OCR ADVANCED SUBSIDIARY GCE IN MATHEMATICS (3890, 389 and 389) OCR ADVANCED GCE IN MATHEMATICS (7890, 789 and 789) Specimen Question Papers and Mark Schemes These specimen question papers and mark schemes are intended to accompany the OCR Advanced Subsidiary GCE and Advanced GCE specifications in Mathematics for teaching from September 004. Centres are permitted to copy material from this booklet for their own internal use. The specimen assessment material accompanying the new specifications is provided to give centres a reasonable idea of the general shape and character of the planned question papers in advance of the first operational examination.
CONTENTS Unit Name Unit Code Level Unit 47: Core Mathematics C AS Unit 47: Core Mathematics C AS Unit 473: Core Mathematics 3 C3 A Unit 474: Core Mathematics 4 C4 A Unit 475: Further Pure Mathematics FP AS Unit 476: Further Pure Mathematics FP A Unit 477: Further Pure Mathematics 3 FP3 A Unit 478: Mechanics M AS Unit 479: Mechanics M A Unit 4730: Mechanics 3 M3 A Unit 473: Mechanics 4 M4 A Unit 473: Probability and Statistics S AS Unit 4733: Probability and Statistics S A Unit 4734: Probability and Statistics 3 S3 A Unit 4735: Probability and Statistics 4 S4 A Unit 4736: Decision Mathematics D AS Unit 4737: Decision Mathematics D A
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 47 Core Mathematics Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are not permitted to use a 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
Write down the exact values of (i) (ii) (iii) 4, [] ( ), [] 3 3 3 ( + + 3 ). [] (i) Express x 8x+ 3 in the form ( x+ a) + b. [3] (ii) Hence write down the coordinates of the minimum point on the graph of y = x 8x+ 3. [] 3 The quadratic equation x + kx+ k = 0 has no real roots for x. (i) Write down the discriminant of x + kx+ k in terms of k. [] (ii) Hence find the set of values that k can take. [4] 4 Find d y dx in each of the following cases: (i) (ii) 3 y = 4x, [] y = x ( x + ), [3] (iii) y = x [] 5 (i) Solve the simultaneous equations y = x 3x+, y = 3x 7. [5] (ii) What can you deduce from the solution to part (i) about the graphs of y = x 3x+ and y = 3x 7? [] (iii) Hence, or otherwise, find the equation of the normal to the curve y= x 3x+ at the point (3, ), giving your answer in the form ax + by + c = 0 where a, b and c are integers. [4] 47 Specimen Paper
3 6 (i) Sketch the graph of y =, where x 0, showing the parts of the graph corresponding to both x positive and negative values of x. [] (ii) Describe fully the geometrical transformation that transforms the curve y = to the curve y x = x +. Hence sketch the curve y = x +. [5] (iii) Differentiate x with respect to x. [] (iv) Use parts (ii) and (iii) to find the gradient of the curve y x at the point where it crosses the y-axis. [3] 7 The diagram shows a circle which passes through the points A (, 9) and B (0, 3). AB is a diameter of the circle. (i) Calculate the radius of the circle and the coordinates of the centre. [4] (ii) Show that the equation of the circle may be written in the form x + y x y+ 47 = 0. [3] (iii) The tangent to the circle at the point B cuts the x-axis at C. Find the coordinates of C. [6] 47 Specimen Paper [Turn over
4 8 (i) Find the coordinates of the stationary points on the curve 3 y = x 3x x 7. [6] (ii) Determine whether each stationary point is a maximum point or a minimum point. [3] (iii) By expanding the right-hand side, show that 3 x 3x x 7 = ( x+ ) (x 7). [] 3 (iv) Sketch the curve y = x 3x x 7, marking the coordinates of the stationary points and the points where the curve meets the axes. [3] 47 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 47 Core Mathematics MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
(i) B For correct value (fraction or exact decimal) 6 (ii) 8 B For correct value 8 only (iii) 6 M For 3 + 3 + 3 3 = 36 seen or implied A For correct value 6 only 4 (i) x 8x+ 3 = ( x 4) 3 B For ( x 4) seen, or statement a = 4 i.e. a = 4, b= 3 M For use of (implied) relation a + b= 3 A 3 For correct value of b stated or implied (ii) Minimum point is (4, 3) Bt For x-coordinate equal to their ( a) Bt For y-coordinate equal to their b 5 3 (i) Discriminant is k 4k M For attempted use of the discriminant A For correct expression (in any form) (ii) For no real roots, k 4k < 0 M For stating their < 0 Hence kk ( 4) < 0 M For factorising attempt (or other soln method) So 0< k < 4 A For both correct critical values 0 and 4 seen A 4 For correct pair of inequalities 6 4 (i) dy x dx n nx A For completely correct answer (ii) 4 y = x + x B For correct expansion dy 3 Hence = 4x + 4x dx M For correct differentiation of at least one term At 3 For correct differentiation of their terms (iii) dy = d x x M For clear differentiation attempt of x A For correct answer, in any form 7 5 (i) x 3x+ = 3x 7 x 6x+ 9= 0 M For equating two expressions for y A For correct 3-term quadratic in x Hence ( x 3) = 0 M For factorising, or other solution method So x = 3 and y = A For correct value of x A 5 For correct value of y (ii) The line y = 3x 7 is the tangent to the curve B For stating tangency y = x 3x+ at the point (3, ) B For identifying x = 3, y = as coordinates (iii) Gradient of tangent is 3 B For stating correct gradient of given line Hence gradient of normal is 3 Bt For stating corresponding perpendicular grad Equation of normal is y = ( x 3) 3 M For appropriate use of straight line equation i.e. x+ 3y 9= 0 A 4 For correct equation in required form 47 Specimen Paper
