Cambridge A Level Mathematics - Further 9231 — 2002 Oct/Nov Paper 1 · Variant 1

CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level FURTHER MATHEMATICS 9231/1 PAPER 1 OCTOBER/NOVEMBER SESSION 2002 3 hours Additional materials: Answer paper Graph paper List of Formulae (MF10) TIME 3hours INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The use of a calculator is expected, where appropriate. Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit. You are reminded of the need for clear presentation in your answers. This question paper consists of 5 printed pages and 3 blank pages. Universrry of CAMBRIDGE @CIE 2002 Local Examinations Syndicate [Turn over

Given that pe ofthe, find x it in terms of N and x. ” n=) Hence determine the set of values of x for which the infinite series uy tu, +H, + is convergent and give the sum to infinity for cases where this exists. The equation 2 Ma +Ar tdy— 0, where A is a constant, has roots a, 8, y, 6. Find a polynomial equation whose roots are Given that P+ PrP eS a find the valuc of A, It is given that, tor 7 = 0, 1, 2,3, ..., a, = 17" + 3(9)" +20. Simplily a, It is given that, for 2 > 0, (i) Find i in terms of c. Gi) Show that n+l 1 m2 7 9 'n 7 De" i) Find 4, in terms of e. 923 ONNiO2 — a,, and hence prove by induction that a, is divisible by 24 for all 1 2 0. [3] 13] m1 (3]

3 The curve C has polar equation r@ = |, for 0 < 6 < 27. (i) Use the fact that ” tends to 1 as @ tends to 0 to show that the line with cartesian equation y = 1 is an asymptote to C. [2] (ii) Sketch C. ul The points P and Q on C cotrespond to @ = in and @ = :7 respectively. (iii) Find the area of the sector OPQ, where O is the origin. (3] (iv) Show that the length of the are PQ is (2] A curve has equation v7 +.xy? (i) Show that there is no point of the curve at which ae [41 ely en . dy @y . Gi) Find the values of — and —, at the point (1, -1). [5] dy dv Given that z = cos @ + isin @, show that (i) z-- = 2ising, en Zz Gi) 2" +27" = 2cosnd. [2] Hence show that sin? @ = 4(10— 15 cos 26 + 6 cos 46 — cos 68). [3] Find a similar expression for cos® @, and hence express cos® @ — sin® @ in the form acos 26 + bcos 68. (3) The value of the assets of a large commercial organisation at time ¢, measured in years, is $(L oy + 10°). The variables y and # are related by the differential equation ay ody . * +52 + 6y = [Scos 3¢ - 3 sin 3r. di- di Find y in terms of 7, given that y = 3 and 2 = -2 when? =0. [9] Show that, for large values of ¢, the value of the assets is less than $9.5 x 10° for about a third of the time. (3) 923 MauNiag [Turn over

4 9 The planes Hi, and I, which meet in the line /, have vector equations 2i+ 4j + Ok + 6 (21+ 3k) + >, (-4) + 5k), 21+ 4j + Ok + 6,(3] +k) + 9,(-1+j + 2k), r respectively. Find a vector equation of the line / in the form r = a + th. [5] Find a vector equation of the plane 7, which contains / and which passes through the point with position vector 4i + 3j + 2k. Find also the equation of IT, in the form av + by + cz = d. 4] Deduce, or prove otherwise, that the system of equations 6x — Sy —4z = -32, Sx- yt3z = 24, 9x — 2y + 5z = 40, has an infinite number of solutions. [3] 10 The linear transformation T : R* > R? is represented by the matrix H, where 1 2 3. -5 -1 4 5 J H= 23 #0 -3 3 5 #7 2 (i) Find the dimension of the range space of T. [3] Gi) Find a basis for the null space of T. [3] It is given that x satisfies the equation 2 Hx = a -I5 Using the fact that 1 2 -3 -10 H rt=t ur le -2 -I5 find the least possible value of |x|. (7) x wv. 2 2 [For the vector x = * ox] = VOT +5 +25 429) xy 923 ONNiO2

il 5 Auswer only one of the following two altematives. EITHER The vector e is an cigenvector of the square matrix G. Show that (i) ¢ is an eigenvector of G + Al, where & is a scalar and I is an identity matrix, ii) e is an cigenvector of G-. (51 Find the eigenvalues, and corresponding eigenvectors, of the matrices A and B’, where OR The curve C has equation where «, &, ¢ are constants, and it is given thatO <a <b <c. (i) Expre: y in the form giving the constants P and Q in terms of a, & and c. XtP+ _ (x-a)lv-h) xe 2 x-o (ii) Find the equations of the asymptotes of C. ) Show that C has cwo stationary points. -3 0 -8 ') . (9] 3 -6 (S] (iv) Given also that @ + & > c, sketch C, showing the asymptotes and the coordinates of the points of intersection of C with the axes. 923 ONNiO2 [4]