which'the volume of liquid in the container
is decreasing at the instant when the depth is 0.6m. (06 marks)
1 :
(a) The lines a J ae perpendicular.
Find the value of k, (04 marks)
(b) Find the coordinates of the point where the line through (3, —4, —5) and
(2,—3, 1) crosses the plane 2x +y +2=7. (08 marks)
2x? +5x—10
in X2.
pe meaisteaer i as far as the te es
Expand Gee Ty Cx Dy sseending powers of 2 as fara the term in
(12 marks)
(a) Sketch the curvey =x? +2. (06 marks)
(b) The area bounded by the curve and the line y = x + 8 is rotated about the
xx —axis through one revolution. Determine the volume of the solid generated.
(06 marks)
3 Turn Over 14,
15.
16.
(a)
(b)
(@)
(b)
Ina culture, the bacteria count is 100,000. The
Jn how many hours will the count reach 200,000, if the rate of growth of bacteria is
proportional to the number present?
sinx
cosx 1+ cosx + st (08 marks,
Prove that; —————_ =
= sinx 1+ cosx —
40sx= 1g’ < x < 360% (4 marks
Solve the equation 2sin x =
tanx
Find the equation of the ellipse whose focus is (1, —2), the directrix
3x — 2y +5 = 0 and accentricity equal to 72. (04 marks)
Points A and B are 10km a part and it is determined from the sound of an explosion
heard at those points at different times that the location of the explosion is 6km
closer to A than B. Show that the location of the explosion is restricted to a hyperbola
2 2
whose equation is = — Fe = (08 marks)
number is increased by 10% in 2 hours.
(12 marks)
END