3 6 (i) B B For correct st quadrant branch For both branches correct and nothing else (ii) Translation of units in the negative x-direction B For translation parallel to the x-axis B For correct magnitude B For correct direction Bt B For correct sketch of new curve 5 For some indication of location, e.g. at y-intersection or at asymptote (iii) Derivative is x M For correct power in answer A For correct coefficient (iv) Gradient of y = at x = is required x B For correctly using the translation This is, which is M For substituting x = in their (iii) 4 A 3 For correct answer 7 (i) AB = (0 ) + (3 9) = 00 M For correct calculation method for AB Hence the radius is 5 A For correct value for radius + 0 9+ 3 Mid-point of AB is, M For correct calculation method for mid-point Hence centre is (6, 6) A 4 For both coordinates correct (ii) Equation is ( x 6) + ( y 6) = 5 M For using correct basic form of circle equn This is x x+ 36 + y y+ 36 = 5 A For expanding at least one bracket correctly i.e. x + y x y+ 47 = 0, as required A 3 For showing given answer correctly (iii) Gradient of AB is 3 9 = 3 0 4 M For finding the gradient of AB 3 A For correct value or equivalent 4 Hence perpendicular gradient is 4 3 At For relevant perpendicular gradient 4 Equation of tangent is y 3 = ( x 0) 3 M For using their perp grad and B correctly Hence C is the point ( 3, 0) 4 M For substituting y = 0 in their tangent eqn A 6 For correct value x = 3 4 3 47 Specimen Paper [Turn over
4 8 (i) dy = 6x 6x dx M For differentiation with at least term OK A For completely correct derivative Hence x x = 0 M For equating their derivative to zero ( x )( x+ ) = 0 x= or M For factorising or other solution method A For both correct x-coordinates Stationary points are (, 7) and (, 0) A 6 For both correct y-coordinates (ii) d y 8 when x x 6 dx + = 8 when x M For attempt at second derivative and at least one relevant evaluation Hence (, 7) is a min and (, 0) is a max A For either one correctly identified A 3 For both correctly identified (Alternative methods, e.g. based on gradients either side, are equally acceptable) (iii) RHS = ( x + x+ )( x 7) M For squaring correctly and attempting 3 = x 7x + 4x 4x+ x 7 complete expansion process 3 = x 3x x 7, as required A For obtaining given answer correctly (iv) B For correct cubic shape B For maximum point lying on x-axis B 3 For x = 7 and y = 7 at intersections 4 47 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 47 Core Mathematics Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphic 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
Expand 4 ( x) in ascending powers of x, simplifying the coefficients. [5] (i) Find dx. [3] x dy (ii) The gradient of a curve is given by =. Find the equation of the curve, given that it passes dx x through the point (, 3). [3] 3 (a) Express each of the following in terms of log x : (i) (ii) log ( x ), [] log (8 x ). [3] (b) Given that y = 7, find the value of log3 y. [3] 4 Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 400. Calculate the publisher s forecasts for (i) the number of copies that will be sold in the 0th week after publication, [3] (ii) the total number of copies sold during the first 0 weeks after publication, [] (iii) the total number of copies that will ever be sold. [] 5 (i) Show that the equation 5cos θ = 3 + sinθ may be written as a quadratic equation in sinθ. [] (ii) Hence solve the equation, giving all values of θ such that 0 θ 360. [6] 47 Specimen Paper
3 6 The diagram shows triangle ABC, in which Calculate AB = 3cm, AC = 5cm and angle ABC =. radians. (i) angle ACB, giving your answer in radians, [] (ii) the area of the triangle. [3] An arc of a circle with centre A and radius 3 cm is drawn, cutting AC at the point D. (iii) Calculate the perimeter and the area of the sector ABD. [4] 7 The diagram shows the curves y = 3x 9x+ 30 and y = x + 3x 0. (i) Verify that the curves intersect at the points A ( 5,0) and B (, 0). [] (ii) Show that the area of the shaded region between the curves is given by ( 4 x x + 40)d x. [] 5 (iii) Hence or otherwise show that the area of the shaded region between the curves is 8. [5] 3 47 Specimen Paper [Turn over
4 8 The diagram shows the curve y =.5 x. (i) A point on the curve has y-coordinate. Calculate its x-coordinate. [3] (ii) Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line x = 4. [4] (iii) State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate. [] (iv) Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region. [] 9 The cubic polynomial 3 x + ax + bx 6 is denoted by f( x ). (i) The remainder when f( x ) is divided by ( x ) is equal to the remainder when f( x ) is divided by ( x + ). Show that b = 4. [3] (ii) Given also that ( x ) is a factor of f( x ), find the value of a. [] (iii) With these values of a and b, express f( x ) as a product of a linear factor and a quadratic factor. [3] (iv) Hence determine the number of real roots of the equation f( x ) = 0, explaining your reasoning. [3] 47 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 47 Core Mathematics MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
