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Practice Makes Improvement ST

The document outlines the instructions and content for the Wakiso-Kampala Teachers’ Association Pure Mathematics mock examination for the Uganda Advanced Certificate of Education, scheduled for July/August 2024. It includes sections A and B, with a total of 68 questions covering various mathematical topics such as geometry, calculus, and algebra, requiring candidates to show all working and adhere to specified guidelines. Candidates must answer all questions in Section A and any five from Section B, with clear instructions on the use of calculators and the format of answers.

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125 views16 pages

Practice Makes Improvement ST

The document outlines the instructions and content for the Wakiso-Kampala Teachers’ Association Pure Mathematics mock examination for the Uganda Advanced Certificate of Education, scheduled for July/August 2024. It includes sections A and B, with a total of 68 questions covering various mathematical topics such as geometry, calculus, and algebra, requiring candidates to show all working and adhere to specified guidelines. Candidates must answer all questions in Section A and any five from Section B, with clear instructions on the use of calculators and the format of answers.

Uploaded by

wekesaronald900
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Pa2s/1 PURE MATHEMATICS Paper 1 Jul/Aug. 2024 3 hours WAKISO-KAMPALA TEACHERS’ ASSOCIATION (WAKATA) WAKATA MOCK EXAMINATIONS 2024 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 mathematica! tables with a list of formulae may be used. Neat work is a must!! D> (©2024 Wakiso Kampala Teachers Association (WAKATA) Turn Over | »> SECTION A (40 MARKS) Answer all questions in this section, 1. Find the square root of 15 + 81 (05 marks) 2. Solve: 3 + 2sin2@ = 2sin@ + 3cos?6 for 0° <$ 8 < 360%. (05 marks) dy -b 3. Given that x = asecO,y = brand. Show tha [7 = = cot? (05 marks) 4. A triangle ABC has position vectors A(2t + 3/ + k), BSL + 4K) and C+ 2j + 12K). Find the area of the triangle. (05 marks) & A point P(2at, at?) lies on the parabola 4ay = x?. Given that the point § is (0,a) and M is the midpoint of PS. Show that the equation of the locus of M is given byx? +a? = 2ay (05 marks) 6. Using the subst 8. Evatutte {? — 7 (05 marks) . Using the substitution x = tan®, Evaluate J, ae marks) 7, The sum of the first n terms of a certain progression is )"= 3” . Find the least value of 7. such that the sum exceeds 10,000. (05 marks) 8. Solve the differential equation xB =xy + e*, given that y = O when x = 1, (05 marks) SECTION B (60 MARKS) Answer any five questions from this section. All questions carry equal marks, 9. (a) Given that 2, = 3~ f, Zp = 3 + i, Find the modulus and argument of Tap (04 marks) (b) Show that Z,= 2 and 2, = 1/. (—1 + #) are roots of the equation. 2z3 — 227. 32-2 = 0. (04 marks) (©) Use DeMoivre’s theorem to solve Z* + 1 = 0. (Leave your answer in surd form) (04 marks) a: 4 @y a4x ‘ 10. (a) Given that y = tan*( 5), show that 2” Gee (06 marks) (in 2x)(In*) (b) Show that © (loge x + log, 2) = Gnaay(in®/p) (06 marks) (nP\inx? 11, A and B are points whose position vectors are a = + k and b =~ j + 3k respectively. (a) Determine the position vector of the point P that divides AB in the ratio ~4 : 1. (b) Given that a = i — 3j + 3k and b = —i —3j + 2k, determine (i) the equation of the plane containing a and b, 1 (i) the angle the line makes with the plane in (i) above (12 marks) 2 1 12. (a) Expand (1 — x)3 as far as the term in x. Use your expansion to deduce (24) correct to 3sf (05 marks) (b) Use Maclaurin’s theorem to expand In Ge) as faras the term x3 (07 marks) ey ae EFDE=8) 13, Sketch the curve: y = ET SRAS (2 marks) in20 - 2sin40 + sin6o 14, (@) Show