Further Mathematics SAMPLE Marking Scheme This marking scheme has been prepared as a guide only to markers. This is not a set of model answers, or the exclusive answers to the questions, and there will frequently be alternative responses which will provide a valid answer. Markers are advised that, unless a question specifies that an answer be provided in a particular form, then an answer that is correct (factually or in practical terms) must be given the available marks. If there is doubt as to the correctness of an answer, the relevant NCC Education materials should be the first authority. Throughout the marking, please credit any valid alternative point. Where markers award half marks in any part of a question, they should ensure that the total mark recorded for the question is rounded up to a whole mark.
Answer ALL questions Question a) Use the factor theorem to show that (x 4) is a factor of f(x) = x 3 x 5x f(4) = (4) 3 (4) 5(4) = 0 (x 4) is a factor of f(x) (M evaluation of f(4)) (A f(4) = 0 and conclusion stated) b) Hence, or otherwise, fully factorise f(x). 3 Method : Algebraic Long Division x + 7x + 3 x 4 x 3 x 5x x 3 8x 7x 5x 7x 8x 3x 3x 0 (M Use of Algebraic Long Divison to obtain Ax + Bx + C where A, B, C 0 ) (x 4)(x + 7x + 3) (x 4)(x + 3)(x + ) (A-correct quadratic) (A correct and fully factorised) Method : Equating Coefficients x 3 x 5x (x 4)(Ax + Bx + C) (M-linear quadratic with three non-zero coefficients) A =, B = 7, C = 3 (A All three correct values) (x 4)(x + 7x + 3) (x 4)(x + 3_(x + ) (A Fully factorised) Page of 4 Further Mathematics NCC Education Limited 07
c) Show that n r= 4r + 6r + = n 3 [4n + 5n + ] 4 n n n r= 4r + 6r + = 4 r= r + 6 r= r + r= (M Separate summations) n = 4n 6 (n + )(n + ) + 6n (n + ) + n (M use of summation formulae) = n[(n + )(n + ) + 9(n + ) + 3] (M Factorises by n) 3 = ⅓n[(4n + 6n + ) + 9n + ] (at least one intermediate step shown) = n 3 [4n + 5n + 4] (A All correct with no errors seen) d) Given that 3 and 4 + i are roots of the cubic equation f(x) = x 3 x + ax + b = 0 i) Find the value of a and the value of b 4 Third root is (4 3i) (B Sight of final root) (x (4 + 3i))(x (4 3i)) = x x(4 3i) x(4 + 3i) + (4 + 3i)(4 3i) (M correct attempt at using complex roots to obtain a quadratic or cubic not containing i) = x 8x + 5 (A Correct quadratic obtained) (x 3)(x 8x + 5) = x 3 8x + 5x 3x + 4x 75 = x 3 x + 49x 75 a = 49, b = 75 (A for both correct) Page 3 of 4 Further Mathematics NCC Education Limited 07
ii) Show the three roots on an Argand diagram B Two complex roots correct and reflection of each other in x-axis B Third root at (3, 0) e) Given f(x) = 3(9x +) (3x+)(3x ) A + B 3x+ + C 3x, i) Find the values of A, B and C 3 3(9x +) (3x+)(3x ) 3 + 6 (3x+)(3x ) (B) 6 A + B (3x+)(3x ) 3x+ 3x A = 3, B = 3 (M Either A or B Correct) 3(9x +) (3x+)(3x ) 3 + 3 3x 3 3x+ (A both A and B found and all correct) ii) Use the quotient rule to differentiate f(x), giving your answer in the form f (x) = Qx (9x ) f (x) = (3x+)(3x )(54x) 3(9x +)(8x) (9x ) (M-correct un-simplified use of quotient rule) f (x) = 08x (9x ) (A for correct simplified quotient with Q=08) Total 0 Page 4 of 4 Further Mathematics NCC Education Limited 07
Question a) A curve has equation f(x) = 5 (+x)(3 x) i) Express f(x) in terms of its partial fractions. 4 5 A + B (+x)(3 x) +x 3 x 5 A(3 x) + B( + x) (M) Let x = 3 5 = 5B B = (A) Let x = 5 = 5A A = (A) 5 + (+x)(3 x) (+x) (3 x) (A) b) i) Use the binomial theorem to expand f(x) in ascending powers of x, up to and including the term in x 3, simplifying each term. 5 +x = ( + x) or 3 x = (3 x) (B ( + x) = x + x x 3 + (3 x) = (3 ) ( 3 x) = 3 ( 3 x) (3 x) = 3 + 4 9 x + 8 7 x + 6 8 x3 + 5 Hence, 5 5 35 x + (+x)(3 x) 3 9 7 x 65 8 x3 for either term) (B-Correct Expansion) (M- Factorising) (A Correct Expansion) (A All Correct) ii) Determine the range of values of x for which the expansion is valid. x < or x < 3 Hence, valid for x < (A considers both and makes conclusion) Page 5 of 4 Further Mathematics NCC Education Limited 07