3 4 8x+ 4x 3x + 6x B For first two terms 8x M For expansion in powers of ( x) M For any correct use of binomial coefficients A For any one further term correct A 5 For completely correct expansion 5 (i) x dx = x + c M For any attempt to integrate x A For correct expression (in any form) B 3 For adding an arbitrary constant (ii) y = x + c passes through (, 3), so 3 = + c c= 4 M For attempt to use (, 3) to evaluate c At For correct value from their equation Hence curve is y = + 4 x A 3 For correct equation 6 3 (a) (i) log x B For correct answer -------------------------------------------------------------------------------------------------------------------------------------- (ii) log (8 x ) = log8 + log x M For relevant sum of logarithms M 3 For relevant use of 8= = 3+ log x A 3 For correct simplified answer (b) log3 y = log37 M For taking logs of both sides of the equation Hence log 3 3 y = A For any correct expression for log3 y A 3 For correct simplified answer 7 4 (i) 400 r = = 0.8 3000 B For the correct value of r 9 Forecast for week 0 is 3000 0.8 43 M n For correct use of ar A 3 For correct (integer) answer (ii) 0 n 3000( 0.8 ) a( r ) = 4 87 M For correct use of 0.8 r A For correct answer (3sf is acceptable) (iii) 3000 a =5 000 M For correct use of 0.8 r A For correct answer 7 5 (i) LHS is 5( sin θ ) M For using the relevant trig identity Hence equation is 5sin θ + sinθ = 0 A For correct 3-term quadratic (ii) (5sinθ + )(3sinθ ) = 0 M For factorising, or other solution method Hence sin θ = or 5 3 A For both correct values So θ = 9.5,60.5, 03.6, 336.4 M For any relevant inverse sine operation A For any one correct value At For corresponding second value At 6 For both remaining values 8 x 47 Specimen Paper
3 6 (i) 3 5 3 = sinc = sin. 5 sinc sin. M For any correct initial statement of the sine rule, together with an attempt to find sin C Hence C = 0.544 A For correct value (ii) Angle A is π. 0.5444 = 0.497 M For calculation of angle A Area is 5 3 sin 0.497 M For any complete method for the area i.e. 3.58 cm At 3 For correct value, following their C (iii) Sector perimeter is 6 + 3 0.497 M For using rθ with their A in radians i.e. 7.49 cm At For correct value, following their A Sector area is 3 0.497 M For using r θ with their A in radians i.e..4 cm At 4 For correct value, following their A 9 7 (i) 75 + 45 + 30 = 0, 5 5 0 = 0 B For checking one point in both equations 8 + 30 = 0, 4 + 6 0 = 0 B For checking the other point in both (ii) Area is {( 3 x 9 + 30) ( + 3 0)}d x 5 M For use of ( y y)dx i.e. ( 4 x + 40)d, as required A For showing given answer correctly 5 3 (iii) EITHER: Area is 4 x 6x 40x + 3 5 M For integration attempt with one term OK A For at least two terms correct A For completely correct indefinite integral = ( 3 4 + 80) ( 500 50 00) M For correct use of limits 3 3 = 8 3 A For showing given answer correctly OR: Area under top curve is M For complete evaluation attempt A For correct indefinite integration (allow for other curve if not earned here) x 3 9 30 7 5 x x + x = Area above lower curve is 3 x 3 x 0x 57 3 5 6 A For correct value + = M For evaluation and sign change So area between is 7 + 57 = 8 A 5 For showing given answer correctly 6 3 9 8 (i).5 = xlog.5 = log B For correct initial use of logs log Hence x = = 3. M For correct log expression for x log.5 A 3 For correct numerical value 0 3 4 (ii) {.5 + (.5 +.5 +.5 ) +.5 } B For correct recognition of h = M For any use of values.5 x for x = 0,, 4 M For use of correct formula Area is 6.49 A 4 For correct answer (iii) The trapezia used in (ii) extend above the curve M For stating or sketching trapezia above curve Hence the trapezium rule overestimates the area A For stating overestimate with correct reason (iv) Use more trapezia, with a smaller value of h B For stating that more trapezia should be used 0 47 Specimen Paper [Turn over
4 9 (i) 8+ 4a+ b 6= 8+ 4a b 6 M For equating f() and f( ) A For correct equation Hence 4b= 6 b= 4 A 3 For showing given answer correctly (ii) + a 4 6= 0 M For equating f() to 0 (not f( ) ) Hence a = 9 A For correct value (iii) f( x) = ( x )( x + 0x+ 6) M For quadratic factor with x and/or + 6 OK A For trinomial with both these terms correct A 3 For completely correct factorisation (iv) The discriminant of the quadratic is 76 M For evaluating the discriminant Hence there are 3 real roots altogether M For using positive discriminant to deduce that there are roots from the quadratic factor A 3 For completely correct explanation of 3 roots 47 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 473 Core Mathematics 3 Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphic 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
Solve the inequality x+ > x. [5] (i) Prove the identity sin( x+ 30 ) + ( 3)cos( x+ 30 ) cos x, where x is measured in degrees. [4] (ii) Hence express cos5 in surd form. [] 3 The sequence defined by the iterative formula with x =, converges to α. x 3 n+ = (7 5 xn), (i) Use the iterative formula to find α correct to decimal places. You should show the result of each iteration. [3] (ii) Find a cubic equation of the form 3 x + cx+ d = 0 which has α as a root. [] (iii) Does this cubic equation have any other real roots? Justify your answer. [] 4 The diagram shows the curve y =. (4x + ) The region R (shaded in the diagram) is enclosed by the curve, the axes and the line x =. (i) Show that the exact area of R is. [4] (ii) The region R is rotated completely about the x-axis. Find the exact volume of the solid formed. [4] 473 Specimen Paper