thar SPO—SSDEE TSE = 1 —sec20 (04 marks) sin20 + 2sin40 + sin60 4, 14sin26)'/2 _ 14 tane (Ss Ero Sit G = nat) T= tand ¥ Hence or otherwise solve for 6, if (+=) 24 V3 =0; ford < @ < 90% (08 marks) 15, (a) Given that r = 3tan is the polar equation of the circle; find its Cartesian form. (03 marks) (b) Prove that the chord joining the points P(ap?, 2ap) and Q(aq?, 2aq) on the parabola y? = 4ax has the equation (p + q)y = 2x + 2apq. A variable chord PQ of the ‘parabola is such that the lines OP and O@ are perpendicular, where O is the origin. (i) Prove that the chord PQ cuts the x — axis at a fixed point, and give the 2 — coordinate of this point, (i) Find the equation of the locus of the mid-point of PQ. (09marks) 3 Turn Over 26. 00pm end was ‘The temperature of a sick student was measured by the schoo! nurs¢ found to be 50°C. The nurse nociced that the temperature of the sick bay at that instant was a ae | 25°C. She again took the temperature of the student after one hour, when it showed 45°C. Assuming that the rate of change of the student's temperature was directly proportional to the difference between student's temperature, T and that of the sick bay. (@) (Write a differential equation to represent the rate of change of temperature of the student. (i) Using the conditions given, solve the differential equati (09 marks) (b) At what time of the day did the temperature of the student reduce to 38°C? (03 marks) END in town, 46 formu P425/1 PURE MATHEMATICS Paper 1 Nov 2020 3hrs ST. MARYS’ KITENDE Uganda Advanced Certificate of Education RESOURCEFUL MOCK EXAMINATIONS 2020 PURE MATHEMATICS Paper 1 3 hours INSTRUCTIONS TO CANDIDATES: Attempt all the eight questions in Section A andNot more than five from Section B. Any additional question(s) will not be marked. All working must be shown clearly. Silent non-programmabe calculators and mathematical tables with a list of formulae may be used. Graph papers are provided. SECTION A: (40MARKS) Answer all the eight questions in this Section. 1. Solve the simultaneous equations; += 4 ; = (Smarks) 2. Prove that; ‘eee 5 =logl0. (5marks) 3. Given the parabola y? = 8x, a) Express a point T on the parabola in parametric form using t as the parameter. (2marks) 2 b) If parameter r gives point R, show that the gradient of chord TR is —. (8marks) 4. Find f x%e*"dx. (Smarks) 5. The line r = () + o(;) meets a plane P perpendicularly at the point (3,1,2) Find the vector equation of the plane. (5marks) 6. Solve sin(120° + 3x) = cos(90° — x) for 0° < x < 90°. (5marks) 7. A roll of fencing material 152m long is used to enclose a rectangular area using two existing perpendicular walls. Find the maximum area enclosed. (marks) d: 8. Solve the differential equation ox —x = given that y =e when x =e. (Smarks) SECTION B : (60MARKS) 9. a) Prove that; "Hic + Thc = +2C_ (6marks) b) Two blue, three red and four black beads are to be arranged on a circular ring made of a wire so that the red are separated. Find the number of different arrangements. (6marks) 10. Given that; f(x) = = a) Find Maclaurin’s expansion of f(x) upto the term in x?. (8marks) 1.02 b) Hence, find the value of = to four significant figures. (4marks) . d: 11, a) Given that; ysinx + xcosy = > find 2. (4marks) b) A square prism is always three times the width in length. If the volume increases at a constant rate of 4cms-!, find the rate of change of the cross- sectional area when the width is 12cm. (8marks) Cc A 12. & a oO B ‘B Fig. 1 Figure 1 shows points A and B with position vectors a and b respectively. 3AC = BO. a) Express each of the following in terms of vectors aand b. i) BA (2marks) ii) BC (8marks) b) Find the ration BP: PC (7marks) 1 13. a) Prove that cos(tan~1x) = (x? +1). (4marks) 2, in? b) i) Prove that cos entcostetsi X= 3 for0

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