# * * MATHEMATICS (MEI) 4751 Introduction to Advanced Mathematics (C1) ADVANCED SUBSIDIARY GCE. Monday 11 January 2010 Morning QUESTION PAPER

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2 Answer all questions on the Printed Answer Book provided. Section A (6 marks) Rearrange the formula c = a + b to make a the subject. [] Solve the inequality 5x < x + 5. [] (i) Find the coordinates of the point where the line 5x + y = 0 intersects the x-axis. [] (ii) Find the coordinates of the point of intersection of the lines 5x + y = 0 and y = 5 x. [] 4 (i) Describe fully the transformation which maps the curve y = x onto the curve y = (x + 4). [] (ii) Sketch the graph of y = x 4. [] 5 (i) Find the value of 44. [] (ii) Simplify Give your answer in the form a + b 7. [] c 6 You are given that f(x) = (x + ) (x 5). (i) Sketch the graph of y = f(x). [] (ii) Express f(x) in the form ax + bx + cx + d. [] 7 When x + x + 5x + k is divided by (x + ), the remainder is 6. Find the value of k. [] 8 Find the binomial expansion of (x + 5 x ), simplifying the terms. [4] 9 Prove that the line y = x 0 does not intersect the curve y = x 5x + 7. [5] OCR Jan0

3 Section B (6 marks) 0 y C D B A O x Fig. 0 Fig. 0 shows a trapezium ABCD. The coordinates of its vertices are A (, ), B (6, ), C (, 5) and D (, ). (i) Verify that the lines AB and DC are parallel. [] (ii) Prove that the trapezium is not isosceles. [] (iii) The diagonals of the trapezium meet at M. Find the exact coordinates of M. [4] (iv) Show that neither diagonal of the trapezium bisects the other. [] A circle has equation (x ) + (y + ) = 5. (i) State the coordinates of the centre of this circle and its radius. [] (ii) Verify that the point A with coordinates (6, 6) lies on this circle. Show also that the point B on the circle for which AB is a diameter has coordinates (0, ). [] (iii) Find the equation of the tangent to the circle at A. [4] (iv) A second circle touches the original circle at A. Its radius is 0 and its centre is at C, where BAC is a straight line. Find the coordinates of C and hence write down the equation of this second circle. [] [Question is printed overleaf.] OCR Jan0 Turn over

5 475 Mark Scheme 475 (C) Introduction to Advanced Mathematics [ a = ]c b www o.e. M for each of complete correct steps, ft from previous error if equivalent difficulty condone = used for first two Ms 5x < x+ 0 M M0 for just 5x < (x + 5) x < M or < x or ft x < o.e. M or ft; isw further simplification of /; M0 for just x < 4. (i) (4, 0) allow y = 0, x = 4 bod B for x = 4 but do not isw: 0 for (0, 4) seen 0 for (4, 0) and (0, 0) both given (choice) unless (4, 0) clearly identified as the x-axis intercept (ii) 5x + (5 x) = 0 o.e. (0/, 5/) www isw M A for subst or for multn to make coeffts same and appropriate addn/subtn; condone one error or A for x = 0/ and A for y = 5/ o.e. isw; condone. or better and.67 or better A for (.,.7) 4 (i) translation 4 by or 4 [units] to left 0 4 (ii) sketch of parabola right way up and with minimum on negative y-axis min at (0, 4) and graph through and on x-axis B B B B 0 for shift/move or 4 units in negative x direction o.e. mark intent for both marks must be labelled or shown nearby 5 (i) or ± M for 44 o.e. or for 44 = soi 5 (ii) denominator = 8 B B0 if 6 after addition numerator = ( 5+ 7) = as final answer M A for M, allow in separate fractions allow B for as final answer 8 www