3 5 At time t minutes after an oven is switched on, its temperature θ C is given by 0.t θ = 00 80e. (i) State the value which the oven s temperature approaches after a long time. [] (ii) Find the time taken for the oven s temperature to reach 50 C. [3] (iii) Find the rate at which the temperature is increasing at the instant when the temperature reaches 50 C. [4] 6 The function f is defined by f: x + x for x 0. (i) State the domain and range of the inverse function f. [] (ii) Find an expression for f ( x). [] (iii) By considering the graphs of y = f( x) and y = f ( x), show that the solution to the equation f( x) = f ( x) is x = (3+ 5). [4] 7 (i) Write down the formula for tan x in terms of tan x. [] (ii) By letting tan x = t, show that the equation becomes 4 tan x+ 3cot xsec x = 0 4 3t 8t 3= 0. [4] (iii) Hence find all the solutions of the equation 4 tan x+ 3cot xsec x = 0 which lie in the interval 0 x π. [4] 473 Specimen Paper [Turn over
4 8 The diagram shows the curve y = (ln x). (i) Find d y dx and d y dx. [4] (ii) The point P on the curve is the point at which the gradient takes its maximum value. Show that the tangent at P passes through the point (0, ). [6] 9 The diagram shows the curve y = tan x and its asymptotes y =± a. (i) State the exact value of a. [] (ii) Find the value of x for which tan x= a. [] The equation of another curve is y = tan ( x ). (iii) Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of a. [3] (iv) Verify by calculation that the value of x at the point of intersection of the two curves is.54, correct to decimal places. [] Another curve (which you are not asked to sketch) has equation y ( tan x) =. (v) Use Simpson s rule, with 4 strips, to find an approximate value for ( tan x) 0 dx. [3] 473 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 473 Core Mathematics 3 MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
EITHER: 4x + 4x+ > x x+ M For squaring both sides i.e. 3x + 6x> 0 A For reduction to correct quadratic So xx+ ( ) > 0 M For factorising, or equivalent Hence x< or x > 0 A For both critical values correct A For completely correct solution set OR: Critical values where x+ = ± ( x ) M For considering both cases, or from graphs i.e. where x = and x = 0 B For the correct value A For the correct value 0 Hence x< or x > 0 M For any correct method for solution set using two critical values A 5 For completely correct solution set 5 (i) sin x( 3) + cos x( ) + ( 3)(cos x( 3) sin x( )) M For expanding both compound angles A For completely correct expansion M For using exact values of sin30 and cos30 = cos x+ 3 cos x= cos x, as required A 4 For showing given answer correctly (ii) sin 45 + ( 3) cos 45 = cos5 M For letting x = 5 throughout + 3 Hence cos5 = A For any correct exact form 6 3 3 (i) x = 7 =.99... B For.9 seen or implied x3 =.957..., x4 =.9346... M For continuing the correct process α =.94 to dp A 3 For correct value reached, following x 5 and x 6 both.94 to dp 3 3 (ii) x = (7 5 x) x + 5x 7 = 0 M For letting xn = xn+ = x (or α) A For correct equation stated 3 (iii) EITHER: Graphs of y = x and y = 7 5x only cross once M For argument based on sketching a pair of graphs, or a sketch of the cubic by calculator Hence there is only one real root At For correct conclusion for a valid reason OR: d ( 3 5 7) 3 x x x 5 0 + = + > M For consideration of the cubic s gradient dx Hence there is only one real root At For correct conclusion for a valid reason 7 x x x 0 0 4 (i) (4 + ) d = (4 + ) = (3 ) = M For integral of the form k(4x+ ) A For correct indefinite integral M For correct use of limits A 4 For given answer correctly shown (ii) d ln(4 ) ln9 4 0 4 0 4x x = π x + = + π M For integral of the form kln(4x+ ) A For correct ln(4 ) 4, with or without π M Correct use of limits and π A 4 For correct (simplified) exact value 8 473 Specimen Paper
3 5 (i) 00 C B For value 00 0.t 0.t (ii) 50 = 00 80e e = 50 80 M For isolating the exponential term Hence 0.t = ln 5 t =.8 8 M For taking logs correctly A 3 For correct value.8 (minutes) (iii) dθ 0.t = 8e dt M For differentiation attempt A For correct derivative 0..8 Hence rate is 8e = 5.0 C per minute M For using their value from (ii) in their θ A 4 For value 5.0(0) 6 (i) Domain of f is x B For the correct set, in any notation Range is x 0 B Ditto (ii) If y = + x, then x = ( y ) M For changing the subject, or equivalent Hence f ( x) = ( x ) A For correct expression in terms of x (iii) The graphs intersect on the line y = x B For stating or using this fact Hence x satisfies x = ( x ) B For either 8 x = f( x) or x = f ( x) 3± 5 i.e. x 3x+ = 0 x = M For solving the relevant quadratic equation So (3 5) as x must be greater than A 4 For showing the given answer fully 8 tanx 7 (i) tan x = B For correct RHS stated tan x 8t (ii) + 3 ( + t ) = 0 B For cot x = seen t t t B For sec x = + t seen Hence 8t + 3( t )( + t ) = 0 M For complete substitution in terms of t 4 i.e. 3t 8t 3= 0, as required A 4 For showing given equation correctly (iii) (3t + )( t 3) = 0 M For factorising or other solution method Hence t =± 3 A For t = 3 found correctly So x = π, π, 4π, 5π A For any two correct angles 3 3 3 3 A 4 For all four correct and no others 9 473 Specimen Paper [Turn over
