C4 Jan 06

OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certi? cate of Education Advanced General Certi? cate of Education MATHEMATICS Core Mathematics 4 Monday 4724 Afternoon 1 hour 30 minutes 23 JANUARY 2006 Additional materials: 8 page answer booklet Graph paper List of Formulae (MF1) TIME 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES • • • • Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 signi? cant ? gures unless a different degree of accuracy is speci? d in the question or is clearly appropriate. You are permitted to use a graphical calculator in this paper. INFORMATION FOR CANDIDATES • • • • The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. You are reminded of the need for clear presentation in your answers. This question paper consists of 3 printed pages and 1 blank page. © OCR 2006 [R/102/2711] Registered Charity Number: 1066969 Turn over 2 1 Simplify x3 ? 3×2 . x2 ? 9 dy in terms of x and y. dx [3] 2 Given that sin y = xy + x2 , ? nd [5] 3 (i) Find the quotient and the remainder when 3×3 ? 2×2 + x + 7 is divided by x2 ? 2x + 5. [4] (ii) Hence, or otherwise, determine the values of the constants a and b such that, when [2] 3×3 ? 2×2 + ax + b is divided by x2 ? 2x + 5, there is no remainder. 4 (i) Use integration by parts to ? nd (ii) Hence ? nd x sec2 x dx. [4] [3] x tan2 x dx. 5 A curve is given parametrically by the equations x = t2 , y = 2t. (i) Find dy in terms of t, giving your answer in its simplest form. dx [2] ii) Show that the equation of the tangent to the curve at (p2 , 2p) is py = x + p2 . [2] (iii) Find the coordinates of the point where the tangent at (9, 6) meets the tangent at (25, ? 10). [4] 6 (i) Show that the substitution x = sin2 ? transforms 1 x dx to 1? x 2 sin2 ? d? . [4] (ii) Hence ? nd 0 x dx. 1? x [5] 7 The expression 11 + 8x is denoted by f(x). (2 ? x)(1 + x)2 (i) Express f(x) in the form (ii) Given that | x | < 1, ? nd the ? rst 3 terms in the expansion of f(x) in ascending powers of x. A B C + + , where A, B and C are constants. 2 ? x 1 + x (1 + x)2 [5] [5] 4724/Jan06 3 8 (i) Solve the differential equation iving the particular solution that satis? es the condition y = 4 when x = 5. (ii) Show that this particular solution can be expressed in the form dy 2 ? x = , dx y ? 3 [5] where the values of the constants a, b and k are to be stated. (iii) Hence sketch the graph of the particular solution, indicating clearly its main features. 9 (x ? a)2 + (y ? b)2 = k, [3] [3] Two lines have vector equations r= where a is a constant. 4 2 ? 6 +t ?8 1 ? 2 and r= ?2 a ? 2 +s ?9 2 ? 5 , (i) Calculate the acute angle between the lines. (ii) Given that these two lines intersect, ? nd a and the point of intersection. [5] [8] 4724/Jan06 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable e ort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 4724/Jan06