c) Use your answer to a) to find f (x) and hence find the x coordinate of the stationary point of f(x) 5 d [ dx 5 (+x)(3 x) ] = d dx [( + x) + (3 x) ] ( + x) or ( )(3 x) (M) 4 (3 x) (+x) (A Both Correct) 4( + x) (3 x) = 0 (M sets derivative equal to zero and attempts to solve for x) 4( + x + x ) (9 x + 4x ) = 0 0x 5 = 0 (M Expands brackets correctly and simplifies to obtain linear equation in x) x = 4 (A) d) Write down the equations of all the asymptotes of f(x) Vertical Asymptotes at x = and x = 3 y = 0 (B both required) (B accept x-axis) e) Sketch the graph of f(x) 3 B - Correct shape in each quadrant B - Asymptotes at x=- and x=3/ correctly drawn and labelled B Minimum point at x = ¼ and 5/3 labelled on the y-axis Total 0 Page 6 of 4 Further Mathematics NCC Education Limited 07
Question 3 a) Given Z = 4 + i and z z = 3 + i, find: i) Z in the form a + ib 5 Z = (4 + i) (3 + i) (M) (M multiply numerator and denominator by (3 i) ( + + 3 i 4i) (A) = ( + ) 0 + (3 4)i 0 (A Real Part, A Imaginary Part) ii) The argument of Z Arg(Z ) = arctan( 3 4 ) (M) + Arg(Z ) = 0. 08(086 ) (A to 3 s.f. or more) iii) The exact value of Z 3 ( + ) = ( + )( + ) = 44 + 4 + = 46 + 4 (3 4) = (3 4)(3 4) = 8 4 + 6 = 34 4 (B Either Correct) Z = (46 4 ) + (34 + 4 ) 0 (M Use of Pythagoras) Z =. 8 = 3 5/5 (A accept equivalent form) b) Given Z 3 = 4 + 3i and Z 4 = 3 + i, find i) The modulus of (Z 4 )(Z 3 ) 4 ( 4) + (3) [= 5] (M) ((3) + () ) = 3 (B) Multiplying together (M) = 5 3 or 90. 4 (A) Page 7 of 4 Further Mathematics NCC Education Limited 07
ii) Show on an Argand diagram the points A and B, where A represents Z 3 and B represents Z 4 B For one correct B For both correct iii) Find Z 3 Z 4 in polar form Arg (Z 3 ) = π arctan ( 3 ) =. 50c 4 Arg(Z 4 ) = arctan ( ) = 0. 59c (B 3 Hence, Z 3 Z 4 = [5 3, 3. 09] for either) (B) iv) Determine angle AO B.50 0.59 =.9 (or 09 degrees) (B) v) Write Z 4 in exponential form Z 4 = 3e 0.59i (B ) Total 0 Page 8 of 4 Further Mathematics NCC Education Limited 07
Question 4 a) Prove that cosh (x) sinh (x) using the exponential definition for cosh(x) and sinh(x) ( ex +e x ) = ex +e x + ( ex e x ) = ex +e x (M Use correct exponential 4 4 forms of cosh(x) and sinh(x) and both brackets squared) A Finishes proof with no errors seen. b) Show that the equation tan (x) = 3 sec(x) + 9 = 0 can be written as sec (x) 3 sec(x) 0 = 0 sec (x) = 3sec(x) + 9 sec (x)) sec (x) 3sec(x) 0 = 0 (M Uses correct identity + tan (x) (A without wrong working) c) Hence, solve tan (x) = 3 sec(x) + 9 giving all values of x in the interval 0 x 360 giving solutions to d.p. where necessary 4 (sec(x) + )(sec(x) 5) = 0 sec(x) = 5 or sec(x) = (M correct factorisation) (A Both) cos(x) = 5 cos(x) = cos(x) = 5 cos(x) = x = 78. 46 o, 8. 54 o, 0 o, 40 o (A two correct, A all four correct) d) Sketch the graph of y = 3 sinh(x) + B Correct shape and in correct quadrants B Passes through (0,) Page 9 of 4 Further Mathematics NCC Education Limited 07
e) Prove that sin(x) = eix e ix i 4 e ix = cos(x) + isin(x) (B correct use of Euler s Formula) e i( x) = cos( x) + isin( x) = cos(x) isin(x) (M Use of odd and even functions) e ix e ix = isin(x) sin(x) = eix e ix i (M Subtracting either way round) (A All correct with no errors seen) f) Find the first three terms in the series expansion of cos(x) using Maclaurin s expansion f(x) = cos(x) f(0) = f (x) = sin(x) f (0) = 0 f (x) = cos(x) f (0) = f (x) = sin(x) f (0) = 0 f v (x) = cos(x) f v (0) = (M correct values from differentiation allow one error) Hence, cos(x) x + x4 4 (A All correct) i) Given ( + x) x + x, find the first three non-zero terms in the expansion of sec(x), in ascending order. 4 sec(x) = cos(x) = x +x4 4 (B FT from previous question) = ( x + x4 4 ) = { ( x + x4 4 ) + ( x + x4 ) 4 } (M Uses given result) = { + x x4 4 + x4 4 + } = + x + 5x4 4 (A first two terms correct) (A last term correct) Total 0 Page 0 of 4 Further Mathematics NCC Education Limited 07