4 8 (i) dy lnx = dx x M For relevant attempt at the chain rule A For correct result, in any form d y x(/ x) lnx lnx = = dx x x M For relevant attempt at quotient rule A 4 For correct simplified answer (ii) For maximum gradient, ln x = 0 x= e M For equating second derivative to zero A For correct value e Hence P is (e,) At For stating or using the y-coordinate The gradient at P is e At For stating or using the gradient at P Tangent at P is y = ( x e) e M For forming the equation of the tangent Hence, when x = 0, y = as required A 6 For correct verification of (0, ) 9 (i) a = π B For correct exact value stated (ii) x = tan( π ) = M For use of x= tan( a) 4 At For correct answer, following their a (iii) B For x-translation of (approx) + B For y-stretch with (approx) factor 0 Asymptotes are y =± a B 3 For correct statement of asymptotes x tan x tan ( x ) (iv).535 0.993 0.983 M For relevant evaluations at.535,.545.545 0.996 0.998 Hence graphs cross between.535 and.545 A For correct details and explanation (v) Relevant values of ( ) tan x are (approximately) M For the relevant function values seen or 0, 0.0600, 0.50, 0.44, 0.669 implied; must be radians, not degrees {0 4(0.0600 0.44) 0.50 0.669} + + + + M For use of correct formula with h = 4 Hence required approximation is 0.45 A 3 For correct ( or 3sf) answer 4 473 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 474 Core Mathematics 4 Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphic 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
Find the quotient and remainder when 4 x + is divided by x +. [4] (i) Expand ( ) in ascending powers of x, up to and including the term in x 3 x. [4] (ii) State the set of values for which the expansion in part (i) is valid. [] 3 Find x xe dx, giving your answer in terms of e. [5] 0 4 As shown in the diagram the points A and B have position vectors a and b with respect to the origin O. (i) Make a sketch of the diagram, and mark the points C, D and E such that OC = a, OD = a+ b and OE = OD. [3] 3 (ii) By expressing suitable vectors in terms of a and b, prove that E lies on the line joining A and B. [4] 5 (i) For the curve x + xy+ y = 4, find d y dx in terms of x and y. [4] (ii) Deduce that there are two points on the curve x + xy+ y = 4 at which the tangents are parallel to the x-axis, and find their coordinates. [4] 474 Specimen Paper
3 6 The diagram shows the curve with parametric equations x = asin θ, y = aθcosθ, where a is a positive constant and π θ π. The curve meets the positive y-axis at A and the positive x-axis at B. (i) Write down the value of θ corresponding to the origin, and state the coordinates of A and B. [3] (ii) Show that d y = θ tanθ, and hence find the equation of the tangent to the curve at the origin. [6] dx 7 The line L passes through the point (3, 6,) and is parallel to the vector i+ 3j k. The line L passes through the point (3,, 4) and is parallel to the vector i j+ k. (i) Write down vector equations for the lines L and L. [] (ii) Prove that L and L intersect, and find the coordinates of their point of intersection. [5] (iii) Calculate the acute angle between the lines. [4] 8 Let I = x( + x) dx. (i) Show that the substitution u = x transforms I to u( + u) du. [3] (ii) Express u( + u) in the form A + B + C. [5] u + u ( + u) (ii) Hence find I. [4] 474 Specimen Paper [Turn over
4 9 A cylindrical container has a height of 00 cm. The container was initially full of a chemical but there is a leak from a hole in the base. When the leak is noticed, the container is half-full and the level of the chemical is dropping at a rate of cm per minute. It is required to find for how many minutes the container has been leaking. To model the situation it is assumed that, when the depth of the chemical remaining is x cm, the rate at which the level is dropping is proportional to x. Set up and solve an appropriate differential equation, and hence show that the container has been leaking for about 80 minutes. [] 474 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 474 Core Mathematics 4 MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
x x 4 + x = + + x + B For correct leading term x in quotient M For evidence of correct division process A For correct quotient x A 4 For correct remainder 4 (i) 3 ( )( ) ( x) = + ( )( x) + ( x) + ( 3 5 3 + x + 3! M For nd, 3rd or 4th term OK (unsimplified) 3 5 3 = + x+ x + x A For + x correct A For 3 x 3 A 4 For 5 x (ii) Valid for x < B For any correct expression(s) 5 3 4 (i) x x x = 0 0 0 xe dx xe e dx M For attempt at parts going the correct way x x x e e 4 0 A For correct terms x x x e e dx = M For consistent attempt at second integration M For correct use of limits throughout 3 = 4 4 e A 5 For correct (exact) answer in any form 5 B B Bt For C correctly located on sketch For D correctly located on sketch 3 For E correctly located wrt O and D (ii) AE = ( a+ b) a= ( b a) M For relevant subtraction involving OE 3 3 A For correct expression for ( ± ) AE or EB Hence AE is parallel to AB A For correct recognition of parallel property i.e. E lies on the line joining A to B A 4 For complete proof of required result 7 5 (i) dy dy 4x x y y 0 dx dx dy x y dx B dy For correct term y dx Hence d y 4 x+ y = M For solving for d y dx x + y dx A 4 For any correct form of expression (ii) dy 0 y 4x dx y 0 dx Hence x + ( 4 x ) + ( 4 x) = 4 M For solving simultaneously with curve equn i.e. x = A For correct value of x (or y ) So the two points are (, 4) and (, 4) A 4 For both correct points identified 8 474 Specimen Paper