6 475 Mark Scheme January 00 6 (i) cubic correct way up and with two turning pts B touching x-axis at, and through it at.5 and no other intersections y- axis intersection at 5 B B intns must be shown labelled or worked out nearby 6 (ii) x x 8x 5 B for terms correct or M for correct expansion of product of two of the given factors 7 attempt at f( ) k = 6 8 k = x + 5x+ + www isw x x or x + 5x + 75x - + 5x - www isw M A A 4 or M for long division by (x + ) as far as obtaining x x and A for obtaining remainder as k 4 (but see below) equating coefficients method: M for (x + )(x x + 8) [+6] o.e. (from inspection or division) eg M for obtaining x x + 8 as quotient in division 5 B for both of x and x and or 5x - isw M for soi; A for each of 5x and 75 x or 75x- isw or SC for completely correct unsimplified answer

8 475 Mark Scheme January 00 0 (iii) [BD eqn is] y = M eg allow for at M, y = or for subst in eqn of AC eqn of AC is y 5 =6/5 (x ) o.e [ y =.x +.4 o.e.] M is (4/, ) o.e. isw 0 (iv) midpt of BD = (5/, ) or equivalent simplified form cao midpt AC = (/, ) or equivalent simplified form cao or M is / of way from A to C conclusion neither diagonal bisects the other M A M M A or M for grad AC = 6/5 o.e. (accept unsimplified) and M for using their grad of AC with coords of A(, ) or C (, 5) in eqn of line or M for stepping method to reach M allow : at M, x = 6/ o.e. [eg =4/] isw A0 for. without a fraction answer seen or showing BM MD oe [BM = 4/, MD = 7/] or showing AM MC or AM MC in these methods A is dependent on coords of M having been obtained in part (iii) or in this part; the coordinates of M need not be correct; it is also dependent on midpts of both AC and BD attempted, at least one correct alt method: show that mid point of BD does not lie on AC (M) and vice-versa (M), A for both and conclusion 4

9 475 Mark Scheme January 00 (i) centre C' = (, ) radius 5 0 for ± 5 or 5 (ii) showing (6 ) + ( 6 + ) = 5 B interim step needed showing that AC = C B = 4 o.e. B or B each for two of: showing midpoint of AB = (, ); showing B (0, ) is on circle; showing AB = 0 or B for showing midpoint of AB = (, ) and saying this is centre of circle or B for finding eqn of AB as y = 4/ x + o.e. and B for finding one of its intersections with the circle is (0, ) or B for showing C B = 5 and B for showing AB = 0 or that AC and BC have the same gradient or B for showing that AC and BC have the same gradient and B for showing that B (0, ) is on the circle (iii) grad AC' or AB = 4/ o.e. grad tgt = /their AC' grad y ( 6) = their m(x 6) o.e. y = 0.75x 0.5 o.e. isw (iv) centre C is at (, 4) cao circle is (x ) + (y + 4) = 00 M M M A B B or ft from their C', must be evaluated may be seen in eqn for tgt; allow M for grad tgt = ¾ oe soi as first step or M for y = their m x + c then subst (6, 6) eg A for 4y = x 4 allow B4 for correct equation www isw B for each coord ft their C if at least one coord correct 5

10 475 Mark Scheme January 00 (i) 0 (ii) [x =] 5 or ft their (i) ht = 5[m] cao (iii) d = 7/ o.e. [y =] /5.5 (0.5) o.e. or ft = 9/0 o.e. cao isw (iv) 4.5 = /5 x(0 x) o.e..5 = x(0 x) o.e. x 0x + 45 [= 0] o.e. eg x 0x +.5 [=0] or (x 5) =.5 0 ± 40 [ x = ] or 5± 0 o.e. 4 width = 0 o.e. eg.5 cao M M A M M A M A not necessarily ft from (i) eg they may start again with calculus to get x = 5 or ft their (ii).5 or their (i).5 o.e. or 7 /5.5 or ft or showing y 4 = /0 o.e. cao eg 4.5 = x( 0.x) etc cao; accept versions with fractional coefficients of x, isw or x 5= [ ± ].5 o.e.; ft their quadratic eqn provided at least M gained already; condone one error in formula or substitution; need not be simplified or be real accept simple equivalents only 6

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