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Pure Mathematics Question Paper
Pure Mathematics
Course:A-Level
Institution: Mock question papers
Exam Year:2011
P425/1
PURE MATHEMATICS
PAPER 1
3 HOURS
RESOURCEFUL-MOCK EXAMINATIONS 2011
Uganda Advanced Certificate of Education
PURE MATHEMATICS
Paper 1
3 hours
INSRUCTIONS TO CANDIDATES:
Answer all the eight questions in section A and any five from section B.
All the necessary working must be shown clearly.
Silent non-programmable scientific calculators and mathematical tables with a list of formulae may be used.
SECTION A: (40 MARKS)
Answer all questions in this section.
Solve the inequality: 1/(x+1) = 1/(2x-4) (05 marks)
In triangle ABC, prove that; bc/(ab+ac) =(csc? (B+C))/(csc? B+csc? C) (05 marks)
One stationary point of the curve y = (ax+b)/(x^2+1) (2, 1). Find the values of a and b. (05 marks)
Part of the line 4x+3y=13 is a chord to the curve represented by the parametric equations, x=(t+5)/(t-1),y=13-4t .find the length of the chord. (05 marks)
The sum of the first 4 terms of the geometric progression 1+2x+(2+p) x^2+(2+q) x^3+?…………..is 0. find the values of p and q and the sum to infinity of the series.
(05 marks)
In a right pyramid with a square base, the sum of its height and the perimeter of its base is 36. Find the maximum volume of the pyramid. (05 marks)
O, A and B are non-collinear points, and OA=a, OB =b and C is the mid-point of AB.D is a point in OB such that OD = 1/4 OB.T is the point of intersection of OC and AD. Find vector OT in terms of a and b. (05 marks)
Show that ?_0^1¦?x^5/(1+x^3 )^2 dx?= -1/3(In0.5+0.5) (05marks)
SECTION B: (60MARKS)
Answer any five questions from this section. All questions carry equal marks.
9. (a) solve the equation :sin?3?+sin?2?=1 for 0°<?<360°. (06marks)
(b) Find the exact value of x in tan^(-1)?2 =2tan^(-1)??(1-x)?.
10. (a) Differentiate x^2+sin??x^2 ? from first principles. (06 marks)
(b)Find the first two non-zero terms of the maclaurin’s expansion of In(1-x^2 ), hence evaluate In(0.96) to 3dp.
11. (a) when the polynomial x^3+4x^2+ax+b is divided by x^2+2x+4, the remainder is 2x-4. Determine the values of a and b, and state the quotient. (06 marks)||
(b) Prove by mathematical induction that
sin??+sin??3?+sin??5?+?…..sin??(2n-1)?=(?sin?^2 n?)/sin?? ? ? ? (06 marks)
12. Given the curve y =45/(x^2+4x-5)
Find the equations of the asymptotes
Find the equation of the line of symmetry; hence deduce the coordinates of the stationary point of the curve.
Sketch the curve; hence state the range of values of y within which the curve does not lie. (12 marks)
13. Sketch the following loci
0=arg?(z-1-2i)<p/4
|z-3| =3
(b)Given that 2+i is a root of the equation z^3-6z^2+aZ+b=0
White down the quadratic factor of the polynomialz^3-6z^2+az+b,where a,b are real numbers (03 marks)
Find the real root of the original equation. (05 marks)
14.L is the line r=[3,1,-2]+t[2,-1,2];
Find the value of k if L lies in the plane r.[1,-2,-2]=k
Calculate the distance of the points (3, 1, 7) from L.
Determine the angle between L and the plane2x+4y-2z=5 (04 marks)
15. Find (a)?¦?x^2v(e?^(?-x?^3 )-4) dx (07 marks)
(b)?¦?(4x(?x^2+2)?^0.5)/(x^4-4) dx? (07 marks)
16. (a) use the method of infinite small changes to estimate tan??(44.95°)? (05 marks)
(b)ABCD is a perimeter fence completely fenced using a chain link of length 90m, forming a trapezium.DO=3X, BC=4X
Show that CD =(45-3x)m.
Find the maximum area of the trapezium.
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