3 6 (i) θ = 0 at the origin B For the correct value A is (0, aπ ) B For the correct y-coordinate at A B is ( a,0) B 3 For the correct x-coordinate at B (ii) dx = acosθ dθ B For correct differentiation of x dy = a(cosθ θsin θ) dθ M For differentiating y using product rule Hence d y cos θ θ sin θ = = θ tanθ M For use of dy = dy dx dx cosθ dx dθ dθ A For given result correctly obtained Gradient of tangent at the origin is M For using θ = 0 Hence equation is y = x A 6 For correct equation 6 (i) L : r = 3 i + 6 j + k + s( i + 3 j k ) M For correct RHS structure for either line L : r = 3 i j + 4 k + t( i j + k ) A For both lines correct (ii) 3+ s = 3 + t,6+ 3s = t, s = 4+ t M For at least equations with two parameters First pair of equations give s =, t = M For solving any relevant pair of equations A For both parameters correct Third equation checks: + = 4 A For explicit check in unused equation Point of intersection is (, 3, ) A 5 For correct coordinates (iii) + 3 ( ) + ( ) = ( 4)( 6)cosθ B For scalar product of correct direction vectors B For correct magnitudes 4 and 6 M For correct process for cosθ with any pair of vectors relevant to these lines Hence acute angle is 56.9 A 4 For correct acute angle 9 8 (i) I = udu = du M For any attempt to find d x d u or u ( + u) u( + u) du dx A For dx = udu or equivalent correctly used A 3 For showing the given result correctly (ii) A( + u) + Bu( + u) + Cu M For correct identity stated A = B For correct value stated C = B For correct value stated 0= A + B (e.g.) A For any correct equation involving B B = A 5 For correct value (iii) lnu ln( + u) + + u Bt For Aln u+ Bln( + u) with their values Bt For C( + u) with their value Hence I = ln x ln( + x) + + c + x M For substituting back A 4 For completely correct answer (excluding c) 474 Specimen Paper [Turn over
4 9 dx = k x dt M For use of derivative for rate of change A For correct equation (neg sign optional here) dx x = 00 and = k = 0. dt M For use of data and their DE to find k Hence equation is d x = 0. x dt A For any form of correct DE x dx = 0. dt x = 0.t+ c M For separation and integration of both sides A For x correct At For ( ± )kt correct (the numerical evaluation of k may be delayed until after the DE is solved) B For one arbitrary constant included (or equivalent statement of both pairs of limits) x = 00, t = 0 c= 00 M For evaluation of c So when x = 00, 00 = 0.t+ 00 M For evaluation of t i.e. t = 8.8 A For correct value 8.8 (minutes) 474 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 475 Further Pure Mathematics Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphic 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
Use formulae for n r and r= n r to show that r= n rr ( + ) = nn ( + )( n+ ). [5] 3 r= The cubic equation 3 x 6x + kx+ 0= 0 has roots p q, p and p+ q, where q is positive. (i) By considering the sum of the roots, find p. [] (ii) Hence, by considering the product of the roots, find q. [3] (iii) Find the value of k. [3] 3 The complex number + i is denoted by z, and the complex conjugate of z is denoted by z *. (i) Express z in the form x+ i y, where x and y are real, showing clearly how you obtain your answer. [] (ii) Show that 4z z simplifies to a real number, and verify that this real number is equal to zz *. [3] z + (iii) Express z in the form x+ i y, where x and y are real, showing clearly how you obtain your answer. [3] 4 A sequence u, u, u 3, is defined by n u = 3. n (i) Write down the value of u. [] (ii) Show that 8 3 n n un u + =. [3] (iii) Hence prove by induction that each term of the sequence is a multiple of 8. [4] 475 Specimen Paper
3 5 (i) Show that r r 4r = +. [] (ii) Hence find an expression in terms of n for + + + +. [4] 3 5 35 4n (iii) State the value of (a) 4r, [] r= (b) 4r. [] r= n+ 6 In an Argand diagram, the variable point P represents the complex number z = x+ i y, and the fixed point A represents a = 4 3i. (i) Sketch an Argand diagram showing the position of A, and find a and arga. [4] (ii) Given that z a = a, sketch the locus of P on your Argand diagram. [3] (iii) Hence write down the non-zero value of z corresponding to a point on the locus for which (a) the real part of z is zero, [] (b) arg z = arga. [] 7 The matrix A is given by A =. (i) Draw a diagram showing the unit square and its image under the transformation represented by A. [3] (ii) The value of det A is 5. Show clearly how this value relates to your diagram in part (i). [3] A represents a sequence of two elementary geometrical transformations, one of which is a rotation R. (iii) Determine the angle of R, and describe the other transformation fully. [3] (iv) State the matrix that represents R, giving the elements in an exact form. [] 475 Specimen Paper [Turn over
4 8 The matrix M is given by a M = 3, where a is a constant. (i) Show that the determinant of M is a. [] (ii) Given that a 0, find the inverse matrix M. [4] (iii) Hence or otherwise solve the simultaneous equations x+ y z =, x+ 3y z =, x y+ z = 0. [3] (iv) Find the value of k for which the simultaneous equations y z = k, x+ 3y z =, x y+ z = 0, have solutions. [3] (v) Do the equations in part (iv), with the value of k found, have a solution for which x= z? Justify your answer. [] 475 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 475 Further Pure Mathematics MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
