Mathematics 9709/32 February/March 2019: question paper and mark scheme

Mark scheme

1(i) Use law for the logarithm of a product or quotient M1 Use log10 100 = 2 or 102 = 100 M1 Obtain x 2 − 4 x − 100 = 0 , or equivalent A1 3 1(ii) Solve a 3-term quadratic equation M1 Obtain answer 12.2 only A1 2

Mark scheme

2(i) Use the iterative formula correctly at least once M1 Obtain answer 1.3195 A1 Show sufficient iterations to 6 d.p. to justify 1.3195 to 4 d.p., or show there is a sign change in A1 (1.31945, 1.31955) 3 2(ii) 2 x 6 + 12 x B1 State x = , or equivalent 3x5 + 8 State answer 5 4 , or exact equivalent B1 2

Mark scheme

3(i) Use trig formulae and obtain an equation in sin θ and cos θ M1 Obtain a correct equation in any form A1 Substitute exact trig ratios and obtain an expression for tan θ M1 2 2 −1 A1 Obtain answer tan θ = , or equivalent 1− 6 4 3(ii) State answer, e.g. θ = 128.4° B1 State second answer, e.g. θ = 308.4° B1 ft 2

Mark scheme

4 − 1 − 1 1 M1* Integrate by parts and reach ax 2 ln x + b ∫ x 2 . dx x − 1 1 − 1 A1 Obtain −2 x 2 ln x + 2 ∫ x 2 . dx , or equivalent x − 1 − 1 A1 Complete the integration, obtaining −2 x 2 ln x − 4 x 2 , or equivalent Substitute limits correctly, having integrated twice M1(dep*) Obtain the given answer following full and correct working A1 5

Mark scheme

5 dy B1 State cos y as derivative of sin y dx State correct derivative in terms of x and y, e.g. sec2 x / cos y B1 sec 2 x B1 State correct derivative in terms of x, e.g. 1 − tan 2 x Use double angle formula M1 Obtain the given answer correctly A1 5

Mark scheme

6 Separate variables correctly and attempt integration of at least one side B1 1 B1 Obtain term − , or equivalent 2 y2 Obtain term – k e− x B1 Use a pair of limits, e.g. x = 0, y = 1 to obtain an equation in k and an arbitrary constant c M1 Use a second pair of limits, e.g. x = 1, y = e , to obtain a second equation and solve for k or M1 for c 1 A1 Obtain k = and c = 0 2 1 x A1 Obtain final answer y = e 2 , or equivalent 7

Mark scheme

7(a) Use quadratic formula to solve for z M1 Use i 2 = −1 throughout M1 Obtain correct answer in any form A1 Multiply numerator and denominator by 1 – i, or equivalent M1 Obtain final answer, e.g. 1 – i A1 5 1 A1 Obtain second final answer, e.g. + i 2 2 6 7(b) Show the point representing u in relatively correct position B1 Show the horizontal line through z = i B1 Show correct half-lines from u, one of gradient 1 and the other vertical B1ft Shade the correct region B1 4

Mark scheme

8(i) B C B1 State or imply the form A + + 2 + x 3 − 2x Use a correct method for finding a constant M1 Obtain one of A = 2, B = – 4 and C = 6 A1 Obtain a second value A1 Obtain the third value A1 5 8(ii) Use correct method to find the first two terms of the expansion of ( 2 + x ) or ( 3 − 2 x ) , or M1 −1 −1 equivalent Obtain correct unsimplified expansions up to the term in x2 of each partial fraction A1ft +A1ft The ft is on B and C Add the value of A to the sum of the expansions M1 7 7 A1 Obtain final answer 2 + x + x2 3 18 5

Mark scheme

9(i) State or imply a correct normal vector to either plane, e.g. B1 2i + 3j – k, or i – 2j + k Carry out correct process for evaluating the scalar product of two normal vectors M1 Using the correct process for the moduli, divide the scalar product of the two normal vectors by M1 the product of their moduli and evaluate the inverse cosine of the result Obtain answer 56.9° or 0.994 radians A1 4 9(ii) EITHER: Carry out a complete strategy for finding a point on the line (call the line l) M1 Obtain such a point, e.g. (1, 1, 4) A1 EITHER: State a correct equation for a direction vector ai + bj + ck for l, B1 e.g. 2a + 3b – c = 0 State a second equation, e.g. a – 2b + c = 0, M1 and solve for one ratio, e.g. a : b Obtain a : b : c = 1 : – 3: – 7, or equivalent A1 State a correct answer, e.g. A1 r = i + j + 4k + λ(i – 3j – 7k) OR1: Attempt to calculate the vector product of the two normal vectors M1 Obtain two correct components A1 Obtain i – 3j – 7k, or equivalent A1 State a correct answer, e.g. A1 r = i + j + 4k + λ(i – 3j – 7k), or equivalent 9(ii) OR2: Obtain a second point on l e.g. (0, 4, 11) B1 Subtract position vectors and obtain a direction vector for l M1 Obtain i – 3j – 7k, or equivalent A1 State a correct answer, e.g. r = 4j + 11k + µ(i – 3j – 7k), or equivalent A1 OR3: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. y = 4 − 3 x A1 Express the third variable in terms of the second M1 Obtain a correct simplified expression, e.g. z = 11 − 7 x A1 Form a vector equation for the line M1 State a correct answer, e.g. r = 4j + 11k + λ(i – 3j – 7k), or equivalent A1 6 9(ii) OR4: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. A1 4 y x= − 3 3 Express the same variable in terms of the third M1 Obtain a correct simplified expression, e.g. A1 11 z x= − 7 7 Form a vector equation for the line M1 Obtain a correct answer, e.g. A1 r = 4j + 11k +µ(i – 3j – 7k), or equivalent 6

Mark scheme

10(i) State or imply du = – sin x dx B1 Using Pythagoras express the integral in terms of u M1 ( Obtain integrand ± u 1 − u 2 ) A1 2 3 2 7 A1 Integrate and obtain − u 2 + u 2 , or equivalent 3 7 3 7 M1 Or substitute original limits Change limits correctly and substitute correctly in an integral of the form au 2 + bu 2 correctly in an integral of the form 3 7 a (cos x ) 2 + b(cos x) 2 8 A1 Obtain answer 21 6 10(ii) Use product rule and chain rule at least once M1 Obtain correct derivative in any form A1 + A1 Equate derivative to zero and obtain a horizontal equation in integral powers of sin x and cos x M1 Use correct methods to obtain an equation in one trig function M1 Obtain tan 2 x = 6 , 7cos 2 x = 1 or 7sin 2 x = 6 , or equivalent, and obtain answer 1.183 A1 6