Mathematics 9709/31 May/June 2019: question paper and mark scheme

Mark scheme

1 State or imply ordinates 3, 2, 0, 4 B1 These and no more Accept in unsimplified form 20 − 4 etc. Use correct formula, or equivalent, with h = 1 and four ordinates M1 Obtain answer 5.5 A1 3

Mark scheme

2 Use law for the logarithm of a product, quotient or power M1 x Condone ln for M1 x −1 Obtain a correct equation free of logarithms A1 e.g. ( 2 x − 3)( x − 1) = x 2 or x 2 − 5 x + 3 = 0 Solve a 3-term quadratic obtaining at least one root M1 Must see working if using an incorrect quadratic  5 ± 13     2  Obtain answer x = 4.30 only A1 Q asks for 2 dp. Do not ISW. Overspecified answers score A0 Overspecified and no working can score M1A0 4

Mark scheme

3 dy B1 State or imply 3 y 2 + 6 xy as derivative of 3xy 2 dx dy B1 State or imply 3 y 2 as derivative of y 3 dx Equate derivative of LHS to zero, substitute (1, 3) and find the M1  dy x 2 + y 2  gradient  = 2  For incorrect derivative need to see the  dx y − 2 xy  substitution 10 A1 3.33 or better. Allow 30 ISW after correct answer seen Obtain final answer or equivalent 9 3 4

Mark scheme

4 Use correct trig formula and obtain an equation in tan θ M1 Allow with 45° e.g. 1 − 1 =3 tan θ tan θ + tan 45° 1 − tan θ tan 45° Obtain a correct horizontal equation in any form A1 e.g. 1 + tan θ − tan θ (1 − tan θ ) = 3tan θ (1 + tan θ ) Reduce to 2tan 2θ + 3 tan θ − 1 = 0 A1 or 3-term equivalent Solve 3-term quadratic and find a value of θ M1 Must see working if using an incorrect quadratic Obtain answer 15.7 ° A1 One correct solution (degrees to at least 3 sf) Obtain answer 119.(3 )° A1 Second correct solution and no others in range (degrees to at least 3 sf) Mark 0.274, 2.082 as MR: A0A1 6

Mark scheme

5(i) Use chain rule M1 ( k cos θ sin −3 θ = − kcosec 2θ cot θ ) Allow M1 for −2 cos θ sin −1 θ Obtain correct answer in any form A1 θ Accept −2sin θ cos θ e.g. −2cosec 2θ cot θ , −2cos sin θ 3 sin 4 θ 2 5(ii) Separate variables correctly and integrate at least one side B1 ∫ x dx = ∫ −cosec θ cot θ dθ 2 1 2 B1 Obtain term x 2 k M1* or equivalent Obtain term of the form sin 2θ 1 A1 or equivalent Obtain term 2sin 2θ 1 DM1 Dependent on the preceding M1 Use x = 4, θ = π to evaluate a constant, or as limits, in a solution 6 2 b with terms ax and , where ab ≠ 0 sin 2θ Obtain solution x = ( cosec θ + 12 ) 2 A1 or equivalent 6

Mark scheme

6(i) State correct expansion of sin ( 2 x + x ) B1 Use trig formulae and Pythagoras to express sin 3x in terms of sin x M1 Obtain a correct expression in any form ( ) ( A1 e.g. 2sin x 1 − sin 2 x + sin x 1 − 2sin 2 x ) Obtain sin 3 x ≡ 3sin x − 4sin 3 x correctly AG A1 Accept = for ≡ 4 6(ii) 3 1 B1 B1 One mark for each term correct Use identity, integrate and obtain − cos x + cos3x 4 12 Use limits correctly in an integral of the form M1  3 1 3 1 11 2  a cos x + b cos 3x, where ab ≠ 0 − − + − = − +   8 12 4 12 24 3  5 A1 Must be exact. Accept simplified equivalent e.g. 15 Obtain answer 72 24 Answer only with no working is 0/4 4

Mark scheme

7(i) State at least one correct derivative B1 −2sin 1 x, 1 2 ( 4 − x )2 Equate product of derivatives to – 1 M1 or equivalent 1 A1 Obtain a correct equation, e.g. 2sin x = ( 4 − x ) 2 2 A1 Rearrange correctly to obtain a = 4 − 2sin a AG 2 4 7(ii) Calculate values of a relevant expression or pair of expressions at M1 a = 2 2 < 2.7027..  0.703   2.317  a = 2 and a = 3 e.g.     a = 3 3 > 2.587..  −0.412   −0.995  Values correct to at least 2 dp Complete the argument correctly with correct calculated values A1 2 7(iii) 1 M1 Use the iterative formula an +1 = 4 − (2sin an ) correctly at least 2 once Obtain final answer 2.611 A1 Show sufficient iterations to 5 d.p. to justify 2.611 to 3 d.p., or show A1 2, 2.70272, 2.60285, 2.61152, 2.61070, 2.61077 there is a sign change in the interval 2.5, 2.62233, 2.60969, 2.61087, 2.61076 (2.6105, 2.6115) 3, 2.58756, 2.61301, 2.61056, 2.61079 Condone truncation. Accept more than 5 dp 3