n n n 6 r= r= r= rr ( + ) = r + r= nn ( + )( n+ ) + nn ( + ) M For considering the two separate sums A For either correct sum formula stated A For completely correct expression = nn ( + )(n+ + 3) = nn ( + )( n+ ) M For factorising attempt 6 3 A 5 For showing given answer correctly (i) ( p q) + p+ ( p+ q) = 6 p = M For use of Σ α = b/ a A For correct answer (ii) ( q)( + q) = 0 Bt For use of αβγ = d/ a Hence 4 q = 5 q = 3 M For expanding and solving for A 3 For correct answer (iii) EITHER: Roots are,, 5 Bt For stating or using three numerical roots + 5+ 5= k M For use of Σ αβ = c/ a i.e. k = 3 At For correct answer from their roots OR: Roots are,, 5 Bt For stating or using three numerical roots Equation is ( x+ )( x )( x 5) = 0 M For stating and expanding factorised form Hence k = 3 At 3 For correct answer from their roots 3 (i) z = (+ i) = 4 + 4i + i = 3+ 4i M For showing 3-term or 4-term expansion A For correct answer (ii) 4z z = 8+ 4i 3 4i= 5 B For correct value 5 zz * = (+ i)( i) = 5 B For stating or using z * = i B 3 For correct verification of given restult (iii) z + 3 + i (3 + i)( i) 4 i = = = = i B For correct initial form 3 + i z + i (+ i)( i) + i M For multiplying top and bottom by i A 3 For correct answer i 4 (i) u = 8 B For correct value stated ( n+ ) n n n n ( n+ ) (ii) 3 (3 ) = 9 3 3 = 8 3 B For stating or using u n + 3 M For relevant manipulation of indices in u n + A 3 For showing given answer correctly (iii) u is divisible by 8, from (i) B For explicit check for u Suppose u k is divisible by 8, i.e. uk = 8a M For induction hypothesis u k is mult. of 8 k k Then uk+ uk 8 3 8( a 3 ) 8b u k + i.e. u k + is also divisible by 8, and result follows by the induction principle A 4 For correct conclusion, stated and justified 5 8 8 8 q 475 Specimen Paper
3 r+ (r ) 5 (i) LHS = = = RHS M For correct process for adding fractions (r )(r+ ) 4r A For showing given result correctly (ii) Sum is ( ) + ( ) + ( ) +... + ( ) 3 3 5 5 7 n n + M For expressing terms as differences using (i) A For at least first two and last terms correct This is n + M For cancelling pairs of terms A 4 For any correct form (iii) (a) Sum to infinity is Bt For correct value; follow their (ii) if cnvgt -------------------------------------------------------------------------------------------------------------------------------------- (b) Required sum is n + Bt For correct difference of their (iii)(a) and (ii) 8 6 (i) (See diagram in part (ii) below) B For point A correctly located a = (3 + 4 ) = 5 B For correct value for the modulus ( ) 3 4 arga = tan = 0.644 M For any correct relevant trig statement A 4 For correct answer (radians or degrees) (ii) 7 (i) B B B For any indication that locus is a circle For any indication that the centre is at A 3 For a completely correct diagram (iii) (a) z = 6i B For correct answer -------------------------------------------------------------------------------------------------------------------------------------- (b) z = 8 6i M For identification of end of diameter thru A A For correct answer 0 0 0 0 = 0 0 0 3 M A A For at least one correct image For all vertices correct 3 For correct diagram (ii) The area scale-factor is 5 B For identifying det as area scale factor The transformed square has side of length 5 M For calculation method relating to large sq. So its area is 5 times that of the unit square A 3 For a complete explanataion (iii) Angle is tan () = 63.4 B For tan (), or equivalent Enlargement with scale factor 5 B For stating enlargement B 3 For correct (exact) scale factor (iv) 5 5 cosθ sinθ M For correct sinθ cosθ pattern 5 5 A For correct matrix in exact form 475 Specimen Paper [Turn over
4 8 (i) det M = a(3 ) ( ( )) ( 6) M For correct expansion process = a A For showing given answer correctly (ii) M = 4 a a a + 8 a+ 4 3a 4 M For correct process for adjoint entries A For at least 4 correct entries in adjoint B For dividing by the determinant A 4 For completely correct inverse (iii) x y = M, with a = z 0 B For correct statement involving inverse So x = 0, y =, z = M For carrying out the correct multiplication A 3 For all three correct values (iv) Eliminating x gives 4y z = M For eliminating x from nd and 3rd equns So for consistency with st equn, k = M For comparing two y-z equations A 3 For correct value for k (v) Solving x+ 3y =,3x y = 0 gives x=, y = 3 5 5 M For using x = z to solve a pair of equns These values check in y x =, so soln exists A For a completely correct demonstration 4 475 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 476 Further Pure Mathematics Specimen Paper Additional materials: Answer booklet Graph paper List of Formulae (MF ) TIME 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 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphic 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 7. 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 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
(i) Starting from the definition of cosh x in terms of e x, show that cosh x = cosh x. [] (ii) Given that cosh x = k, where k >, express each of cosh x and sinh x in terms of k. [4] The diagram shows the graph of x + 3x+ 3 y =. x + (i) Find the equations of the asymptotes of the curve. [3] (ii) Prove that the values of y between which there are no points on the curve are 5 and 3. [4] 3 (i) Find the first three terms of the Maclaurin series for ln( + x). [4] (ii) Write down the first three terms of the series for ln( x), and hence show that, if x is small, then + x ln x. [3] x 476 Specimen Paper