Question 5 a) Describe the transformation represented by the matrix ( ) B Rotation B - 45 o clockwise about the origin i) Find A A = ( ) ( ) (M Decent Attempt At Multiplying) A = ( 0 ) (A - All Correct) 0 ii) Describe the transformation represented by A Rotation 90 o clockwise about the origin (A all required) b) Given A = ( a 6 a + 6 ) i) Find det(a) giving your answer in terms of a. det(a) = a(a + 6) ( 6 ) det(a) = a + 6a + (M correct use of determinant) (A) ii) Show that the matrix A is non-singular for all values of a. a + 6a + (a + 3) + 3 (M Attempts to complete the square) (a + 3) + 3 > 0 x, hence det(a) 0 (A Needs Conclusion) iii) Find A, when a=3 A = ( 3 6 ) det(a) = 7 + = 39 9 (M) A = ( 9 6 ) 39 3 (A All Correct) Page of 4 Further Mathematics NCC Education Limited 07
c) A rectangular hyperbola H has parametric equations given by x = 3t, y = 3, t 0. 3 t i) The line L has equation 6y = 4x 5. Show that L intersects H when 4t 5t 6 = 0 H: x = 3t, y = 3 t L: 6y = 4x 5 6 ( 3 t ) = 4(3t) 5 8 = t 5t (M- Attempts to substitute x and y into line A all correct) t 5t 8 = 0 4t 5t 6 = 0 (A) ii) Find dy dx at the point where t= 3 dx = 3, dy = 3 dt dt t dy = dy dt = 3 dx dt dx t 3 (M for both) (M Use of the chain rule) At t =, dy dx = 3 4 3 = 4 (A) ii) The line intersects the hyperbola at two points, a and b. Find the coordinates of a and the coordinates of b. 3 (t )(4t + 3) = 0 t =, t = 3 4 (M factorises quadratic in t) When t =, x = 6, y = 3 (6, 3 ) (A) When t = 3, x = 9, y = 4 ( 9, 4) (A) 4 4 4 Total 0 End of paper Page of 4 Further Mathematics NCC Education Limited 07
Learning Outcomes matrix Question Learning Outcomes assessed,, 5, 6 Yes,4,6 Yes 3,8 Yes 4 6,7,8 Yes 5 3,4 Yes Grade descriptors Marker can differentiate between varying levels of achievement Learning Outcome Pass Merit Distinction Understand different to solve cubic equations and write expressions in terms of their partial fractions adequate robust Be able to work with complex numbers, perform arithmetic calculations using complex numbers, solve higher order polynomials with complex roots and sketch regions in the complex plane Be able to perform arithmetic operations using matrices, understand basic transformations using matrices and, in addition, understand which matrices represent linear transformations and calculate the inverse of a matrix Understand the properties of rational functions and understand conic sections Understand how to use sigma notation to calculate the sum of simple finite series, and appreciate the relationship between the roots of polynomials and their coefficients ability to perform the tasks ability to perform adequate adequate ability to perform the tasks consistently well ability to perform consistently well robust robust Grade descriptors continue on next page highly comprehensive ability to perform the tasks to the highest standard ability to perform to the highest standard highly comprehensive highly comprehensive Page 3 of 4 Further Mathematics NCC Education Limited 07
Learning Outcome Pass Merit Distinction Understand further in calculus to differentiate combinations of functions, how to use these to solve problems involving functions given parametrically and how to derive Maclaurin and Taylor series adequate robust Understand further trigonometry and hyperbolic functions Understand Euler s relation and De Moivre s theorem and derive relations between trigonometric functions and hyperbolic functions adequate adequate level of understanding robust robust level of understanding highly comprehensive highly comprehensive highly comprehensive level of understanding Page 4 of 4 Further Mathematics NCC Education Limited 07