Mark scheme

8(i) A B C B1 State or imply the form + + 2+ x 3− x ( 3 − x )2 Use a correct method to obtain a constant M1 Obtain one of A =2, B = 2, C = – 7 A1 Obtain a second value A1 Obtain the third value A1 A Dx + E [Mark the form + , where A = 2, D = – 2 and 2+ x ( 3 − x )2 E = – 1, B1M1A1A1A1.] 5 8(ii) Use a correct method to find the first two terms of the expansion of M1 −1 ( 2 + x ) , ( 3 − x ) or ( 3 − x ) , or equivalent, e.g. 1 + x  −1 −1 −2 1  2  Obtain correct unsimplified expansions up to the term in x 2 of each A1 FT on A, B and C partial fraction A1 x x2 2  x x 2  7  2 x 3x 2  A1 1 − 2 + 4 3 1 + 3 + 9  − 9 1 + 3 + 9  8 43 7 2 A1 Obtain final answer − x+ x 9 54 108 For the A, D, E form of fractions give M1A1ftA1ft for the expanded partial fractions, then, if D ≠ 0, M1 for multiplying out fully, and A1 for the final answer. 5 JJJG

Mark scheme

9(i) Obtain a vector parallel to the plane, e.g. CB = 2i + j B1 Use scalar product to obtain an equation in a, b, c, M1 e.g. 2a + b = 0, a + 5c = 0, a + b – 5c = 0 Obtain two correct equations in a, b, c A1 Solve to obtain a : b : c, M1 or equivalent Obtain a : b : c = 5 : – 10 : – 1, A1 or equivalent Obtain equation 5 x − 10 y − z = −25 , A1 or equivalent Alternative method 1 JJJG JJJG Obtain a vector parallel to the plane, e.g. CD = i + 5k B1 BD = −i − j + 5k Obtain a second such vector and calculate their M1 vector product, e.g. ( 2i + j) × ( i + 5k ) Obtain two correct components A1 Obtain correct answer, e.g. 5i – 10j – k A1 Substitute to find d M1 Obtain equation 5 x − 10 y − z = −25 , A1 or equivalent 9(i) Alternative method 2 JJJG Obtain a vector parallel to the plane, e.g. DB = i + j − 5k B1 Obtain a second such vector and form correctly a M1 2-parameter equation for the plane State a correct equation, e.g. A1 r = 3i + 4 j + λ ( i + 5k ) + µ ( i + j − 5k ) State three equations in x, y, z, λ and µ A1 Eliminate λ and µ M1 Obtain equation 5 x − 10 y − z = −25 A1 or equivalent Alternative method 3 Substitute for B and C and obtain 3a + 4b = d and B1 a + 3b = d Substitute for D to obtain a third equation and M1 eliminate one unknown (a, b, or d) entirely Obtain two correct equations in two unknowns, e.g. a, b, c A1 Solve to obtain their ratio, e.g. a : b : c M1 Obtain a : b : c = 5 : – 10 : – 1, A1 or equivalent a : c : d = 5 : – 1: – 25, or b : c : d = 10 : 1 : 25 Obtain equation 5 x − 10 y − z = −25 A1 or equivalent 9(i) Alternative method 4 Substitute for B and C and obtain 3a + 4b = d and B1 a +3b = d Solve to obtain a : b : d M2 or equivalent Obtain a : b : d = 1 : – 2: – 5 A1 or equivalent Substitute for C to obtain c M1 Obtain equation 5 x − 10 y − z = −25 A1 or equivalent 6 9(ii) State or imply a normal vector for the plane OABC is k B1 Carry out correct process for evaluating a scalar product of two M1 i.e. correct process using k and their normal relevant vectors, e.g. (5i – 10j – k).(k) Using the correct process for calculating the moduli, divide the M1 Allow M1M1 for clear use of an incorrect vector that has been scalar product by the product of the moduli and evaluate the inverse stated to be the normal to OABC cosine of the result Obtain answer 84.9° or 1.48 radians A1 4

Mark scheme

10(i) State or imply r = 2 B1 Accept 4 1 B1 State or imply θ = π 6 Use a correct method for finding the modulus or the argument of u 4 M1 Allow correct answers from correct u with minimal working shown Obtain modulus 16 A1 2 A1 i 2π Obtain argument π Accept 16e 3 3 5 10(ii) Substitute u and carry out a correct method for finding u 3 M1 ( u = 8i ) Follow their u if found in part (i) 3 3 Verify u is a root of the given equation A1 State that the other root is 3 −i B1 Alternative method State that the other root is 3 −i B1 Form quadratic factor and divide cubic by quadratic M1 ( z − 3 − i )( z − 3 + i )( = z − 2 3z + 4) 2 Verify that remainder is zero and hence that u is a root of the given A1 equation 3 10(iii) Show the point representing u in a relatively correct position B1 Show a circle with centre u and radius 2 B1 FT on the point representing u. Condone near miss of origin Show the line y = 2 B1 Im shaded y=2 u Re Shade the correct region B1 Show that the line and circle intersect on x = 0 B1 Condone near miss 5