3 4 The equation of a curve, in polar coordinates, is r = cos θ ( π < θ π). (i) Find the values of θ which give the directions of the tangents at the pole. [3] One loop of the curve is shown in the diagram. (ii) Find the exact value of the area of the region enclosed by the loop. [5] 5 The diagram shows the curve y = x + together with four rectangles of unit width. (i) Explain how the diagram shows that + + + < 3 4 5 4 0 d x + x. [] The curve y = passes through the top left-hand corner of each of the four rectangles shown. x + (ii) By considering the rectangles in relation to this curve, write down a second inequality involving + + + and a definite integral. [] 3 4 5 (iii) By considering a suitable range of integration and corresponding rectangles, show that 000 ln(500.5) < < ln(000). [4] r r= 476 Specimen Paper [Turn over
4 6 (i) Given that n In = x ( x) dx, prove that, for n, 0 (n+ 3) In = ni. [6] n (ii) Hence find the exact value of I. [4] 7 The curve with equation has one stationary point for x > 0. y = x cosh x (i) Show that the x-coordinate of this stationary point satisfies the equation xtanh x = 0. [] The positive root of the equation xtanh x = 0 is denoted by α. (ii) Draw a sketch showing (for positive values of x) the graph of y = tanh x and its asymptote, and the graph of y =. Explain how you can deduce from your sketch that α >. [3] x (iii) Use the Newton-Raphson method, taking first approximation x =, to find further approximations x and x 3 for α. [5] (iv) By considering the approximate errors in x and x, estimate the error in x 3. [3] 8 (i) Use the substitution t = tan x to show that π 0 cosx t dx= dt. [4] + sin x ( + t)( + t ) 0 (ii) Express t ( + t)( + t ) in partial fractions. [5] (iii) Hence find 0 π cosx + sinx d x, expressing your answer in an exact form. [4] 476 Specimen Paper
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS 476 Further Pure Mathematics MARK SCHEME Specimen Paper MAXIMUM MARK 7 This mark scheme consists of 4 printed pages. OCR 004 Registered Charity Number: 066969 [Turn over
x x x x RHS = (e + e ) = (e + e ) = LHS M For correct squaring of (e + e ) (i) ( ) A For completely correct proof cosh x = k cosh x= ( + k) M For use of (i) and solving for cosh x (ii) ( ) A For correct positive square root only sinh x+ = k sinh x=± ( k ) M For use of cosh x sinh x =, or equivalent ( ) A 4 For both correct square roots x x (i) x = is an asymptote B For correct equation of vertical asymptote y = x+ + x + M For algebraic division, or equivalent Hence y = x+ is an asymptote A 3 For correct equation of oblique asymptote (ii) EITHER: Quadratic x + (3 y) x+ (3 y) = 0 M For using discriminant of relevant quadratic has no real roots if (3 y) < 8(3 y) A For correct inequality or equation in y Hence (3 y)( 5 y) < 0 M For factorising, or equivalent So required values are 3 and 5 A For given answer correctly shown OR: dy = d x ( x + ) = 0 M For differentiating and equating to zero Hence ( x + ) = A For correct simplified quadratic in x So x = and 0 y = 5 and 3 M For solving for x and substituting to find y A 4 For given answer correctly shown 6 7 3 (i) EITHER: If f( x) = ln( x+ ), then f( x) = + x M For at least one differentiation attempt and f( x) = ( + x) A For correct first and second derivatives f(0) = ln, f (0) =, f (0) = At For all three evaluations correct 4 8 Hence ln( x+ ) = ln + x x +... A For three correct terms OR: ln( + x) = ln[( + x)] M For factorising in this way = ln + ln( + x) A For using relevant log law correctly ( x) = ln + x +... M For use of standard series expansion = ln + x x +... A 4 For three correct terms 8 (ii) ln( x) ln x x Bt For replacing x by x 8 + x ln (ln + x x ) (ln x x ) 8 8 x M For subtracting the two series x, as required A 3 For showing given answer correctly 7 476 Specimen Paper
3 4 (i) r = 0 cosθ = 0 θ =± π, ± 3 π M For equating r to zero and solving for θ 4 4 A For any two correct values A 3 For all four correct values and no others (ii) Area is i.e. π π 4 4 4cos θ dθ M For us of correct formula π 4 4 cos4 d sin4 π 4 4 π 4 π Bt For correct limits from (i) r d + θ θ = θ + θ = π M For using double-angle formula A For 4 sin 4θ A 5 For correct (exact) answer θ 5 (i) LHS is the total area of the four rectangles B For identifying rectangle areas (not heights) RHS is the corresponding area under the curve, which is clearly greater B For correct explanation (ii) 4 + + + > d 3 4 5 0 x + x M For attempt at relevant new inequality A For correct statement (iii) Sum is the area of 999 rectangles M For considering the sum as an area again Bounds are 999 0 d x + x and So lower bound is [ ] 999 and upper bound is [ ] 999 0 999 0 d x + x M 8 For stating either integral as a bound ln( x + ) = ln(500.5) A For showing the given value correctly ln( x + ) = ln(000) A 4 Ditto 0 6 (i) 3 3 n n 3 3 0 0 In = x ( x) + n x ( x) dx M For using integration by parts A For correct first stage result n 3 0 = n x ( x) ( x) dx M For use of limits in integrated term M For splitting the remaining integral up = ni ( 3 n In) A For correct relation between In and In Hence (n+ 3) In = nin, as required A 6 For showing given answer correctly (ii) I = 4I = 4 I M For two uses of the recurrence relation 7 7 5 0 Hence I 3 8 6 = ( x) 35 = 3 05 0 8 A For correct expression in terms of I 0 M For evaluation of I 0 A 4 For correct answer 0 476 Specimen Paper [Turn over